Resource-Based Fitness Sharing

Jeffrey Horn

Northern Michigan UniversityDepartment of Mathematics and Computer Science

Marquette, MI USAjhorn@nmu.edu

http://cs.nmu.edu/~jeffhorn

PPSN VII September 10, 2002

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The Problem

• We want to exploit the “covering” capabilities of niching/speciation. Idea is to make fitness a function of converage.

• Example applications: shape nesting, cutting stock trim minimization, layout, packing, etc.

• Goal is to cover a finite, uniform surface (the substrate) with the maximum number of shapes (or pieces).

• All pieces are identical.

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Resource Sharing Defined

Example Scenario:Three overlapping

Niches A, B, C

.CABAA nn

fnn

fn

- f - fff

ACABAC AB A sh,A

Shared fitness

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Resource Sharing on One-Dimension Nesting Problem

(selection only)Bold rectangle is substrate to be covered by small squares

All squares represented initially

Final coverage still contains overlapping squares, and is missing some globals

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Fitness Sharing Define

Pj

ish,i

jiSh

f f

),(

otherwise.

),(for ),(

0

1),( shjid

jid

sh jiSh

where

is the sharing function

The shared fitness is

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02 RESOURCE SHARINGRESOURCE SHARING

+ + FITNESS SHARINGFITNESS SHARING

==RESOURCE-BASED FITNESS SHARINGRESOURCE-BASED FITNESS SHARING

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Resource-based Fitness Sharing Defined

.

species Y

Sh,XXYX

X

fn

f f

ACAB A

A

fnfnfn

ff

CBA

Sh,A

Shared fitness

Example for threeOverlapping niches

Note how RFS combines the simpler structure(a ratio) of fitness sharing with the resource-basedniche overlap calculation of resource sharing.

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RFS on the One-Dimension Shape Nesting Problem

Note edgeeffect

All squares represented initially

(selection only)

PerfectCoverage(indicates

highselectionpressure)

Blue rectangle is substrate to be covered by smaller, green, squares

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Fitness Sharing on a “Hat” Function

 

f(x)1  0

  

 

Initial populationcovers entire domain

“off-substrate”Individuals have

died off. Niches at edges do well

(the edge effect)

Edge effectspropogate

toward center,reinforce each

other

Ideal solution.Nine remainingspecies exactly

cover the “top” ofThe “hat”

Success of FS in One Dimension Nesting

(selection only)

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Here linesconnect species

that seem to cooperate in

trying to cover the surface

together(they do not

overlap)

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Generation 0

0

1000

Generation 25

GlobalsOnly

Distributionof

EntirePopulation

RFS in Two Dimensions

Blue squareis the

substrateto be

covered

All

overhanging pieces

have

been

eliminated

Initial distribution, including globals, is uniform. Beginning of corner effect…

(selection only)

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Generation 130

0

1000

Generation 400

All 16,000 population slots are filled (fairly evenly) with copies of the 16 globals

Still some overlap left in the population…

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Distribution

of

Entire

Population

RFS with Mutation

Much smaller pop size (N=500).Some globals mustbe discovered by

mutation(some are NOT in

Initial pop.)

Pop has converged on the 16

globals, withmutation still

producingsome “misfits”

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The Approaches

• FITNESS SHARING (FS) established– Fast and simple– Some success (e.g., niching on the Pareto front in multi-objective

EC)– LIMITATION: fixed niche radius implies spherically-shaped

niches/pieces ONLY (also constrains shape of substrate)

• RESOURCE SHARING (RS) natural– Based on actual, arbitrarily shaped pieces and substrate– Natural– LIMITATION: introduces complex dynamics that often prevent

convergence to “optimal” equilibrium distribution

• RESOURCE-BASED FITNESS SHARING (RFS) new– Combines benefits of both FS and RS– Overcomes above limitations of FS and RS– Simpler dynamics (so more robust convergence) than RS, but based

directly on actual coverage of resources (substrate) by pieces (shapes)

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Summary

• RFS seems to have the simplicity and efficiency of fitness sharing

• But also has the “natural fit” of resource sharing (with niches based entirely on resource coverage)

• Potential for success on harder shape nesting problems (e.g., irregular shapes, irregular substrates, rotated pieces, etc.)