1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 20 Network Problems

2. Network problems Network representations are used for many real-world problems –Transportation and distribution –Communication networks –Project planning –Facility location –Assignment problems Linear programming can be used to solve large scale network problems Oct 23, 2012Wood 492 - Saba Vahid2

3. Network Terminology A network has a set of points (called Nodes, e.g. node A) and a set of lines (calles Arcs, e.g. arc AD, CB) connecting those points –Node examples: cut blocks, log yards, sawmills, factories, … –Arc Examples: logging roads, water routes, conveyor belts,…. If only one direction of flow is allowed through an arc, its called a directed arc (e.g. DE), otherwise, its anundirected arc (e.g. AC) A network with only directed arcs, is a directed network, a network with all undirected arcs, is an undirected network A sequence of arcs between two nodes, is a path, which can be directed or undirected (e.g. BC-CA-AD is an undirected path between B and D) Oct 23, 2012Wood 492 - Saba Vahid3 A D C B E

4. Network Terminology A path that begins and ends with the same node is a cycle, (e.g. AB-BC-CA) If there is at least one path (undirected or directed) between two nodes, those nodes are said to be connected (e.g. A and C are connected) If all nodes in a network are connected, that network is a connected network (e.g. the network in the top right figure) A connected network that contains no (directed or undirected) cycles is called a spanning tree (e.g. bottom right figure) Oct 23, 2012Wood 492 - Saba Vahid4 A L C B M O N

5. 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 23, 2012Wood 492 - Saba Vahid5

6. Shortest Path Problem - example Seervada Park: has a narrow road system with a tram system to deliver hikers from Entrance O to Backcountry Gate T Problem: Find the shortest travel route for trams Solution mechanism: –Exhaustive search: Search all of the possible arcs in the network and pick the best one –Special network algorithm Oct 23, 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 From: Hillier & Lieberman Distance in miles

7. Shortest Path Solution Oct 23, 2012Wood 492 - Saba Vahid7 O A BD T EC 2 4 4 7 5 7 2 5 3 4 1 1 Park Entrance Backcountry Gate Solved node Two options for the shortest paths

8. Shortest Path Solution with Simplex Oct 23, 2012Wood 492 - Saba Vahid8 LP Matrix

9. Lab 7 preview A facility Location problem Integer formulation Locating wood pallet plants to serve 5 districts –Minimize annual transportation and investment costs –Control CO2 emissions –Must serve all districts (process all the residues to produce pallets) Oct 23, 2012Wood 492 - Saba Vahid9 Lab 7