Spatial Thinking and Modeling of Network-Based Problems Presentation at the SPACE Workshop Columbus, Ohio, July 1, 25 Shih-Lung Shaw Professor Department of Geography University of Tennessee Knoxville, TN 37996-925 E-mail: sshaw@utk.edu
Spatial Thinking! Spatial data is embedded in many social science studies. We could choose to ignore the spatial component in our data sets, but it does not remove the fact that many of our data sets are spatial in nature.! Some examples of spatial concepts: location, distance, adjacency, connectivity, direction, hinterland, interaction,
Different Types of Network! Transportation networks: highway, railroad, waterway, air, pedestrian, bike,! Utility networks: electricity, water, sewer,! Communications networks: telephone, Internet,! Social/Economic/Cultural networks: church members, colleagues, supply chain,! Natural networks: rivers/streams, animal migration routes,
A Network Example! Let s start with an example of airline networks.! The following slides show six airline passenger networks of American, Continental, Delta, Northwest, United, and USAir among the 1 most populous cities in the 48 contiguous states (Source: Shaw, S-L., 1993. Hub Structures of Major U.S. Passenger Airlines, Journal of Transport Geography, 1(1), 47-58.)! What are some spatial concepts embedded in these airline networks?
Note that the rankings change with the index applied to the BNA: Nashville, TN IND: Indianapolis, IN SDF: Louisville, KY MEM: Memphis, TN TYS: Knoxville, TN MKE: Milwaukee, WI
H i : standardized index measuring total number of direct connections. S i : standardized index measuring total number of shortest-path direct and indirect connections. V i : index measuring total distance of shortestpath direct and indirect connections.
H i & S i : measure only connectivity pattern of topological distance and do not consider location. V i : measures linkage pattern of real distance. It takes into account the location of nodes on the network and therefore favors those centrally located nodes.
ArcGIS 9 Airline Networks Demo
Reflection of Spatial Thinking:! Accessibility levels at the nodes in a network depend on their direct and indirect connections with other nodes and their locations in the network.! Geographic centrality alone does not explain the variations between the networks. It appears that network centrality also plays an important role in determining the rankings.
! What are the implications of these different spatial structures chosen by individual airlines? Are some spatial patterns better than others?! Accessibility levels can be measured by different variables (e.g., topological distance that emphasizes connectivity vs. actual distance that emphasizes relative location with respect to other nodes). Is it better to choose the major hubs such that they have consistent top rankings among H i, S i, and V i?
Some spatially relevant questions: 1 Description: Where are the hubs? What are the hub-and-spoke network structures? 2 Explanation: Why does a particular network have x number of hubs? Why do airlines choose hubs at certain locations? 3 Prediction: How do we forecast interactions (flows) between the hubs? 4 Optimality: What is the optimal design of a hub-and-spoke network? " How many hubs? (numbering problem) " Where should the hubs be placed? (location problem) " How to allocate nodes to different hubs? (allocation problem) " How to route flows through the network? (routing problem)
Spatial Analysis/Spatial Modeling! spatial pattern and spatial analysis can rarely if ever be used to confirm theories, though they can certainly be used to deny false ones and to justify controlled experiments or longitudinal analysis where these are possible. In summary, spatial analysis is perhaps best seen as an exploratory technique, more suitable for the generation of hypotheses and insights than to strict confirmation of theory. (Source: Goodchild M.F. and Janelle, D.G., 24. Thinking Spatially in the Social Science, Spatially Integrated Social Science. New York: Oxford University Press, p. 7)
Network Analysis! Network analysis places an emphasis on the structures formed by linkages and nodes in a network.! Two relevant mathematical subfields are graph theory and topology.! Two common representation methods are: Network as a Graph: A network is represented as a set of nodes interconnected by a set of linkages. Network as a Matrix: A network is represented as a matrix of data between a set of origins and a set of destinations.
(Source: Taaffe, Gauthier, and O Kelly, 1996. Geography of Transportation.)
Representation of graphs and networks:! Undirected graph: implies that relationships are hold in both directions.! Directed graph: specifies relationships that hold only in particular directions.! Connectivity matrix (or adjacency matrix): Matrix rows and columns correspond to network nodes. If an arc exists between node i and node j, then x ij = 1 Otherwise, x ij =
N6 N5 N4 N3 N2 N1 N6 N5 N4 N3 N2 N1 N1 N2 N3 N4 N5 N6 Connectivity Matrix 1 N6 1 1 N5 1 N4 1 1 1 N3 1 1 N2 1 N1 N6 N5 N4 N3 N2 N1 1 2 1 3 2 1 Degree of a node
! Node-arc incident matrix: Matrix rows correspond to nodes, while matrix columns correspond to arcs. x ij = 1 if node i is the start node of arc j x ij = -1 if node i is the end node of arc j Otherwise, x ij = N1 A1 N2 A2 A5 N6 A3 N3 A4 N5 N4
N6 N5 N4 N3 N2 N1 A5 A4 A3 A2 A1 A1 A2 A3 A4 A5 N1 N2 N3 N4 N5 N6 Node-Arc Incident Matrix -1 N6 1-1 N5 1 N4 1-1 -1 N3 1-1 N2 1 N1 A5 A4 A3 A2 A1
! Shortest-path matrix: Shortest-path matrix considers both direct and indirect linkages along the shortest path between each pair of nodes in a network.
