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Non-dominated Sorting Genetic Algorithm (NSGA-II) Karthik Sindhya, PhD Postdoctoral Researcher Industrial Optimization Group Department of Mathematical Information Technology Karthik.sindhya@jyu.fi http://users.jyu.fi/~kasindhy/

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Objectives The objectives of this lecture is to: Understand the basic concept and working of NSGA-II Advantages and disadvantages

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Non-dominated sorting genetic algorithm –II was proposed by Deb et al. in 2000. NSGA-II procedure has three features: – It uses an elitist principle – It emphasizes non-dominated solutions. – It uses an explicit diversity preserving mechanism NSGA-II

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NSGA-II ƒ1ƒ1 ƒ2ƒ2 Crossover & Mutation NSGA-II

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Crowded tournament selection operator – A solution x i wins a tournament with another solution x j if any of the following conditions are true: If solution x i has a better rank, that is, r i < r j. If they have the same rank but solution x i has a better crowding distance than solution x j, that is, r i = r j and d i > d j. NSGA-II Objective space

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Crowding distance – To get an estimate of the density of solutions surrounding a particular solution. Crowding distance assignment procedure – Step 1: Set l = |F|, F is a set of solutions in a front. Set d i = 0, i = 1,2,…,l. – Step 2: For every objective function m = 1,2,…,M, sort the set in worse order of f m or find sorted indices vector: I m = sort(f m ). NSGA-II

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Step 3: For m = 1,2,…,M, assign a large distance to boundary solutions, i.e. set them to ∞ and for all other solutions j = 2 to (l-1), assign as follows: i i+1 i-1 NSGA-II

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Advantages: – Explicit diversity preservation mechanism – Overall complexity of NSGA-II is at most O(MN 2 ) – Elitism does not allow an already found Pareto optimal solution to be deleted. Disadvantage: – Crowded comparison can restrict the convergence. – Non-dominated sorting on 2N size. NSGA-II

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