Genetic Algorithms

1. Genetic Algorithms

2. Introduction To Genetic Algorithms (GAs)

3. What Are Genetic Algorithms (GAs)? Genetic Algorithms are search and optimization techniques based on Darwin’s Principle of Natural Selection.

4. Characteristics of GA • Parallel-search procedures that can be implemented on parallel processing machines for speeding operations • Applies to both continuous and discrete optimization problems • Stochastic in nature and less likely to get caught in local minima • Facilitates both structure and parameter identification

5. Darwin’s Principle Of Natural Selection • I • IF there are organisms that reproduce, and • IF offsprings inherit traits from their parents, and • IF there is variability of traits, and • IF the environment cannot support all members of a growing population, • THEN those members of the population with less-adaptive traits (determined by the environment) will die out, and • THEN those members with more-adaptive traits (determined by the environment) will thrive The result is the evolution of species.

6. Working of GA • GA encodes each point in a parameter space into a binary bit called chromosome • Each point is associated with a fitness function • Gene pool is a population of all such points • In each generation GA constructs a new population using genetic operators • Crossover • Mutation

7. Components of GA • Encoding schemes • Crossover operators • Mutation operators

8. Encoding schemes • Transforms points in parameter space into string representations • Eg (11,6,9) is represented as 101101101011 • Encoding schemes provide a way of translating problem-specific knowledge directly into GA framework • After this the fitness function is evaluated • Next selection is based on the fittest survivor

9. Fitness evaluation

10. How is it different from other optimization and search procedures? • Works with a coding of the parameter set, not the parameters themselves • Search for a population of point and not a single point • Use objective function information and not derivatives or other auxiliary knowledge

11. How GA is used and different from other optimization techniques? • The first step in GA is to code the parameter x as a finite length string • Example 1 can be code as string of 5 bits with an output f=f(s) , where s=string of bits • Successive populations are generated using the GA • For effective check GA requires only objective functions associated with individual strings

12. Simple genetic algorithm • Reproduction:- individual strings are copied according to their objective fn: values f(FITNESS FUNCTION) • Crossover:- Members of the newly reproduced strings are mated at random. Each pair of strings undergoes crossing overs. • Mutation:-supplements reproduction and crossover and acts as an insurance policy against premature loss of important notions

13. Basic Idea Of Principle Of Natural Selection “Select The Best, Discard The Rest”

14. Example 1Maximize f(x) =x2 on the integer scale from 0-31 1000 f(x) 0 31 x

15. Example 1

16. Roulette wheel with slots sized according to fitness

17. Crossover A1=0 1 1 0 1 A2=1 1 0 0 0 A1’=0 1 1 0 0 A2’=1 1 0 0 1

18. Simple GA by Hand(Reproduction)

19. Crossover Probability of mutation in this test is 0.001. With 20 transferred bit positions we should expect 20*0.001=0.02

20. Grist for the search mill • How does the directed search guide help improvement? • Seeking similarities among strings in population • Causal relationships between similarities and high fitness SCHEMATA

21. Evolution in the real world • Each cell of a living thing contains chromosomes - strings of DNA • Each chromosome contains a set of genes - blocks of DNA • Each gene determines some aspect of the organism (like eye colour) • A collection of genes is sometimes called a genotype • A collection of aspects (like eye colour) is sometimes called a phenotype • Reproduction involves recombination of genes from parents and then small amounts of mutation (errors) in copying • The fitness of an organism is how much it can reproduce before it dies • Evolution based on “survival of the fittest”

22. Basic Idea Of Principle Of Natural Selection “Select The Best, Discard The Rest”

23. Algorithm Generate Initial Population do Calculate the Fitnessof each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutatecurrent population till result is obtained }

24. Population population Chromosomes could be: • Bit strings (0101 ... 1100) • Real numbers (43.2 -33.1 ... 0.0 89.2) • Permutations of element (E11 E3 E7 ... E1 E15) • Lists of rules (R1 R2 R3 ... R22 R23) • ... any data structure ...

