1. Resource-Based Fitness Sharing Jeffrey Horn Northern Michigan University Department of Mathematics and Computer Science Marquette, MI USA jhorn@nmu.edu http://cs.nmu.edu/~jeffhorn PPSN VII September 10, 2002

2. 2 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.

3. 3 PPSN VII September 10, 2002 Resource Sharing Defined Example Scenario: Three overlapping Niches A, B, C Shared fitness

4. 4 PPSN VII September 10, 2002 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

5. 5 PPSN VII September 10, 2002

6. 6 Fitness Sharing Define where is the sharing function The shared fitness is

7. 7 PPSN VII September 10, 2002 RESOURCE SHARING + FITNESS SHARING = RESOURCE-BASED FITNESS SHARING

8. 8 PPSN VII September 10, 2002 Resource-based Fitness Sharing Defined. Shared fitness Example for three Overlapping niches Note how RFS combines the simpler structure (a ratio) of fitness sharing with the resource-based niche overlap calculation of resource sharing.

9. 9 PPSN VII September 10, 2002 RFS on the One-Dimension Shape Nesting Problem Note edge effect All squares represented initially (selection only) Perfect Coverage (indicates high selection pressure) Blue rectangle is substrate to be covered by smaller, green, squares

10. 10 PPSN VII September 10, 2002 Fitness Sharing on a “Hat” Function f(x)1 0 f(x)1 0 Initial population covers entire domain “off-substrate” Individuals have died off. Niches at edges do well (the edge effect) Edge effects propogate toward center, reinforce each other Ideal solution. Nine remaining species exactly cover the “top” of The “hat” Success of FS in One Dimension Nesting (selection only)

11. 11 PPSN VII September 10, 2002

12. 12 PPSN VII September 10, 2002 Cooperative Behavior? Here lines connect species that seem to cooperate in trying to cover the surface together (they do not overlap)

13. 13 PPSN VII September 10, 2002 Globals Only Distribution of Entire Population RFS in Two Dimensions Blue square is the substrate to be covered All overhanging pieces have been eliminated Initial distribution, including globals, is uniform.Beginning of corner effect… (selection only)

14. 14 PPSN VII September 10, 2002 All 16,000 population slots are filled (fairly evenly) with copies of the 16 globals Still some overlap left in the population…

15. 15 PPSN VII September 10, 2002 Distribution of Entire Population 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”

16. 16 PPSN VII September 10, 2002 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)

17. 17 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.)