Monte Carlo Methods and the Genetic Algorithm Applications and Summary John E. Nawn MAT 5900 April 5 th, 2011

Review of the Genetic Algorithm Heuristic approach Fitness function Chromosomes Generations Mutations Cross Overs

The Traveling Salesman Problem First formulated in 1930 Given a collection of cities and the cost of travel between each pair of them, the traveling salesman problem (TSP) is to find the cheapest way of visiting all of the cities and returning to the starting point Important for testing computational strategies Solution potentially involves length, costs, specific penalties etc.

Components of the TSP GA Initial Conditions Selection Criteria Mutation Mask Cross Over Mask Counter

R Code Seeding the initial distance matrix Computational speed: “gen” vs. “n” Number of Computations Best solution ◦ Generations ◦ Length floor Addition of Cost

Several Considerations Markov chain similarities ◦ N th step depends only on (n – 1) th step ◦ Critique: storing best solution Multiple criteria Selective solution “breeding” ◦ Ex: if solutions a, b, …, n all provide a suitable solution, mutate and cross over together in order to generate better solutions

The Future of the GA Increasing theory No Free Lunch Theorem (NFL) ◦ Complex algorithms vs. random searches Stability in population selection ◦ Local optima vs. absolute optium Landscape and neighborhood functions ◦ Linking mutation and crossing over Variety of Applications ◦ Exs. GA Tech’s World TSP, Rankings

Selected Bibliography Craig, Nancy L. et. al. Molecular Biology: Principles of Genome Function. New York: Oxford University Press, Print. Krzanowski, Roman and Jonathan Raper. Spatial Evolutionary Modeling. New York: Oxford University, Inc., Print. Reeves, Colin R. and Johnathan E. Rowe. Genetic Algorithms: Principles and Perspectives: A Guide to GA Theory. Boston: Kluwer Academic Publishers, Print. Russell, Peter J. iGenetics: A Mendelian Approach. San Francisco: Pearson Education, Inc., Print. “The Traveling Salesman Problem.” Traveling Salesman Problem. Georgia Tech. Janurary April 2011.