Management Science 461Lecture 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 distanceTimeDependencies

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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

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Basic Metrics

Two fundamental metrics: Euclidean and rectilinear

Rectilinear or right-angle distance metric

Euclidean or straight-line distance metric

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Do maps help us?

Google Earth Pick two points – is the path between them

a grid, straight line, or some combination?

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Fit a Distance Metric

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x1 x2

y1

y2

Distances

Rectilinear: |x1-x2| + |y1-y2|

Euclidean: [(x1-x2)2 + (y1-y2)2]1/2

Can we combine these two into a single formula?

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k and s Distance Metric

If s=1: Rectilinear metric If s=2: Euclidean metric k is a scaling factor

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1/( , ) | | | |

ss sij i j i jd k s k x x y y

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!

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Estimate k and s

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i j

ijij skdLskSSE 2,,

0 and 1subject to

),(minimize

ks

skSSE

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

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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

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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

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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

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