N6 N5 N4 N3 N2 N1 N6 N5 N4 N3 N2 N1 N1 N2 N3 N4 N5 N6 1 1 2 3 5 Shortest-path Matrix 5 45 35 55 65 N6 5 4 3 5 6 N5 45 4 1 3 4 N4 35 3 1 2 3 N3 55 5 3 2 1 N2 65 6 4 3 1 N1 N6 N5 N4 N3 N2 N1 25 185 165 125 165 25 Total
Network Representations in Commercial GIS! Almost all commercial GIS packages implement the network as a graph representation approach.! However, many commercial GIS packages do not support network as a matrix representation approach.! Also, most commercial GIS packages offer limited transportation analysis functions.
Some Basic Network Concepts:! Planar and Nonplanar Networks: Planar networks: * Linkages cannot cross each other without creating new intersection nodes. * e.g., most surface transport networks (exceptions include overpass/underpass, tunnel). Nonplanar networks: * Linkages can cross each other without creating new intersection nodes. * e.g., airline networks
Address Geocoding:! Address geocoding (also known as address matching) is a process that creates a point GIS map layer from street addresses. To match street addresses, GIS compares the address components in an address data file and the address range data in a GIS map layer.! What are components of addresses and what is a match? Address Data Address Range Data in GIS Layer a) 62 CENTER STREET 1. 51 599 5 598 CENTER AV b) 71 PALM AV W 2. 61 699 6 698 CENTER ST c) 68 1ST ST 3. 71 799 7 798 PALM AV W 4. 71 799 7 798 PALM AV E 5. 651 699 65 698 FIRST ST
Example of Address Geocoding Applications:! A company can match customer addresses against a street network to find out where the customers are located. The information can be used to perform marketing analysis.! Emergency dispatch operators can use geocoding to enter an address, determine who should respond, and route emergency vehicles and personnel to the specific address.! A school district can use geocoding to match student addresses against a GIS street map database. It can then develop school busing plans.! Environmental engineers can identify potential impacts of hazardous material storage facilities in populated areas by matching the site addresses to a GIS database and then perform spatial analysis with census data.
Address Geocoding/Network Analysis Exercise:! To download exercise files, please visit my web site for the 25 SPACE Summer Workshop at http://web.utk.edu/~sshaw/space25.htm. There are two files: Instructions for the exercise (Microsoft Word file) Data file (a self-extracting zip file)! You will need ESRI ArcView 3 with the Network Analyst extension to complete this exercise.
ArcView 3 Address Geocoding Demo
Reflection of Spatial Concepts:! Spatial data exist in many forms. Street addresses, ZIP codes, census unit IDs, telephone numbers (how about cell phones?) all include information about locations.! GIS offer geocoding functions to help convert implicit spatial data into explicit locations on a map.
Solving Network-based Problems:! Shortest-Path Problem: Find the shortest (or best) path between two or more nodes on a network.! Traveling Salesman Problem (TSP): Given n nodes in a network, find the shortest (or best) tour of visiting each of the n nodes only once and returning to the starting node.
ArcView 3 Shortest-Path and TSP Demos
Reflection of Spatial Concepts:! The best tour of visiting a set of given nodes in a network is different from finding a sequence of shortest paths between adjacent nodes.! In other words, spatial proximity is not the only consideration in solving some network-based problems.! Connectivity between linkages is critical to solving many network-based problems. GIS representations must properly represent network connectivity.
Solving Network-based Problems: (cont.)! Vehicle Routing Problem (VRP): Vehicle Routing Problem is essentially a fleet version of the TSP. VRP is to find the best routes for a fixed number of vehicles from their respective depots to serve a set of demand locations such that the total cost is minimized and vehicle capacity constraints are not violated.
TransCAD VRP Demo
Reflection of Spatial Concepts:! VRP finds multiple best tours to serve subsets of the demand nodes in a network. This is a very hard (NP-hard) problem to solve because it needs to partition the demand locations into subsets and find the best tour for each subset at the same time.! VRP uses different spatial information (e.g., proximity, connectivity, direction), along with nonspatial information (e.g., number of vehicles, vehicle capacity, open time windows, demand levels) to solve the problem (mostly based on heuristic algorithms).
Table 5-1 (part 1): Algorithm complexity (Miller & Shaw, 21, p. 138)
Table 5-1 (part 2): Algorithm complexity (Miller & Shaw, 21, p. 138)
Solving Network-based Problems: (cont.)! Service Areas Delineation and Network Partitioning Problems: Service Area Delineation Problem identifies the accessible streets within a specified travel time or distance in a network and the service areas that encompass the accessible streets. Network Partitioning Problem partitions a network layer into zones or districts based on seed locations.
ArcView 3 Service Areas Delineation and TransCAD Network Partitioning Demos
Solving Network-based Problems: (cont.)! Facility Location Problems: Facility Location Problems attempt to find optimal or near-optimal locations for activities such as warehouses, retail stores, public libraries, emergency service facilities to meet a demand pattern. There are many variations of facility location problems. Each is designed to solve a particular type of facility location problem according to its specific criteria.
TransCAD Facility Location Problem Demos
Reflection of Spatial Concepts:! A comprehensive facility location-routing problem involves solving numbering problem, location problem, allocation problem, and routing problem simultaneously.! Spatial data, in their implicit or explicit forms, provide critical information to solve network-based problems.
Thank You! Again, here is the information to download Address Geocoding/Network Analysis Exercise files:! To download exercise files, please visit my web site for the 25 SPACE Summer Workshop at http://web.utk.edu/~sshaw/space25.htm. There are two files: Instructions for the exercise (Microsoft Word file) Data file (a self-extracting zip file)! You will need ESRI ArcView 3 with the Network Analyst extension to complete this exercise.