Resource-Based Fitness Sharing
Jeffrey Horn
Northern Michigan University
Department of Mathematics and Computer Science
Marquette, MI USA
[email protected]
http://cs.nmu.edu/~jeffhorn
PPSN VII
September 10, 2002
PPSN VII September 10, 2002
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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PPSN VII September 10, 2002
Resource Sharing Defined
Example Scenario:
Three overlapping
Niches A, B, C
Shared fitness
fsh,A

fA - fAB - fAC
nA
fAB
fAC

nAnB  nA  nC .
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Resource Sharing on One-Dimension
Nesting Problem
PPSN VII September 10, 2002
(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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5
PPSN VII September 10, 2002
Fitness Sharing Define
PPSN VII September 10, 2002
The shared fitness is
fsh,i 
fi
 Sh(i, j )
j P
where
d (i , j )

1 
 sh
Sh(i, j )  

0
for d (i , j )   sh
otherwise.
is the sharing function
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PPSN VII September 10, 2002
RESOURCE SHARING
+
FITNESS SHARING
=
RESOURCE-BASED FITNESS SHARING
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Resource-based Fitness Sharing Defined
PPSN VII September 10, 2002
Shared fitness
fSh,X 
fX
 nX fXY
 species Y
.
Example for three
Overlapping niches
fA
fSh,A 
nA fA  nB fAB  nC fAC
Note how RFS combines the simpler structure
(a ratio) of fitness sharing with the resource-based
niche overlap calculation of resource sharing.
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RFS on the One-Dimension Shape Nesting
Problem
(selection only)
Blue rectangle is substrate to be covered by smaller, green, squares
PPSN VII September 10, 2002
All squares represented initially
Note
edge
effect
Perfect
Coverage
(indicates
high
selection
pressure)
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Fitness Sharing on a “Hat” Function
PPSN VII September 10, 2002
(selection only)
Initial population
covers entire domain
f(x)
1
“off-substrate”
Individuals have
died off. Niches at
edges do well
(the edge effect)
0
Ideal solution.
Nine remaining
species exactly
cover the “top” of
The “hat”
Edge effects
propogate
toward center,
reinforce each
other
Success of FS in One Dimension Nesting
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11
PPSN VII September 10, 2002
PPSN VII September 10, 2002
(they do not
overlap)
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RFS in Two Dimensions
PPSN VII September 10, 2002
(selection only)
All
Distribution
of
Entire
Population
overhanging
pieces
Blue square
is the
substrate
to be
covered
have
been
eliminated
Generation 25
Generation 0
Globals
Only
20
1000
10
0
Initial distribution, including globals, is uniform.
0
Beginning of corner effect…
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PPSN VII September 10, 2002
Generation 400
Generation 130
1000
1000
0
0
Still some overlap left in the population…
All 16,000 population slots are filled
(fairly evenly) with copies of the 16 globals
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PPSN VII September 10, 2002
RFS with Mutation
Much smaller
pop size (N=500).
Some globals must
be discovered by
mutation
(some are NOT in
Initial pop.)
Pop has
converged
on the 16
globals, with
mutation still
producing
some “misfits”
Distribution
of
Entire
Population
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The Approaches
PPSN VII September 10, 2002
• 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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PPSN VII September 10, 2002
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.)
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