Themen der Lehrveranstaltung
- Linear Programming (LP): LP Definition, Model Manipulations, Standard and Conic Forms of LP, and Geometric Solutions and Analysis
- Modelling with LP: Fundamental Models in Production Planning, Resource Allocation, Diet Planning, Transportation, and Cutting Stock Problem
- Geometry of LP: Vector and Matrix Spaces, Special Matrix Formats, Linear Systems, Elementary Matrix Operations, Gaussian Reduction, Convexity of Sets, Cones and Functions, Hyperplanes and Half-Spaces, and Polyhedral Geometry
- Simplex Algorithm for Solving LPs: Basic Feasible Solutions, Key Idea of the Simplex Method, LP Representation, Geometric and Algebraic Aspects of the Simplex Method, Termination of the Simplex Method, The Simplex Algorithm in Tableau Format, Initialization with Artificial Variables, Two-Phase and Big-M Methods, Degeneracy and Cycling, and Pivoting Rules
- Duality Theory of LP: Dual Formulation, Relationships Between Primal and Dual Models, Weak and Strong Duality, Complementary Slackness, Dual Simplex Method, and Sensitivity (Post-Optimal) Analysis
- Transportation and Assignment Models: Transportation Model Definition, Characterization of a Basis in the Transportation Tableau, Simplex Algorithm for the Transportation Problem, Assignment Model Definition, Reduced Assignment Matrix, and the Hungarian Algorithm for the Assignment Problem
- Network Flow Models: Basic Graph Theory, Minimum-Cost Network Flow Problem, Network Simplex Algorithm, Maximum Flow and Minimum Cut Problems, and Shortest Path Problem
- Integer Programming (IP): Definitions and Fundamental Models in Capital Budgeting, Project Selection, Set-Covering, Job Sequencing, and Facility Location
- IP Algorithms: Cutting-Plane and Branch-and-Bound Methods
- Dynamic Programming: Definition, Dynamic Programming Method for Binary and Integer Knapsack Problems, and Bellman–Ford Method for the Shortest Path Problem
- Metaheuristic Algorithms: Basic Concepts of Soft Computing, Traveling Salesman Problem (TSP), Encoding, Genetic Algorithms, and Simulated Annealing Method
- Goal Programming: Multiobjective Optimization and the Weighted-Sum and Preemptive Methods
- Nonlinear Programming (NLP): Definitions, Geometric Solutions and Analysis, Nonlinear Facility Location Problem, Linear Regression, Portfolio Optimization, Optimality Conditions, Lagrange Multipliers, Karush–Kuhn–Tucker (KKT) Conditions, and the Modified Simplex Algorithm for Quadratic Programming
- Complementary Topics: Parametric Linear Programming, Decision Making under Uncertainty, and Game Theory
Unterrichtsform
Lectures: The course is delivered using lecture slides, which are regularly uploaded to Microsoft Teams. Additional explanations and derivations are provided on the board to clarify concepts and complement the slide material.
Exercises: Topic-oriented exercises are solved, primarily on the board, during class to reinforce students' understanding and help them internalize the theoretical concepts.
Software Laboratory: After introducing the main categories of OR models and the underlying theoretical concepts, basic real-world decision-making problems are modeled in the OPL software environment and solved primarily using the CPLEX solver. Particular emphasis is placed on interpreting the main output components and understanding the information provided by the solver.