2. An-Najah N. University Faculty of Engineering and Information Technology Department of Management Information systems Operations Research and Applications MIS:10676213 Prepared by Mr.Maher Abubaker Fall 2015/2016 Resources Daniel J. Epstein Department of Industrial and Systems Engineering University of Southern California Andrew and Erna Viterbi School of Engineering http://www.eecs.qmul.ac.uk/~eniale/teaching/ise330/index.html Operations Research: An Introduction, 9/EHamdy A. Taha, University of ArkansasISBN-10: 013255593X ISBN-13: 9780132555937©2011 Prentice Hall Cloth, 832 ppPublished 08/29/2010 http://www.pearsonhighered.com/educator/product/Operations-Research-An- Introduction/9780132555937.page#downlaoddiv ™ INFORMS – www.informs.org ™ ORMS - www.lionhrtpub.com/ORMS.shtml ™ Science of Better - www.scienceofbetter.org

3. 2 Network Analysis Links, nodes, trees, graphs, paths and cycles what does it all mean? Become networked! Span a tree Real OR in action! Minimal spanning tree shortest route maximum flow

4. 3 Network Terminology Graph - set of points (nodes) connected by lines (arcs) network - graph with numbers assigned to the arcs chain - sequence of arcs connecting two nodes connected graph - a chain exists for all pairs of nodes path - directed chain directed network - network with all arcs directed cycle - chain (path) connecting a node to itself tree - graph with no cycles spanning tree - tree with all nodes connected capacitated network - network with arc capacities

5. 4 A Graph

6. 5 A Network 2 5 5 34 6 7 8 10 4 7 8 12 4 Distances are in miles.

7. 6 Are there any real examples of these so called arcs and nodes? NodesArcsFlows intersectionsroadsvehicles airportsair lanesaircraft switching pointswires, channelsmessages pumping stationspipesfluid or gas work centersmaterials-handlingjobs

8. 7 Minimal Spanning Tree Select those branches of the network having the shortest total length while providing a path between each pair of nodes. The greedy algorithm: 1. Select any node arbitrarily 2. Find the unconnected node nearest to a connected node and connect the two nodes. 3. Repeat step 2 until all nodes are connected.

9. 8 A 2 5 5 34 6 7 8 10 4 7 8 12 4 Roadways connecting towns in Putrid County. Distances are in miles. The County desires to connect all the towns to a single sewage system. This will require having each town tied to a piping system. How should the system be designed to minimize the total length of pipe.

10. 9 2 5 5 34 6 7 8 10 4 7 8 12 4 1. Arbitrarily pick a starting node. 2. Connect it to the nearest node. 3. Find the nearest unconnected node to a connected node and connect it. 4. Repeat step 3 until all nodes are connected. Distance = 4 + 3 + 2 + 5 + 4 + 6 +7 + 4 = 35 miles Animated

11. 10 Shortest Route Problem Find the shortest path from one node to another node or to all other nodes in a directed network. Solution procedures 1. Tabular Method 2. Dijkstra's algorithm Shortest Path

12. 11 2 5 5 34 6 7 8 10 4 7 8 12 4 Roadways connecting towns in Putrid County. Distances are in miles. Find the shortest route from the County Seat to each of the other towns. A B C D E F G H County Seat

13. 12 SeatABCDEFGH SA-3AB-2BA-2CB-5DF-8EG-4FH-4GE-4HF-4 SC-4BC-5CF-7DH-10EB-5FC-7GF-7HD-10 SD-6BE-5EF-8FG-7GH-12HG-12 FE-8 FD-8

14. 13 SeatABCDEFGH SA-3AB-2BA-2CB-5DF-8EG-4FH-4GE-4HF-4 SC-4BC-5CF-7DH-10EB-5FC-7GF-7HD-10 SD-6BE-5EF-8FG-7GH-12HG-12 FE-8 FD-8 3 4 5 6101114 15 Seat - C - F - H 4 + 7 + 4 = 15 miles Seat - A - B - E - G 3 + 2 + 5 + 4 = 14 miles Animated

