WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 20 Network Problems

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 Saba Vahid2

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 Saba Vahid3 A D C B E

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 Saba Vahid4 A L C B M O N

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 Saba Vahid5

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 Saba Vahid6 O A BD T EC Park Entrance Backcountry Gate From: Hillier & Lieberman Distance in miles

Shortest Path Solution Oct 23, 2012Wood Saba Vahid7 O A BD T EC Park Entrance Backcountry Gate Solved node Two options for the shortest paths

Shortest Path Solution with Simplex Oct 23, 2012Wood Saba Vahid8 LP Matrix

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 Saba Vahid9 Lab 7