Management Science 461 Lecture 1b - Distance Metrics September 9, 2008

Distance Metrics Without distances, DM problems usually aren’t DM problems at all If not distances, then metrics based on distance  Time  Dependencies 2

Location Problems In large-scale location problems, it may be hard to obtain all distances Consider a problem with 1000 nodes: we need a 1000x1000 distance matrix (or do we?) Distance metrics allow us to estimate with relative accuracy, without resorting to more complicated methods 3

Basic Metrics Two fundamental metrics: Euclidean and rectilinear Rectilinear or right-angle distance metric Euclidean or straight-line distance metric 4

Do maps help us? Google Earth Pick two points – is the path between them a grid, straight line, or some combination? 5

Fit a Distance Metric 6 x1x1 x2x2 y1y1 y2y2

Distances Rectilinear: |x 1 -x 2 | + |y 1 -y 2 | Euclidean: [(x 1 -x 2 ) 2 + (y 1 -y 2 ) 2 ] 1/2 Can we combine these two into a single formula? 7

k and s Distance Metric If s=1: Rectilinear metric If s=2: Euclidean metric k is a scaling factor 8

Fit a Distance Metric Determine the actual distances for a subset and estimate parameters k and s Estimate k and s by minimizing the sum of squared differences between actual and estimate distances Choose as large and diverse a sample as possible; bigger sample means better fit  Be careful of overselecting one type! 9

Estimate k and s 10

Straight to the answer It may seem that the straight-line metric would be a poor approximation in most cases … … but straight-line metric provides a surprisingly good approximation of the total distance between many pairs of points 11

Some real-world examples Within the state of Wisconsin, road distances between cities are, on average, 18% longer than the straight-line metric In Ontario, they are about 30% longer So – when we set s = 2, then the optimal k comes out to be 1.18 and 1.30, assuming we use the same scale on the grid to generate the x and y coordinates as the scale on the map 12

Distances in Alberta To model highway distances in Alberta, it is a good idea to use the rectilinear metric (s = 1).  Rural road network is a grid (two miles between E-W roads and one mile between N- S roads, with corrections) Travel distances can be approximated quite accurately using the rectilinear metric 13

Distance Metrics – Final Note In some instances, actual distances will be longer (due to rivers, mountains), and in other instances, actual distances will be shorter (interprovincial highways) Note the difference between distance and time travelled 14