1 Methodological Basics

1. 1 Methodological Basics

2. 1.1 Complexity • We solve problems in production and logistics by using: • exact methods • heuristics: • if exact methods are available but too time consuming to be applied • for „NP-hard“ problems • Selection of method depends on: • available software • cost-benefit • problem complexity Layout and Design

3. Example I • LP-Problems (average case) with polynomial complexity • number of iterations increases linear with the number of constraints • each interation causes quadratic effort • LP-Problems with integervariablessolved by Branch and Bound (B&B) • solve a LP-Model in each iteration • number of iterations increases exponentially with the number of integer variables • -> these problems cannot be solved with polynomial effort Layout and Design

4. Example II • For some problems due to their special structure (e.g. TP, Linear Assignment Problem) integer/binary property of the decision variables is guaranteed automatically -> low problem complexity • Some problems with integer/binary variables can (by using special exact methods) be solved with polynomial effort Layout and Design

5. Heuristics • Starting heuristics (quick generation of a feasible solution) • Improvement heuristics (start with a feasible solution and try to find a better one) • Combinations of starting and improvement heuristics Layout and Design

6. 1.2 Costs and distances • The majority of problems is solved based on costs (distances) cij: • costs are determined based on given technical parameters (machine setup,..) • or based on distances (e.g. distance between object i and object j) • common distances: • Euclidean distance • Manhattan distance • Maximum distance Layout and Design

7. Euklidean distance • Straight line distance between two points x and y. Layout and Design

8. Manhattan distance • The distance between two points measured along axes at right angle Layout and Design

9. Maximum distance • Drilling plates, movement of cranes,.. Layout and Design

10. 1.3 Basics on Graph Theory graph (Graph): points (nodes, vertices, Knoten) are connected with each other using lines (edges, arcs, Kanten) Graph Layout and Design

11. Chain chain (Kette): between nodes i and j: sequence of edges connecting these two nodes path(Weg): chain where the direction is clear (oriented) oriented edges are usually called arrows (or arcs) Layout and Design

12. cycle cycle(Zyklus): chain that connects a node with itself, while no edge is traversed more than once Layout and Design

13. tree tree(Baum): connected graph without cycles connected graph(verbunden), graph where for each pair of nodes there exists a path connecting these two Layout and Design

14. basic result from graph theory A graph with n nodes is a tree, if it contains (n-1) edges but no cycles Layout and Design

15. Definition – Graph II arrow(Pfeil, arc, directed edge) if orientation is given (one way street) directed graph (digraph) … contains only directed arcs undirected graph … contains only undirected edges mixedgraph … contains directed and undirected edges Layout and Design