Network Optimization Models

1. Chapter 10 Network Optimization Models

2. Network Optimization Models

3. 10.1 Prototype Example • The road system for Seervada Park • Location O: park entrance • Location T: a scenic wonder • Trams transport sightseers from park entrance to location T and back

4. 10.1 Prototype Example

5. Prototype Example • Summary:Park management faces three problems • Determine the route with the smallest total distance • A shortest-path problem • Determine where telephone lines should be laid • A minimum spanning tree problem • Determine how to route tram to maximize number of trips during peak season • A maximum flow problem

6. 10.2 The Terminology of Networks • Network consists of a set of points and a set of lines connecting points • Node: point (vertex) in the network • Lines: links, arcs, edges, or branches • Labeled by naming the node at each end • From node precedes the to node • Have a flow of some type through them • Directed arcs have unidirectional flow • Undirected arcs (links) allow bidirectional flow

7. The Terminology of Networks

8. The Terminology of Networks • Directed network • Network has only directed arcs • Undirected network • Network has only undirected arcs • Path between two nodes • A sequence of distinct arcs connecting the nodes • Directed path from node i to node j • Sequence of connecting arcs toward node j

9. The Terminology of Networks • Undirected path from node i to node j • Sequence of connecting arcs whose direction can be with toward or away from node j (note a directed path also satisfies the definition of an undirected path, but not vice versa). Example: in Fig. 10.2, A->B->C->E is a directed path and B->C->A->D is an undirected path • Cycle: A path begins and ends at the same node; DE-ED is a directed cycle; AB-BC-CA is an undirected cycle • Connected network: Every pair of nodes in the network has at least one undirected path between them • Tree (spanning tree): Connected network with no undirected cycles; every spanning tree has n – 1 arcs (the minimum number of arcs needed to have a connected network and the maximum number of arcs possible without having undirected cycles).

10. The Terminology of Networks

11. The Terminology of Networks There are several alternative choices of adding an arc at each stage of the process.

12. The Terminology of Networks • Arc capacity • Maximum amount of flow that can be carried on a directed arc • Supply node (or source node) • Flow out exceeds flow in • Demand node (sink node) • Flow in exceeds flow out • Transshipment node (intermediate node) • Flow in equals flow out

13. 10.3 The Shortest-Path Problem • Consider an undirected, connected network • Contains origin and destination nodes • Each link has a nonnegative distance • The problem • Find the shortest path from origin to destination

14. The Shortest-Path Problem This is a special type of linear programming problem (strictly speaking it is an integer programming problem, but due to the special structure of the model, the integer restriction on the decision variables can be relaxed). Each node can be thought of as having a unit flow passing through if it is on the selected path, no flow otherwise. Thus, O is the source node, T is the demand node, and other nodes are transshipment nodes.

15. The Shortest-Path Problem Minimize Z = 2xOA + 5xOB + 4XOC + 2XAB + 7xAD + xBC + 4xBD + 3xBE + xCB + 4xCE + xDE + 5xDT + xED + 7xET s.t.(net flow constraints) xOA+ xOB + xOC = 1 (source node O) -xDT – xET = -1 (sink node T) -xOA + xAB + xAD = 0 (node A) xBC + xBE + xBD – xOB – xAB – xCB = 0 (node B) xCB + xCE – xOC – xBC = 0 (node C) xDE + xDT – xAD – xBD – xED = 0 (node D) xED + xET – xDE – xBE – xCE = 0 (node E) xij ≥ 0 The optimal solution by Excel Solver (see course website) is xOA = xAB = xBE = xED = xDT = 1 (and other xij = 0) with Z = 13.

16. The Shortest-Path Problem The Excel Solver uses the general simplex method. However, using the shortest path algorithm to solve the problem is much more efficient.

17. The Shortest-Path Problem • Algorithm (using a table or directly applying the algorithm on the graph) • Objective of nth iteration: find the nth nearest node to the origin • Repeat for n = 1, 2… until destination is reached • Input for nth iteration: n − 1 nearest nodes to the origin (these nodes are called solved nodes and other nodes are called unsolved nodes), including shortest path and distance from the origin

18. The Shortest-Path Problem • Algorithm (cont’d.) • Candidates for nth nearest node: for each n – 1 solved node, find an unsolved node (if any) with shortest connecting link to the solved node and make this unsolved node a candidate node (ties provide additional candidates) • Calculation of nth nearest node • For each solved node and its candidate, add the distance between them and the distance of the shortest path from the origin to this solved node • Candidate with smallest total distance is the nth nearest node

20. The Shortest-Path Problem • Shortest path for the Seervada park problem • Looking at last column in Table 10.2, two potential shortest paths exist from the destination to the origin • T→ D → E → B → A → O or T → D → B → A → O • Total of 13 miles on either path

