1. 1 Methodological Basics

2. Layout and DesignKapitel 1 / 2 (c) Prof. Richard F. Hartl 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

3. Layout and DesignKapitel 1 / 3 (c) Prof. Richard F. Hartl 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 integer variables solved 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

4. Layout and DesignKapitel 1 / 4 (c) Prof. Richard F. Hartl 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

5. Layout and DesignKapitel 1 / 5 (c) Prof. Richard F. Hartl 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

6. Layout and DesignKapitel 1 / 6 (c) Prof. Richard F. Hartl 1.2 Costs and distances The majority of problems is solved based on costs (distances) c ij :  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

7. Layout and DesignKapitel 1 / 7 (c) Prof. Richard F. Hartl Euklidean distance Straight line distance between two points x and y.

8. Layout and DesignKapitel 1 / 8 (c) Prof. Richard F. Hartl Manhattan distance The distance between two points measured along axes at right angle

9. Layout and DesignKapitel 1 / 9 (c) Prof. Richard F. Hartl Maximum distance Drilling plates, movement of cranes,..

10. Layout and DesignKapitel 1 / 10 (c) Prof. Richard F. Hartl 1.3 Basics on Graph Theory graph (Graph): points (nodes, vertices, Knoten) are connected with each other using lines (edges, arcs, Kanten) Graph

11. Layout and DesignKapitel 1 / 11 (c) Prof. Richard F. Hartl 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)

12. Layout and DesignKapitel 1 / 12 (c) Prof. Richard F. Hartl cycle cycle (Zyklus): chain that connects a node with itself, while no edge is traversed more than once

13. Layout and DesignKapitel 1 / 13 (c) Prof. Richard F. Hartl tree tree (Baum): connected graph without cycles connected graph (verbunden), graph where for each pair of nodes there exists a path connecting these two

14. Layout and DesignKapitel 1 / 14 (c) Prof. Richard F. Hartl basic result from graph theory A graph with n nodes is a tree, if it contains (n-1) edges but no cycles

15. Layout and DesignKapitel 1 / 15 (c) Prof. Richard F. Hartl 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 mixed graph … contains directed and undirected edges