25. Algorithm Generate Initial Population do Calculate the Fitnessof each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutatecurrent population till result is obtained }

26. Algorithm Generate Initial Population do Calculate the Fitnessof each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutatecurrent population till result is obtained }

27. Fitness Function A fitness function quantifies the optimality of a solution so that that particular solution may be ranked against all the other solutions. • A fitness value is assigned to each solution depending on how close it actually is to solving the problem. • Ideal fitness function correlates closely to goal + quickly computable.

28. Algorithm Generate Initial Population do Calculate the Fitnessof each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutatecurrent population till result is obtained }

29. Algorithm Generate Initial Population do Calculate the Fitnessof each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutatecurrent population till result is obtained }

30. Crossover Mimics biological recombination Some portion of genetic material is swapped between chromosomes Typically the swapping produces an offspring

31. CROSSOVER (1 2 9 3 0 7 ) (1 2 9 7 9 5) (4 6 1 7 9 5 ) (1 2 9 3 0 7 ) ( 4 6 1 7 9 5) (4 6 1 7 9 5 )

32. Algorithm Generate Initial Population do Calculate the Fitnessof each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutatecurrent population till result is obtained }

33. Algorithm Generate Initial Population do Calculate the Fitnessof each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutatecurrent population till result is obtained }

34. Mutation Selects a random locus – gene location – with some probability and alters the allele at that locus The intuitive mechanism for the preservation of variety in the population

35. Mutation: Local Modification Before: (1 0 1 1 0 1 1 0) After: (0 1 1 0 0 1 1 0) Before: (1.38 -69.4 326.44 0.1) After: (1.38 -67.5 326.44 0.1)

36. The Problem The Traveling Salesman Problem is defined as: Given: 1) A set of cities 2) Symmetric distance matrix that indicates the cost of travel from each city to every other city. Goal: 1) Find the shortest circular tour, visiting every city exactly once. 2) Minimize the total travel cost, which includes the cost of traveling from the last city back to the first city’.

37. Traveling Salesperson Problem

38. Encoding • Represent every city with an integer . • Consider 6 Indian cities – Mumbai, Nagpur , Calcutta, Delhi, Bangalore and Pune assign a number to each. Mumbai 1 Nagpur 2 Calcutta 3 Delhi 4 Bangalore 5 Pune 6

39. Encoding • Thus a path would be represented as a sequence of integers from 1 to 6. • The path [1 2 3 4 5 6] represents a path from Mumbai to Nagpur - Nagpur to Calcutta - Calcutta to Delhi - Delhi to Bangalore - Bangalore to Pune and pune to Mumbai.

40. Fitness Function • The fitness function will be the total cost of the tour represented by each chromosome. • This can be calculated as the sum of the distances traversed in each travel segment. The Lesser The Sum, The Fitter The Solution Represented By That Chromosome.

41. Distance/Cost Matrix For TSP

42. Fitness Function (contd.) • So, for a chromosome [4 1 3 2 5 6 ], the total cost of travel or fitness will be calculated as shown below • Fitness = 1407 + 1987 + 1124 + 1049 + 841 = 6408 kms. • Since our objective is to Minimize the distance, the lesser the total distance, the fitter the solution.

43. Initial Population for TSP (5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2) (2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

44. Select Parents (5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2) (2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4) Try to pick the better ones.

45. Create Off-Spring – 1 point (5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2) (2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4) (3,4,5,6,2)

46. Create More Offspring (5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2) (2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4) (3,4,5,6,2) (5,4,2,6,3)

47. Mutate (5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2) (2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4) (3,4,5,6,2) (5,4,2,6,3)

48. Mutate (5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2) (2,3,4,6,5) (2,3,6,4,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4) (3,4,5,6,2) (5,4,2,6,3)

49. Eliminate (5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2) (2,3,4,6,5) (2,3,6,4,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4) (3,4,5,6,2) (5,4,2,6,3) Tend to kill off the worst ones.

50. Integrate (5,4,2,6,3) (5,3,4,6,2) (2,4,6,3,5) (3,4,5,6,2) (2,3,6,4,5) (3,4,5,2,6) (3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6) (4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)