Objectives and competences
- Understanding of basic techniques of mathematical modeling using principles of design thinking.
- Acquaintance with the theoretical background of mathematical modeling.
- Find and apply technological tools to develop and investigate a mathematical model.
- Understanding of basic applications of algorithms and heuristics to solve mathematical problems.
- Apply the knowledge from other mathematical areas in analysis of practical problems.
- Gain experience in developing a mathematical model, useful in their future career.
- Learn about sources of bibliography on problems related to studying mathematical models.
- Learn to distinguish the relevant data for the model under study.
- Gain experience in explaining and presenting the mathematical model and defending its assumptions.
Content (Syllabus outline)
Mandatory content that familiarizes the students with fundamentals of mathematical modeling:
? Overview of mathematical model types. Process of mathematical model creation. Variable types.
? Mathematical model and inovation process.
? Mathematical models of knowledge and learning: reinforcement learning, interface theory of perception, universal process model.
? Process modelling. Sequence diagrams, flow charts, Petri nets.
? Data modelling. Entity relationship diagrams, Class diagrams.
? Decision modeling. Decision tree. Regression tree, game tree, acyclic decision process.
? Linear program and its dual. Farkash lema. Shadow prices. Sensitivity analysis.
? Data analysis, probability, Monte Carlo simulations.
Within the coursework, the students select problems whose result is a coursework report that is presented in two intermediate and one final presentation in front of the class. The problems are related to their future career (practical problems from industry and business, theoretical problems from the areas of optimization, algorithms, modelling). The content of the remaining lectures is selected according to these projects from the following list:
? Introduction to game theory. Nash equilibria. Matrix zero sum games. Optimization models with centralized decision making. Game theory models with distributed decision making.
? Simulation models. Modeling changes with diference and diferential equations.
? Mathematical behaviour of dynamic systems.
? Applications of mathematical models in science and engineering.
? Relational models applied to timetabling and scheduling problems.
? Deterministic, stohastic, robust problems. Stohastic linear program (discrete variable). Decomposition. Deterministic and stohastic models of portfolio optimization.
? Diet problem. Simplex method.
? Applications of game theory: optimal strategy in two competitor market.
? Queues.
? Other material from the field of mathematical modeling, related to students' projects.
The students are familiarized with open-source and commercial technological solutions for treatment of the studied mathematical models. Excel is used for initial data organization, presentation, and analysis. Students are introduced to different linear programming solvers: Python, R, AMPL, Matlab, depending on the environment the student's problem is coming from.
Learning and teaching methods
? At the lectures, the students are familiarized with the required contents of the course. Applying flipped learning approach, they discuss their coursework projects in relation to the material of the course.
? Within the coursework, the students deepen their understanding of the material on projects, related to their future careers. They are organized in smaller groups who apply the principles of project based learning on three smaller projects.
? At the seminar, the students apply, analyse, and evaluate their projects in order to create new solutions desired by the environment the problems are coming from.
? Applying embedded mathematical modelling, the students measure and reflect upon their progress thus using its mathematical model to experience improvement coming from mathematically supported decision making.
? Three presentations of their results help students acquire confidence with public presentation and defending their results.
? Focused two-day workshop-style organized hackathon helps students experience the joy of focused research and collaboration with peers.
Intended learning outcomes - knowledge and understanding
Knowledge and Understanding:
? To deepen the knowledge of mathematical methods of optimization.
? To deepen the knowledge of applications of mathematical modelling in research and practice.
? To deepen the knowledge of details of advanced applications of mathematical modelling in financial optimization.
Transferable/Key Skills and other attributes:
? Direct applications in finacial mathematics, economy, business, engineering, chemistry, and numerous other social and natural sciences. Also, principles of linear optimization are foundations for mathematical programming.
? Preparation of a detailed technical report or focused report paper describing a mathematical model of a specific mathematical problem the students encounter while investigation their possible future careers.
Intended learning outcomes - transferable/key skills and other attributes
• Direct applications in finacial mathematics, economy, business, engineering, chemistry, and numerous other social and natural sciences. Also, principles of linear optimization are foundations for mathematical programming.
• Preparation of a detailed technical report or focused report paper describing a mathematical model of a specific mathematical problem the students encounter while investigation their possible future careers.
Readings
Osnovno / basic:
- Osais, Yahya Esmail. Computer Simulation: A Foundational Approach Using Python. Chapman and Hall/CRC, 2017.
- R. Rardin. Optimization in Operations Research. Prentice Hall, Inc., Upper Saddle River, New Jersey, 2000.
- J. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems. Classics in Applied Mathematics 37, SIAM, 2002.
- Dossey, Giordano, McCrone, Weir, Mathematics Methods and Modelling for today's Mathematics Classroom, Brooks/Cole, Pacific Grove, 2002.
Dodatno / additional:
- E. Zakrajšek, Matematično modeliranje, DMFA – Založništvo, Ljubljana, 2004.
- J.D. Murray, Mathematical biology I. An introduction, Springer,New York, 2002.
- G. Polya, Kako rešujemo matematične probleme, DMFA, 1989.
Prerequisits
Knowledge of simple algorithms.
Knowledge of basic linear algebra and calculus.
Additional information on implementation and assessment Type (oral examination, coursework - project):
Coursework report, approx. 90 hours of individual work. (75%)
Oral exam (25%)
Each of the mentioned commitments must be assessed with a passing grade.
Passing grade of the coursework report is required for taking the exam.
