Complexity and Optimization in Operations Management: Graphs and Heuristics

OST: Chapter 1 Basics, Complexity, Graphs

Content • Strategic and tactical planning • Basics on graphs • Location problems • Warehouse location problem • Production layout • Job shop production • Assembly line balancing • Cellular manufacturing OST – Chapter 1

Grading • 40% Midterm exam • 12.11.2019 HS 13 • 40% Final exam • 28.01.2020, HS4 • 20% Homework • in moodle – can be done from home • Anwesenheitspflicht: according to the curriculum, this course has „immanenten Prüfungscharakter“ (i.e. not just 1 exam) -> therefore mandatory attendence - but not regularly checked! OST – Chapter 1

Grading • Minimum requirement: • 50% of total score • Slides and additional material: • provided via Moodle • Deregistration • is possible until 15.10.2019 OST – Chapter 1

Basics, Optimization and Graphs Complexity Heuristics Costs and distances Graphs Optimization on graphs Minimal spanning tree (MST) Shortest paths OST – Chapter 1

Complexity Solving problems in production & logistics: • Exact approaches (Linear Program, Mixed Integer Program, Dynamic Optimization, …) • Heuristics: simple to understand, yield a (hopefully) good solution relatively fast, no guarantee for optimality • Partly also used if exact methods were available(but too expensive or to much work to implement) • Mainly used if exact methods would take too long (typically operations problems) OST – Chapter 1

Complexity • Choice of method depends on: • Available software • Cost/benefit analysis • Compexity of the planning problem • Heuristics for „np-hard“ problems • Computational effort increases with dimension faster than any polynomial OST – Chapter 1

Example I Solution of: • LP-Problems (average case) - polynomial effort • Number of itertions of Simplex method increases linearly in number of constraints • Each iteration about quadratic effort • example: XLS ALB • LP-Problems with integer variables: IP, MIP using Branch and Bound (B&B) • In each iteration solve a normal LP • Number of iterations increases exponentially in number of integer variables • Extremely diffcult -> more than polynomial effort OST – Chapter 1

Example II Exceptions: • In some problems (z.B. Transportation, Linear assignment, …) integrality constraints are satisfied automatically -> problem remains simple • Also possible that other optimization algorithms are available (DP, …) so that the problem can be solved efficiently. OST – Chapter 1

Heuristics • Starting heuristics, constructive heuristics(quickly generate a hopefully good initial solution)e.g.: Vogel, Prio-rules, … • Improvement heuristics (starting from a feasible initial solution find a better one)e.g.: 2-opt, exchanges, … • Combination of both • Typically get caught in a local optimum • Metaheuristics: general, not problem specific, generic principles to generate and control heuristics, attempt to excape local optima OST – Chapter 1

Costs and Distances • Many decision problems in operations strategy and operations management use cost or distances between 2 objects: cij: • Can be technically given data such set setup cost between jobs i and j • Often measure of distance between objects i und j • Most common distance measures: • Euklidean distance • Manhattan distance • Maximum distance OST – Chapter 1

Euklidean Distance • Beeline distance (LuftlinienEntfernung) between points x and y in space / plane unit circle OST – Chapter 1

Manhattan Distance • Distance in a rectangular road network • Often also applicable in buildings with aisles/corridors unit circle OST – Chapter 1

Maximum Distance • Movement of cranes, plotters, drilling machines, … • Several motors can operate at the same time unit circle OST – Chapter 1

Graphs Example: Road network with cost (length): national park with entrance O main viewpoint T: Source: Hillier-Liebermann: Operations Research. OST – Chapter 1

Definition – Graph I Graph:several points (Knoten, nodes, vertices) are connected with each other by lines (Kanten,edges, arcs) Complete Graph: each pair of nodes is connected by a direct edge OST – Chapter 1

Definition – Graph II • An edge of a graph is called directed (gerichtet ) oder Arrow (Pfeil), if the direction is given (one way street). • If all edges in a graph are directed, the Graph is called directed graph (gerichteter Graph, Digraph) • If all edges in a graph are not directed, the Graph is called non-directed graph (ungerichteter Graph) • A mixed(gemischter) Graph contains directed and undirected edges OST – Chapter 1

Examples of Graphs Undirected Graph Directed Graph (Digraph) Mixed Graph OST – Chapter 1

Definition – Chain • Chain (Kette) between nodes i and j: sequence of edges connecting these 2 • path (Weg): directed chain Chain between O and D Path from O to D OST – Chapter 1

Definition – Cycle Cycle (Zyklus): chain connecting a node with itself, where no edge is traversed more than once Cycle OST – Chapter 1

Definition – Tree I Tree (Baum): connected Graph, that contains no cycles A Graph is connected (verbunden), if for each pair of nodes there exists a chain connecting these two. Tree OST – Chapter 1

