Optimal Spatial Partitioning for Resource Allocation

1. Optimal Spatial Partitioning for Resource Allocation Kostas Kolomvatsos, Kakia Panagidi, Stathes Hadjiefthymiades Pervasive Computing Research Group (http://p-comp.di.uoa.gr) Department of Informatics and Telecommunications National and Kapodistrian University of Athens ISCRAM 2013 Baden Baden, Germany

2. Outline • Introduction • Problem Formulation • Data Organization • Proposed approach • Case Study

3. Introduction • Spatial Partitioning Problem • Segmentation of a geographical area • Optimal allocation of a number of resources • Resources could be vehicles, rescue teams, items, supplies, etc • The allocation is done according to: • Population patterns • Spatial characteristics of the area • The process is affected by the following issues: • Where to locate the resources • Which area each resource will cover • The number of resources • Final objective: to maximize the area that the limited number of resources will cover under a number of constraints.

4. Problem Formulation • Nj (j=1, 2, …, R, R is the resources number) resources are available to be allocated in an area A • Each resource is of type Tj • The area has an orthogonal scheme (width: W0, height: H0) • A number of constraints should be fulfilled (Cjk, k=1,2, …, K) • In the optimal solution, we have: where Al is the area covered by the lth resource. • The shape of each sub-area is not defined • Overlaps should be eliminated

5. Data Organization • Area related parameters • Population attributes, density of population • Type of area (hilly, flat, etc) • Roads – road segments (length, speed limit, width, type, etc), traffic • Places of interest - PoIs (schools, hospitals, fuel stations, etc) • Resource related parameters • Type (e.g., vehicle, rescue team, supplies, etc) • Maximum speed in emergency and maximum travel distance • Crew or personnel • Current Location • Examples: • Open Street Map could be the basis • OSM data could be retrieved by CloudMade or Mapcruzin.com

6. Proposed Approach (1/2) • Split the area • Area A is defined by [(xUL, yUL), (xLR, yLR)] – upper left and lower right corners • Area A is divided into NcX Nc cells • Size of each cell • Define cell weights • Use of AHP for attributes priority • Users define the relative weight for each attribute - criterion • Cell weight calculation where wi is the ith attribute weight defined by AHP, Aijis the ith attribute value in cell j (e.g., schools, hospitals, fuel stations, etc), NA is the attributes number

7. Proposed Approach (2/2) • Particle Swarm Optimization • We generate M particles (M vectors p of all resources coordinates) p = [(x1, y1), (x2, y2), …, (xN, yN)] • Coordinates are the center of a specific cell • Fitness Function F(p): Covered Area by each particle (each resource) • The best solution p* maximizes F(p*) • If we consider that resources are vehicles • Area covered by a resource T: time restriction, S: maximum speed, wi: the weight of each cell in the neighbor, NH: number of neighbors • Total covered area by the particle , |Nsi|: neighbors number

8. Case Study (1/2) • Suppose Nj = 5 ambulances are available • Their characteristics are: • We define maximum response time T = 5 minutes • We select the desired area

9. Case Study (2/2) • Resource locations are presented in the map • Numerical Results

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