1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 22 Network Problems

2. Example 13: shortest path problem Oct 29, 2012Wood 492 - Saba Vahid2 HW T3 T2 T1T4 T7 T5 T6 T8T9 4.9 6.2 2.0 8.0 2.5 9.0 6.0 2.45.0 8.0 6.0 3.6 4.06.0 Example 13

3. Network Problems There are many types of network problems, we will focus on three types: –Shortest path problems Find the shortest route between the origin and the destination point –Minimum spanning tree Find the smallest network that has a path between each two points –Critical path method (CPM) for time-cost trade-off Find the optimal plan to expedite some activities within a project in order to minimize the costs while meeting the project deadline Oct 29, 2012Wood 492 - Saba Vahid3

4. Minimum Spanning Tree We have: –a set of nodes –a set of potential arcs and their lengths (undirected arcs) Objective: –Insert enough arcs so there is a path between every pair of nodes –Minimize the total length of the inserted arcs Note: for a network with n nodes, a minimum spanning tree can be found with only n-1 arcs A few applications –Design of telecom networks (fiber optic networks, cable TV networks) –Network of pipelines to connect a number of locations –Designing railway networks Oct 29, 2012Wood 492 - Saba Vahid4

5. Minimum Spanning Tree – solution algorithm Being greedy works! It gives us the optimal solution The greedy algorithm: 1.Select any node arbitrarily and connect to the nearest node 2.Identify the unconnected node that is closest to a connected node and then connect these two nodes* 3.Repeat until all nodes have been connected * In case of ties, choose a node arbitrarily. Such ties usually (not always) mean there are multiple optimal solutions. Oct 29, 2012Wood 492 - Saba Vahid5

6. Minimum Spanning Tree – Example 14 Seervada Park example The arcs now represent potential links Problem: Find the minimum spanning tree for the park network Oct 29, 2012Wood 492 - Saba Vahid6 O A BD T EC 2 4 4 7 5 7 2 5 3 4 1 1 Park Entrance Backcountry Gate

7. Example 14 1.We arbitrarily choose node O to start. Closest unconnected node is A 2.The closest unconnected node to either O or A is node B (closest to A) 3.Closest unconnected node to O, A or B is node C (closest to B) 4.Closest unconnected node to O, A, B or C is node E (closest to B) 5.Closest unconnected node to O, A, B,C or E is node D (closest to E) 6.The only remaining node is T, and it’s closest to D 7.The resulting network is the minimum spanning tree with length of 14 Important: The selection of the first node will not impact the final solution Oct 29, 2012Wood 492 - Saba Vahid7

8. Critical Path Method (CPM) A network used to represent a project is called a “project network” Three types of information are needed before we can create a project network: –Project activities: break down the project into individual tasks –Precedence relationships: Identify immediate predecessors of each activity (which activities must be finished before each activity can start) –Time information: estimate activity durations To visualize the network: Activity-on-Node (AON) project networks are common. –Each activity is represented with a node and arcs are used to depict precedence relationships Oct 29, 2012Wood 492 - Saba Vahid8

9. Example 15 – Project network “Reliable Constructions Co.” has identified the activities within a plant construction project The deadline is in 40 weeks The total of all estimated durations will be 79 weeks, but some activities can be done in parallel How long will the project take? First, we need to visualize the project network Oct 29, 2012Wood 492 - Saba Vahid9 Example 15