This study unit is made of two main parts. The first part deals with various currently used optimization methods. Students are first introduced to general Optimization theory. Then various forms of Mathematical Programming are dealt with. After the introduction into Linear and Integer Programming, the attention is given to Nonlinear Programming from its backgrounds like Convexity Theory and optimization conditions to overview and practical use of various optimization algorithms. A practically oriented introduction into Dynamic Programming with finite horizon follows. Students are also introduced into backgrounds of Stochastic Programming and Dynamic programming with infinite horizon, in particular to methods of solving Bellman's equations. The first part is closed by introduction to heuristic optimization algorithms.
The second part of the unit deals with the Queuing Theory. Various models of single queue systems and queuing networks are derived. The theory is then used by solving practical problems. Students are also introduced into simulation methods that are the only feasible solution method when a theoretical model is not available.

Topics

1. Optimization Theory. Terminology, various types and existence of solutions (Weierstrass theorem). Methods based on Calculus.

2. Linear Programming. Theory and Simplex Method.

3. Integer Programming. Solution methods and use of indicator variables in building models that are out of scope of Linear Programming (models with logical conditions, disjunctive constraints, and similar.)

4. Theory of Nonlinear Programming. Convex sets and functions, optimality conditions.

5. Optimization algorithms of Nonlinear Programming and their application.

6. Dynamic Programming with finite horizon. Introduction to recursion, solution of various practical problems by the methods of Dynamic Programming.

7. Introduction to Stochastic Programming. Terminology, basic forms of Deterministic Equivalents and their solution.

8. Introduction to Dynamic Programming with infinite horizon. Terminology, Markov Decision Process, Bellman's equations and their solution.

9. Heuristic optimization algorithms as a method to solve problem of local optima (genetic and similar algorithms based on populations of solutions).

10. Basics of Queuing Theory, introduction to stochastic processes, Poisson process in detail.

11. Models of simple single queue systems (model M/M/1 and similar).

12. Advanced single queue models (M/G/1, G/M/1 and similar). Network models, Jackson theorem.

13. Simulation methods and their use in analysis of queuing systems.