WOOD 492 MODELLING FOR DECISION SUPPORT

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 Wood 492 - Saba Vahid

3. Network Terminology • A network has a set of points (called Nodes, e.g. node A) and a set of lines (callesArcs, 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, it’s called a “directedarc” (e.g. DE), otherwise, it’s an “undirectedarc” (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) D A C E B Wood 492 - Saba Vahid

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) B C A L M N O Wood 492 - Saba Vahid

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 Wood 492 - Saba Vahid

6. A T B D O Backcountry Gate 7 2 2 5 5 4 C E 7 3 1 ParkEntrance 4 1 4 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 Distance in miles From: Hillier & Lieberman Wood 492 - Saba Vahid

7. A T B D O Backcountry Gate 7 2 2 5 5 4 C E 7 3 1 ParkEntrance 4 1 4 Shortest Path Solution Solved node Two options for the shortest paths Wood 492 - Saba Vahid

8. Shortest Path Solution with Simplex • Formulate as a BIP • Assume we have n nodes • dij is the distance of the arc between node i and j • We define a trip on arc as the “flow” of the arc • So if an arc is selected to be in the shortest path, its flow will be “1” • Binary decision variables: • Objective: Minimize the total distance travelled = sumproduct (distances * binary variables) LP Matrix Wood 492 - Saba Vahid

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) Lab 7 Wood 492 - Saba Vahid