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By Tetsuya Higuchi, Xin Yao

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Extra info for Evolutionary Algorithms for Solving Multi-Objective Problems

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Fk (x)). The phrase Pareto Optimal is taken to mean with respect to the entire decision variable space unless otherwise specified. 9 1 cost Fig. 3. An example of a problem with two objective functions: cost and efficiency. The Pareto front or trade-off surface is delineated by a curved line. In words, this definition says that x∗ is Pareto optimal if there exists no feasible vector x which would decrease some criterion without causing a simultaneous increase in at least one other criterion (assuming minimization).

5 EA Basics 27 Crossover Point Parent 1: Parent 2: Offspring 1: Offspring 2: Fig. 11. Single-Point Crossover String S1 S2 S3 S4 mean Fitness 12 12 12 12 12 String S1 S2 S4 S3 S1 S2 S3 S4 mean Fitness 20 10 5 5 10 S1 S4 S2 S3 Equal Fitness Unequal Fitness Fig. 12. Roulette Wheel Selection ranking, tournament, and (µ, λ) selection [72, pg. 180]. Finally, an EA’s decision function determines when execution stops. 2 highlights the major differences between the three major EC instantiations. It is beyond the scope of this book to provide an in-depth analysis of general EVOPs and EA components.

Note that countable includes finite. 1: Relationships between the P ∗ and PF ∗ set size. |PF ∗ | Mappings of sets having size 1, n ¨ 5 , and u ¨6 |P ∗ | 1. Countable → Countable {(1 → 1), (¨ n → 1), (¨ n→n ¨ )} 2. Uncountable → Countable {(¨ u → 1), (¨ u→n ¨ ), (¨ u→n ¨ )} 3. Uncountable → Uncountable {(¨ u→u ¨)} P ∗ and PF ∗ of course represent the theoretical goal sets for a MOEA search algorithm. However, as indicated before, they may not be computationally achievable in any circumstance. 1 reflect an MOP that cannot be optimally solved by a digital computer.

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