Chapter 6: Genetic Algorithms

1. Chapter 6:Genetic Algorithms July 15, 2009 Rob Bilger

2. History & Applications • Biological Definition • Mathematical Definition • Sample problem Overview

3. “Modern” Heuristics • Holland, 1960 • Useful for complex objective functions • If problems are too difficult to solve using current methods History

4. Applications • p-Median Problem • Chapter 6, Facility Location text • Traveling Salesman problem (TSP) • Scheduling problems

5. Biological Definition A random search technique designed to imitate the selective breeding or evolution of organisms It contains chromosomes, crossover, mutation, and the survival of the fittest principle

6. Biological Definition (cont’d) Initially, there is a random pool (P, population) of solutions (chromosomes) P = 5 P = k * (n/p) where k > 1 Each chromosome (feasible solution) is typically encoded as a binary string

7. Biological Definition (cont’d) Each of these chromosomes has a fitness value that corresponds to the objective function objective min = a+2b+3c+4d Next, two solutions (or parents) from this pool (population) are selected for mating to produce two new solutions (offspring)

8. Biological Definition (cont’d) How the offspring are produced is the core part of GA Rank objective function solutions, i probability for selection = 2i / [P(P+1)] where i = (worst) 1, 2, 3, ……. P (best) Parents are selected at random using these probabilities to ensure best

9. Biological Definition (cont’d) Copying parts of the parent chromosomes to the offspring is done by identifying a random crossover point Chromosome sections – after crossover point – are exchanged to create offspring Source: http://library.thinkquest.org/18242/gaexample.shtml

10. Biological Definition (cont’d) • Parents are replaced by the offspring to keep the population size constant • Replacements are referred to as the new generation • This technique can be repeated until a stopping criteria is satisfied: • Fixed number of generations, or • Quality of solution discovered

11. Biological Definition (cont’d) • Mutation rate, M = 1% typically • Done to enrich the gene pool through diversification • Randomly change one element • Invasion frequency, IR% • More intense version of mutation where randomly generated solutions replace IR% of the population every 1/IF generations.

12. Mathematical Definition P = size of initial population G = number of generations O = number of overlapping solutions M = mutation rate Ci = crossover operator I Zj = fitness value of solution j IF = frequency of invasions IR = % of population replaced by invasion

13. Sample Problem • p-Median problem • Locate p facilities to minimize the demand weighted distance between each demand node and its assigned facility Min ΣΣ hi dij yij s.t. Σ xj = p Σ yij = 1 yij - xj ≤ 0 xi {0, 1} yij {0, 1}

14. Sample Problem • p = 2 # of facilities to locate • P = 6 population • n = 5 elements in solution (genes)

15. Closing • Disadvantage of GA is the number of parameters involved: • Size of initial pool • Method for selecting parents for mating • The crossover operator • The number of offspring to produce • The replacement technique • The mutation rate • GA’s are best when there are a small number of constraints (or none)

16. Q & A Questions

17. Homework What are the different crossover methods for creating new solution sets (offspring). Explain each. Again using the p-Median example with P=6, p=2, and n=5 create a new generation of offspring using the Template Operator [ 1,0,1,0,1 ] and initial pool [0,1,1,0,0], [1,1,0,0,0], [0,0,1,0,1], [0,0,1,1,0], [1,0,0,0,1], [0,1,0,0,1] with i values of 1, 2, 3, 4, 5, 6 respectively. For purposes of this exercise assume parents for mating are randomly selected with i values 5, 6; 3, 5; and 4, 6. What problems do you notice with the new generation? Which crossover operator did the text choose for the p-Median problem? Why? Describe a problem that may arise from using one of the other crossover operators.