15. 14 When will Harry meet Sally? Numbers are travel times in hours. O B E H K D C A F G I J L Sally Harry 8 5 6 2 8 10 4 6 8 6 7 5 3 3 2 5 12 4 3 2 Not to scale

16. 15 0ABCDEFGH OA-5AC-2BE-5CA-2DE-2ED-2FI-3GC-3HJ-4 OB-8AI-12BD-6CD-2DC-2EB-5FC-4GJ-3HK-6 CG-3DB-6EH-8HE-8 CF-4DJ-10 IJKL IF-3JG-3KH-6LK-6 IJ-5JH-4KL-6LI-7 IL-7JI-5KJ-8 IA-12JK-8 JD-10 7 8 5 9 10 11 13 14 17 21 0 - A - C - G - J - K Harry meets Sally in 21 hours! Animated

17. 16 Maximal Flow Problem Determine the maximum flow from an origin node to a sink node through a capacitated network.

18. 17 The Algorithm 1. Find a path from origin to sink having nonzero capacities on all of its arcs. 2. Assign a flow along the path equal to the minimum arc capacity. 3. Revise all arc capacities along the path as a result of the assigned flow. 4. Repeat 1-3 until no new paths can be found.

19. 18 A Simple First Example source sink A B 5 4 4 PathFlow So - B - A - Si3 5 2323 3 3030 1 3 Animated

20. 19 A Simple First Example source sink A B 5 5 4 4 3 PathFlow So - B - A - Si3 2323 3030 1 3 So - B - Si2 0505 2 Animated

21. 20 A Simple First Example source sink A B 5 5 4 4 3 PathFlow So - B - A - Si3 2323 3030 1 3 So - B - Si2 0505 2 S So - A - Si1 4141 0 4 Animated

22. 21 A Simple First Example source sink A B 5 5 4 4 3 PathFlow So - B - A - Si3 2323 3030 1 3 So - B - Si2 0505 2 S So - A - Si1 4141 0 4 So - A - B - Si2 2323 0404 1212 Animated

23. 22 But what is the solution? source sink A B 5 5 4 4 3 PathFlow So - B - A - Si3 2323 3030 1 3 So - B - Si2 0505 2 S So - A - Si1 4141 0 4 So - A - B - Si2 2323 0404 1212 Max flow = 8 3 5 4 4 1 Animated

24. 23 Yet another example? The city of Maxiflow has the following main highways leading from the stadium where the Maxiflow Tigers play football to the junction of I-111 (interstate). Mr. Bot L. Necke, the city planner, must determine the proper direction of travel on three streets that are currently bi-directional. Flow capacities in vehicles per 5 minutes are shown. Determine the direction of flow and the maximum flow out of the stadium. S I-111 A B C D 60 35 15 30 20 50 40 25 50

25. 24 S I-111 A B C D 60 35 15 30 20 50 40 25 S - A - C - I30 50 30 020 S - B -D - I35 0 15 5 S - A - D - I 5 25 15 0 S -A - B - D -C -I15 10 0 0 5 S - A - D - C - I 5 5 0 105 90 Animated

26. 25 0 10 5 5 0 0 0 0 0 S I-111 A B C D 60 35 15 30 20 50 40 25 50 S - A - C - I30 S - B -D - I35 S - A - D - I 5 S -A - B - D -C -I15 S - A - D - C - I 5 Animated

27. 26 55 35 15 S I-111 A B C D S - A - C - I30 S - B -D - I35 S - A - D - I 5 S -A - B - D -C -I15 S - A - D - C - I 5 Max flow = 90 vehicles per 5 mins or 18 vehicles per minute. 50 40 10 30 20 50 This stuff really works! Animated

28. 27 The Max Flow as an LP Let x ij = flow from node i to node j i = 1 origin i = n sink c ij = capacity of arc i-j (0 if i,j is not an arc) Conservation of flow

29. 28 Generalized Network Flow Arc gain or loss

30. 29 The End This has been a fast past journey into the wonderful and mysterious world of graphs, networks, arcs, and branches. We hope your visit was enjoyable. Please come see us again and have a safe trip home. Animated Return