Mathematical Tools for Engineering and Management

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1 Mathematical Tools for Engineering and Management Lecture 8 8 Dec 0

2 Overview Models, Data and Algorithms Linear Optimization Mathematical Background: Polyhedra, Simplex-Algorithm Sensitivity Analysis; (Mixed) Integer Programming MIP Modelling; Mathematical Background: Branch & Bound Branch & Bound, Cutting Planes; More Examples; Combinatorial Optimization Combinatorial Optimization: Examples, Graphs, Algorithms Complexity Theory Nonlinear Optimization Scheduling Lot Sizing Multicriteria Optimization Exam

3 Minimum Spanning Tree Problem (MST) Given a graph G = (V,E) with non-negative edge-weights w e for all e E......find a minimum spanning tree for G, that is: a subset E of the edges such that the edges in E form a tree all vertices of G are in the tree the total weight of the tree edges is minimal total weight:

4 Minimum Spanning Tree Problem (MST) Given a graph G = (V,E) with non-negative edge-weights w e for all e E......find a minimum spanning tree for G, that is: a subset E of the edges such that the edges in E form a tree all vertices of G are in the tree the total weight of the tree edges is minimal total weight:

5 Minimum Spanning Tree Problem (MST) Given a graph G = (V,E) with non-negative edge-weights w e for all e E......find a minimum spanning tree for G, that is: a subset E of the edges such that the edges in E form a tree all vertices of G are in the tree the total weight of the tree edges is minimal total not allowed: weight: not a tree!

6 Minimum Spanning Tree Problem (MST) Given a graph G = (V,E) with non-negative edge-weights w e for all e E......find a minimum spanning tree for G, that is: a subset E of the edges such that the edges in E form a tree all vertices of G are in the tree the total weight of the tree edges is minimal total not allowed: weight: not misses a tree! vertices!

7 Minimum Spanning Tree Problem (MST) Given a graph G = (V,E) with non-negative edge-weights w e for all e E......find a minimum spanning tree for G, that is: a subset E of the edges such that the edges in E form a tree all vertices of G are in the tree the total weight of the tree edges is minimal not total allowed: weight: not misses a tree! vertices! Real-world problem: Connect a set of given computers to form a local network, at minimal cost

8 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree

9 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight:

10 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight:

11 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight:

12 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight:

13 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight:

14 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight:

15 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight:

16 Kruskal s algorithm Idea: select cheap edges, as long as they don t result in a cycle (greedy) Set of potential edges := E, tree T := empty Until all vertices are in the tree T......determine cheapest remaining potential edge: e...if adding edge e to the tree T does not result in a cycle: add e to T...remove e from the set of potential edges T is a minimum spanning tree total weight: T contains all vertices done!

17 Kruskal s algorithm summary Kruskal s algorithm is fast (polynomial runtime) and relatively easy to implement (greedy algorithm) Still it always computes an optimal tree! (Proof by contradiction) Published by Joseph B. Kruskal in 956 Joseph B. Kruskal (928 0)

18 Symmetric Travelling Salesman Problem Problem formulation: Given a set of cities together with travel times to travel between every two cities, find a tour leading through every city such that the total travel time is minimized. Cville (symmetric TSP) Epolis 05 Atown New D San F Bcity

19 TSP approximation algorithm using MST MST can be used for an approximation algorithm for the (symmetric, euclidean) TSP: Compute an MST for the graph, using distances as edge weights Create a fake tour by going to and back for every edge of the tree Start traversing the tour at some city Whenever the tour returns to an already visited city, replace the edge by the shortcut to the following city The found solution misses the optimum by a factor of at most 2 (approximation factor) Proof: L sol L fake tour 2 L MST 2 L opt\edge 2 L opt A C 25 D E 45 A C D E A C D E B 80 F B F B F

20 Combinatorial optimization summary Great flexibility in formulating real-life problems Usually integer-programming formulation is possible, but inefficient Specially designed algorithms Wide variety of algorithms: Primal algorithms (heuristics): Provide feasible solutions, but without guarantee of optimality Dual algorithms (branch & bound, approximation algorithms): Provide upper/lower bounds on the optimal solution, but without explicitly giving one Combination: Primal-dual algorithms maximize dual bound optimum primal objective time

21 Example: data flow Consider a data network with central offices, routers and users Data has to be sent from the central offices to the users via the network Restrictions given by capacities of links and devices Various costs depending on links, devices and data volume

22 Network flow problem Given a network i.e. a directed graph, possibly with more parameters, such as capacities and costs for nodes and arcs and certain demands......compute a flow through the network satisfying the demand, respecting the capacities, with minimal total cost cap: 42 cost: 09 v v 3 v 2 cap: 42 cost: 622 cap: 63 cost: 26 cap: 84 cost: cap: cost: 340 v 4 cap: 42 2 cost: cap: 2 cost: 6 v 5 63 cap: 84 cost: 222 v 6 v 7 cap: 63 cost: 572 v 8 62 cap: 2 cost: 078 cap: 2 cost: 459 v 9 2 cap: 42 cost: 59 22