Definition – Tree II From graph theory it is known that a graph consisting of n nodes is connected, if it has (n-1) edges and no cycles.  Tree Not a tree: not connected Not a tree because of cycle OABO OST – Chapter 1

Planning Problem 1 on Graphs: MST Laying of cables so that all stations (nodes) are connected somehow.  Total length of all laid cables should be minimal  Where should the cables be run in order to minimize cost?? • Minimal spanning tree (MST), minimaler spannender Baum, Minimalgerüst OST – Chapter 1

Planning Problem 2 : Shortest Path From the entrance of the park, O, find the shortest path to the view point T • shortest path Known from navigation systems… OST – Chapter 1

Planning Problem 3: Maximal Flow • Each edge has a certain capacity (max number of trips per day). More people want to visit view point T than can be transported on the shortest path from O to T. • Use different paths, not just the shortest • Which paths should be used and how many trips/vehicles should be planned along each path in order to maximize the number of tourists, that can visit point T • Maximal flow OST – Chapter 1

Planning Problem 4: TSP Each node must be visited at the end of each day (empty trash cans, checks, …)  In which order should nodes be visited in order to minimize total driving distance (or cost)? • TSP, travelling salesman problem Rundreiseproblem OST – Chapter 1

Minimal Spanning Tree, MST Since the graph has n = 7 nodes, the network needs to have exactly (n-1) = 6 edges and no cycles in order to be a spanning tree. Applications: e.g. sparse transportation networks, telephone cables, … OST – Chapter 1

Kruskal Algorithm • Choose the shortest edge in the graph and connect the adjacent edges (or start with an arbitrary node and choose the shortest edge connected to this node) • Choose the node that is closest (shortest edge) to an already connected node. Repeat until all nodes are connected (n-1 iterations) In case of ties choose arbitrarily. Computer implementation : • Sort all edges w.r.t. increasing length • Add the edges in this order one by one, skipping all those, that would lead to cycles OST – Chapter 1

Kruskal Algorithm - Example OST – Chapter 1

Shortest Path • Find the shortest path in a road network from a source (Quelle )O to a sink(Senke) T • Needed for many other logistics probems • Various algorithms available • Two main classes: • Tree algorithms: Find the shortest path from one node (source) to one other node, or to all other nodes. In doing so, a tree is constructed. Basic idea is dynamic programming: in each iteration one node is finally „marked“, i.e. its shortest distance to source is finally found -> Dijkstra, Bellman • Algorithms finding the shortest distance between all pairs of nodes.Whole distance matrix is constructed simultaneously -> tripel algorithm OST – Chapter 1

Bellman Algorithm The simplesttreealgorithmistheBellmanAlgorithm: • Start with source as the only marked node • From each marked node (part of subtree) find the non-marked node with the shortest distance (to this node) • Among all these candidates choose the non-marked node with the shortest total distance to the origin OST – Chapter 1

Bellmann Algorithm II 2 1 A 2 O O A O 2 O C 4 C 4 A B 4 B 4 A 3 4 A D 9 B E 7 E 7 B E 8 C 5 D A 9 D 8 D 8 B B E D 8 T 13 13 6 D D T E T 14 OST – Chapter 1

Bellmann Algorithm III 2 8 13 4 7 4 1st solution: O - A – B – D – T 2nd solution: O - A – B – E – D – T OST – Chapter 1

Dijkstra Algorithm • A more efficient way to find the shortest path (same steps, just avoiding duplicate sorting efforts) is the Dijkstra algorithm • Let dij = length of direct edge from i to jIf no direct edge -> dij =  Initialize: n = 0: All nodes iget a preliminary value of D[i] = d0ii.e. the length of direct edge from the origin – can be . Only the source has final value zero. The immediate predecessor if i is denoted by V[i]. Initially V[i] = O OST – Chapter 1

Dijkstra Algorithm - Iteration n • Choose the node (not finally marked) with the smallest preliminary value D[i] and mark this as the final value (shortest distance to origin). This is the n-th next node to the origin. • Its shortest distance to O is now D[i] and the immediate predecessor is V[i]. • Find all nodes j, that can be reached from i using a direst edge i-j.If D[i] + dij < D[j] then first going to i (at cost D[i]) and then using the edge i-j is better than the previously best path with cost D[j]Then, set D[j] = D[i] + dij and V[j] = i. • If sink (or all other nodes) is/are finally marked -> stop. OST – Chapter 1

Dijkstra Algorithm - Example 2 Opt. solution: 8 13 O – A – B – D – T 4 7 4 , O , O , O 2, O 5, O 4, O A, 2 , O , O B, 4 9, A 4, A 4, O , O 8, B 7, B C, 4 4, O , O 8, B 7, B E, 7 14, E D, 8 8, B, E 13, D T, 13 OST – Chapter 1

Optimal Solution Two shortest paths with total length 13: OST – Chapter 1

Complexity and Optimization in Operations Management: Graphs and Heuristics