1. METAHEURISTICS Genetic Algorithm Jacques A. Ferland Department of Informatique and Recherche Opérationnelle Université de Montréal ferland@iro.umontreal.ca

2. Genetic Algorithm (GA) Population based algorithm At each iteration (generation) three different operators are first applied to generate a set of new (offspring) solutions using the N solutions of the current population: selection operator: selecting from the current population parent-solutions that reproduce themselves crossover (reproduction) operator: producing offspring-solutions from each pair of parent-solutions mutation operator: modifying (improving) individual offspring-solution A fourth operator (culling operator) is applied to determine a new population of size N by selecting among the solutions of the current population and the offspring-solutions according to some strategy

3. Two variants of GA At each iteration of the Classical genetic algorithm: - N parent solutions are selected and paired two by two - A crossover operator is applied to each pair of parent-solutions according to some probability to generate two offspring-solutions. Otherwise the two parent-solutions become their own offspring-solutions - A mutation operator is applied according to some probability to each offspring-solution. - The population of the next iteration includes the offspring-solutions At each iteration of the Steady-state population genetic algorithm: - An even number of parent-solutions are selected and paired two by two - A crossover operator is applied to each pair of parent-solutions to generate two offspring-solutions. - A mutation operator is applied according to some probability to each offspring-solution. -The population of the next iteration includes the N best solutions among the current population and the offspring- solutions

4. Encoding the solution The phenotype form of the solution x є ℝ n is encoded (represented) as a genotype form vector z є ℝ m (or chromozome) where m may be different from n. For example in the assignment type problem: let x be the following solution: for each 1≤ i ≤ n, x ij(i) = 1 x ij = 0 for all other j x є ℝ nxm can be encoded as z= [j(1), j(2), …, j(i), …, j(n)] є ℝ n where z i = j(i) i = 1, 2, …, n i.e., z i is the index of the resource j(i) assigned to activity i

5. Selection operator This operator is used to select an even number (2, or 4, or …, or N) of parent-solutions. Each parent-solution is selected from the current population according to some strategy or selection operator. Note that the same solution can be selected more than once. The parent-solutions are paired two by two to reproduce themselves. Selection operators: Random selection operator Proportional (or roulette whell) selection operator Tournament selection operator

6. Random selection operator Select randomly each parent-solution from the current (entire) population Properties: Very straightforward Promotes diversity of the population generated

7. Proportional (Roulette whell) selection operator Each parent-solution is selected as follows: i) Consider any ordering of the solutions z 1, z 2, …, z N in P ii) Select a random number α in the interval [0, ∑ 1≤k≤ N ( 1 / f( z k ) )] iii) Let τ be the smallest index such that ∑ 1≤k≤ τ (1 / f( z k ) ) ≥ α iv) Then z τ is selected 1 / f( z 1 ) 1 / f( z 2 ) 1 / f( z 3 ) 1 / f( z N ) | | | | … | | τ α The chance of selecting z k increases with its fittness 1 / f( z k ) For the problem Min f (x) where x is encoded as z 1/f (z) measures the fittness of the solution z

8. Tournament selection operator Each parent-solution is selected as the best solution in a subset of randomly chosen solutions in P: i) Select randomly N’ solutions one by one from P (i.e., the same solution can be selected more than once) to generate the subset P’ ii) Let z’ be the best solution in the subset P’: z’ = argmin z є P’ f(z) iii) Then z’ is selected as a parent-solution

9. Elitist selection The main drawback of using elitist selection operator like Roulette whell and Tournament selection operators is premature converge of the algorithm to a population of almost identical solutions far from being optimal. Other selection operators have been proposed where the degree of elitism is in some sense proportional to the diversity of the population.

10. Crossover (recombination) operators Crossover operator is used to generate new solutions including interesting components contained in different solutions of the current population. The objective is to guide the search toward promissing regions of the feasible domain X while maintaining some level of diversity in the population. Pairs of parent-solutions are combined to generate offspring- solutions according to different crossover (recombination) operators.

11. One point crossover The one point crossover generates two offspring-solutions from the two parent-solutions z 1 = [ z 1 1, z 2 1, …, z m 1 ] z 2 = [ z 1 2, z 2 2, …, z m 2 ] as follows: i) Select randomly a position (index) ρ, 0 ≤ ρ ≤ m. ii) Then the offspring-solutions are specified as follows: oz 1 = [ z 1 1, z 2 1, …, z ρ 1, z ρ+1 2, …, z m 2 ] oz 2 = [ z 1 2, z 2 2, …, z ρ 2, z ρ+1 1, …, z m 1 ] Hence the first ρ components of offspring oz 1 (offspring oz 2 ) are the corresponding ones of parent 1 (parent 2), and the rest of the components are the corresponding ones of parent 2 (parent 1)