23 Flow variables & capacities Network: directed graph (V, A) Nodes of the network: i V (vertices of the graph) Links of the network: (i,j) A (arcs (edges) of the graph) i j (i, j) S Variables: f (i,j) 0 amount of flow along the arc (i,j) V Capacity constraints: f (i,j) u (i,j) for all arcs (i,j) A C u (i,j) capacity of link (i,j) P Possibly: constraints given by node capacities... Objective: minimize total cost c (i,j) f (i,j) (i,j) A c (i,j) cost per flow unit on arc (i,j) P

24 Flow conservation constraints Flow may not leak from the network! outgoing flow must equal incoming flow f (v,j) f (i,v) = 0 for all v V (v,j) A (i,v) A 7 23 v ?...except for nodes with a demand d v f (v,j) f (i,v) = d v (v,j) A (i,v) A for all v V 7 23 v In general: f (v,j) f (i,v) = (v,j) A (i,v) A d v if v is a demand node s v if v is a supply node 0 otherwise C d v s v demand of node v supply of node v P

25 (Capacitated) Min-Cost Flow LP formulation Objective: minimize (total cost) c (i,j) f (i,j) (i,j) A C (link capacity) f (i,j) u (i,j) for all arcs (i,j) A (flow conservation) f (v,j) d v if v is a demand node f (i,v) = s v if v is a supply node (v,j) A (i,v) A 0 otherwise V (flow on arcs) f (i,j) 0 for all arcs (i,j) A S P Nodes: V Arcs: A {(i,j) i,j V, i j} Link capacities: u (i,j) 0 for all arcs (i,j) A Costs: c (i,j) for all arcs (i,j) A Demand of demand nodes: d v 0 Supply of supply nodes: s v 0

26 Example: IP routing Data are IP packets with a source and a target node for each packet Separate (disaggregated) flow for every pair of nodes Demand matrix with entries d u,v 0 v. v 4. v 9 v v 2 v v : 24 v 9 v 4 : 38 v 9 v 9 : 2 v 2

27 Disaggregated flow Disaggregated flow variables: f (i,j),u v 0 V amount of flow for demand u v along the arc (i,j) Objective: minimize total cost c (i,j) (i,j) A u v f (i,j),u v Capacity constraints: f (i,j),u v u (i,j) for all arcs (i,j) A C u v Flow conservation constraints: for all w V, u,v V,u v, f (w,j),u v d u,v if w = u f (i,w),u v = d u,v if w = v (w,j) A (i,w) A 0 otherwise NOTE: flow for a given demand might be split 32 at nodes! v 4 v 9 28 v9 4 v 9

28 Unsplittable flow How to avoid splitting flow at nodes? 32 Idea: replace flow variables f (i,j),u v with binary decision variables: v 9 v 4 28 v9 4 v 9 y (i,j),u v {0,} y (i,j),u v = complete flow from u to v uses arc (i,j) Set partitioning constraints for outgoing flows at each node w: y (w,j),u v = for all demands u v, u v (w,j) A C Replace flow f (i,j),u v with d u,v y (i,j),u v in all other constraints and in the objective function... Similarly: k-splittable flow allow splitting into at most/exactly/at least k parts at each node

29 Example: Shortest Path Problem: find a shortest path from PTZ to Kreuzburger

30 Shortest Path Problem Given a network i.e. a directed graph with a length for each arc, a start node A and a destination B......compute a shortest path through the network from A to B 26 v 5 A v 3 v v 8 v v B v 9 v 6 v 7 5

31 Shortest Path Problem as network flow model Shortest Path Problem can be formulated as a network flow problem: Only one demand: d A,B =, all other demands d u,v = 0 Unsplittable flow No capacities d A,B = 26 v 5 Optimal solution: y (A,v ) = y (v,v 6 ) = y (v6,v 7 ) = y (v7,b) =, all other variables 0 v 3 57 A v 8 v 34 v v 6 v B

32 Overview Models, Data and Algorithms Linear Optimization Mathematical Background: Polyhedra, Simplex-Algorithm Sensitivity Analysis; (Mixed) Integer Programming MIP Modelling; Mathematical Background: Branch & Bound Branch & Bound, Cutting Planes; More Examples; Combinatorial Optimization Combinatorial Optimization: Examples, Graphs, Algorithms Shortest Path; Complexity Theory Nonlinear Optimization Scheduling Lot Sizing; Multicriteria Optimization Exam

Fundamentals of Integer Programming

Fundamentals of Integer Programming Di Yuan Department of Information Technology, Uppsala University January 2018 Outline Definition of integer programming Formulating some classical problems with integer