21. The Shortest-Path Problem • Network simplex method • An alternate option for solving shortest-path problems • Three categories of applications • Minimize total distance traveled • Minimize total cost of a sequence of activities • Minimize total time of a sequence of activities • Other shortest path problems can be solved by slightly modify the algorithm For example: - shortest directed path (only directed arcs should be considered) - shortest path from the origin to all other nodes (stop the algorithm until all nodes are solved nodes)

22. The Shortest-Path Problem

23. 10.4 The Minimum Spanning Tree Problem • Given: nodes of a network, potential links, and positive length of each link if it is inserted into the network • Design the network by inserting links • A path must exist between every pair of nodes • Problem: minimize total length of links inserted into the network • Network of n nodes requires only n−1 links • Choose the links to form a spanning tree

24. The Minimum Spanning Tree Problem

25. The Minimum Spanning Tree Problem • Applications • Design of telecommunications networks • Design of a lightly-used transportation network to minimize cost of providing links • Design network of power transmission lines • Electrical equipment wiring • Piping systems

26. The Minimum Spanning Tree Problem • Algorithm • Select any node arbitrarily and then add a link to connect it to its nearest node • Identify the unconnected node that is closest to a connected node, and add a link between them • Repeat until all nodes have been connected • Ties may be broken arbitrarily • There may be multiple optimal solutions

27. The Minimum Spanning Tree Problem • Example of graphical approach to implementing the algorithm • Problem: installing telephone lines in Seervada park

28. The Minimum Spanning Tree Problem The optimal solution does not depend on the initial node chosen. Try the algorithm again using a different initial node.

29. 10.5 The Maximum Flow Problem • General problem description • All flow through a directed, connected network originates at a source, and terminates at a sink • Remaining nodes are transshipment nodes • Flow through an arc is allowed in only one direction (indicated by the arrowhead) • Maximum flow is given by arc capacity • Objective: maximize total flow from source to sink

30. The Maximum Flow Problem • Applications • Maximize flow through company’s distribution network from factories to customers • Maximize flow through company’s supply network from vendors to factories • Maximize oil flow through a system of pipelines • Maximize water flow through aqueducts • Maximize flow of vehicles through a transportation network

31. The Maximum Flow Problem • Algorithms • Simplex method can be used • Augmenting path algorithm is more efficient • Residual network • Remaining arc capacities after some flows have been assigned For example, if 5 units flow was sent from O to B with capacity 7, then the residual capacity of arc from O to B is 2. Note that the residual capacity of arc from B to O is 5 (for cancelling some previously assigned flow from O to B).

32. The Maximum Flow Problem • Augmenting path • Directed path from source to sink in residual network such that every arc on path has positive residual capacity. The minimum of these residual capacities is called the residual capacity of the augmenting path.

33. The Maximum Flow Problem

34. The Maximum Flow Problem • Algorithm (each iteration follows these steps) • Identify an augmenting path • If none exists, net flows already constitute an optimal flow pattern • Identify the residual capacity, c* of this augmenting path • It will equal the minimum residual capacity of the arcs on this path • Increase the flow in this path by c*

35. The Maximum Flow Problem • Algorithm (cont’d.) • Decrease by c* the residual capacity of each arc on this augmenting path • Increase by c* the residual capacity of each arc in the opposite direction on this augmenting path • Return to the first step • Example: Seervada park transportation problem

36. The Maximum Flow Problem

37. The Maximum Flow Problem

38. The Maximum Flow Problem

39. The Maximum Flow Problem

40. The Maximum Flow Problem • Use Excel Solver to solve the problem (see the course website)

41. The Maximum Flow Problem

42. The Maximum Flow Problem

43. 10.6 The Minimum Cost Flow Problem • General description of the minimum cost flow problem • The network is directed and connected • At least one of the nodes is a supply node, and one of the other nodes is a demand node • All remaining nodes are transshipment nodes • Flow is only allowed in direction of the arrowhead • Arc capacity gives maximum allowable flow

44. The Minimum Cost Flow Problem • General description (cont’d.) • Network has enough arcs with sufficient capacity to enable all flow generated at supply nodes to reach all demand nodes • Cost of flow through each arc is proportional to the amount of flow • Objective: minimize total cost of sending available supply through the network to meet the given demand

45. The Minimum Cost Flow Problem

46. The Minimum Cost Flow Problem • Linear programming problem formulation

47. The Minimum Cost Flow Problem • Linear programming problem formulation

48. The Minimum Cost Flow Problem • Feasible solutions property • Integer solutions property • For minimum cost flow problems where every bi and uijhave integer values, all the basic variables in every basic feasible solution also have integer values

49. The Minimum Cost Flow Problem • Example in Section 3.4 (Distribution Unlimited Co.) One of the constraints is redundant due to summation of bi’s = 0.

50. The Minimum Cost Flow Problem