12. Two points crossover The two points crossover generates two offspring-solutions from the two parent-solutions z 1 = [ z 1 1, z 2 1, …, z m 1 ] z 2 = [ z 1 2, z 2 2, …, z m 2 ] as follows: i) Select randomly two positions (indices) μ,ν, 1 ≤ μ ≤ ν ≤ m. ii) Then the offspring-soltions are specified as follows: oz 1 = [ z 1 1, …, z μ-1 1, z μ 2, …, z ν 2, z ν+1 1, …, z m 1 ] oz 2 = [ z 1 2, …, z μ-1 2, z μ 1, …, z ν 1, z ν+1 2, …, z m 2 ] Hence the offspring oz 1 (offspring oz 2 ) has components μ, μ+1, …, ν of parent 2 (parent 1), and the rest of the components are the corresponding ones of parent 1 (parent 2)

13. Uniform crossover The uniform crossover requires a vector of bits (0 or 1) of dimension m to generate two offspring-solutions from the two parent-solutions z 1 = [ z 1 1, z 2 1, …, z m 1 ], z 2 = [ z 1 2, z 2 2, …, z m 2 ] : i) Generate randomly a vector of bits, for example [0, 1, 1, 0, …, 1, 0] ii) Then the offspring-solutions are specified as follows: parent 1: [ z 1 1, z 2 1, z 3 1, z 4 1,…, z m-1 1, z m 1 ] parent 2: [ z 1 2, z 2 2, z 3 2, z 4 2,…, z m-1 2, z m 2 ] Vector of bits: [ 0, 1, 1, 0, …, 1, 0 ] Offspring oz 1 : [ z 1 1, z 2 2, z 3 2, z 4 1,…, z m-1 2, z m 1 ] Offspring oz 2 : [ z 1 2, z 2 1, z 3 1, z 4 2,…, z m-1 1, z m 2 ]

14. Uniform crossover The uniform crossover requires a vector of bits (0 or 1) of dimension m to generate two offspring-solutions from the two parent-solutions z 1 = [ z 1 1, z 2 1, …, z m 1 ], z 2 = [ z 1 2, z 2 2, …, z m 2 ] : i) Generate randomly a vector of bits, for example [0, 1, 1, 0, …, 1, 0] ii) Then the offspring-solutions are specified as follows: parent 1: [ z 1 1, z 2 1, z 3 1, z 4 1,…, z m-1 1, z m 1 ] parent 2: [ z 1 2, z 2 2, z 3 2, z 4 2,…, z m-1 2, z m 2 ] Vector of bits: [ 0, 1, 1, 0, …, 1, 0 ] Offspring oz 1 : [ z 1 1, z 2 2, z 3 2, z 4 1,…, z m-1 2, z m 1 ] Offspring oz 2 : [ z 1 2, z 2 1, z 3 1, z 4 2,…, z m-1 1, z m 2 ] Hence the i th component of oz 1 (oz 2 ) is the i th component of parent 1 (parent 2) if the i th component of the vector of bits is 0, otherwise, it is equal to the i th component of parent 2 (parent 1)

15. Ad hoc crossover operator The preceding crossover operators are sometimes too general to be efficient. Hence, whenever possible, we should rely on the structure of the problem to specify ad hoc problem dependent crossover operator in order to improve the efficiency of the algorithm.

16. Recovery procedure Furthermore, whenever the structure of the problem is such that the offspring-solutions are not necessarily feasible, then an auxiliary procedure is required to recover feasibility. Such a procedure is used to transform the offspring-solution into a feasible solution in its neighborhood.

17. Mutation operator Mutation operator is an individual process to modify offspring-solutions In traditional variants of Genetic Algorithm the mutation operator is used to modify arbitrarely each componenet z i with a small probability: For i = 1 to m Generate a random number β є [0, 1] If β < βmax then select randomly a new value for z i where βmax is small enough in order to modify z i with a small probability Mutation operator simulates random events perturbating the natural evolution process Mutation operator not essential, but the randomness that it introduces in the process, promotes diversity in the current population and may prevent premature convergence to a bad local minimum

18. Hybrid Methods Hybrid methods specified by combining two or more heuristic methods to improve their efficiency For instance, using a Neighborhood Search Technique as the mutation operator of a Genetic Algorithm to improve the offspring-solutions. This is a good strategy since it is well known that in general, Genetic Algorithms (and population based algorithms in general) are very time consuming and generate worse solution than NST Strength of hybrid methods comes from combining complementary search strategy to take advantage of their respective strength. For instance, - Intensify the search in a promissing region with the NST - Diversify the search through the selection operator, crossover operator of the GA