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GA Approaches to Multi-Objective Optimization Scott Noble Fred Iskander 18 March 2003

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Multi-Objective Optimization Problems (MOPs) Multiple, often competing objectives In the case of a commensurable variable space, can often be reduced to a single objective function (or sequence thereof) and solved using standard methods Some problems cannot be reduced and must be solved using pure MO techniques

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Three General Approaches Preemptive Optimization sequential optimization of individual objectives (in order of priority) Composite Objective Function weighted sum of objectives Purely Multi-Objective Population-Based Pareto-Based

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Preemptive Optimization Steps 1.Prioritize objectives according to predefined criteria (problem-specific) 2.Optimize highest-priority objective function 3.Introduce new constraint based on optimum value just obtained 4.Repeat steps 2 & 3 for every other objective function, in succession

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Composite Objective Functions 1.Assign weights to each function according to predefined criteria (problem-specific) MAX and MIN objectives receive opposite signs 2.Sum weighted functions to create new composite function 3.Solve as a regular, single-objective optimization problem

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Transformation Approaches Advantages: Easy to understand and formulate Simple to solve (using standard techniques) Disadvantages: A priori prioritization/weighting can end up being arbitrary (due to insufficient understanding of problem): oversimplification Not suited to certain types of MOPs

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Pure MOPs: Population-Based Solutions Allow for the investigation of tradeoffs between competing objectives GAs are well suited to solving MOPs in their pure, native form Such techniques are very often based on the concept of Pareto optimality

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Pareto Optimality MOP tradeoffs between competing objectives Pareto approach exploring the tradeoff surface, yielding a set of possible solutionstradeoff surface Also known as Edgeworth-Pareto optimality

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Pareto Optimum: Definition A candidate is Pareto optimal iff: It is at least as good as all other candidates for all objectives, and It is better than all other candidates for at least one objective. We would say that this candidate dominates all other candidates.

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Dominance: Definition Given the vector of objective functions we say that candidate dominates, (i.e. ) if: (assuming we are trying to minimize the objective functions). (Coello Coello 2002)

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Pareto Non-Dominance With a Pareto set, we speak in terms of non- dominance. There can be one dominant candidate at most. No accommodation for “ties.” We can have one or more candidates if we define the set in terms of non-dominance.

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Pareto Optimal Set The Pareto optimal set P contains all candidates that are non-dominated. That is: where F is the set of feasible candidate solutions (Coello Coello 2002)

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Examples (Fonseca and Fleming 1993)

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Examples Candidatef1f1 f2f2 f3f3 f4f4 1 (dominated by: 2,4,5)56310 2 (dominated by: 5)46310 3 (non-dominated)55211 4 (non-dominated)56210 5 (non-dominated)4539

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Example: Pareto Ranking (Fonseca and Fleming 1993) (1) (3) (2) (6)

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Pareto Front The Pareto Front is simply values of the optimality vector evaluated at all candidates in the Pareto Optimal Set

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Pareto Front (Tamaki et al. 1996)

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Non-Pareto Selection VEGA (Parallel Selection) Vector Evaluated Genetic Algorithm Next-generation sub-populations formed from separate objective functions Tournament Selection Pair wise comparison of individuals w.r.t. objective functions (pre-prioritized or random) Random Objective Selection Repetitive selection using a randomly selected objective function (predetermined probabilities)

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Pareto-Based Selection Pareto Ranking Tournament Selection with Dominance pair wise comparison against a comparison set based on dominance Pareto Reservation (Elitism) carry non-dominated candidates forward from previous generation use additional selection method to regulate population size Pareto-Optimal Selection

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Diversity Lack of genetic diversity is an inherent issue with GAs Fitness sharing encourages diversity by penalizing candidates from the same area of the solution or function space

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Summary There are multiple approaches to MOPs. GAs are well suited to exploring a multi- objective solution space. They provide insight into the tradeoffs associated with MOPs, not necessarily a particular solution.

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Further Reading Coello Coello, C.A. 2002. “Introduction to Evolutionary Multiobjective Optimization.” www.cs.cinvestuv.mx/~EVOCINV/download/class1-emoo- eng.pdfwww.cs.cinvestuv.mx/~EVOCINV/download/class1-emoo- eng.pdf Fonseca, C.M. and P.J. Fleming. 1993. Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization. Genetic Algorithms: Proceedings of the Fifth International Conference. S. Forrest, ed. San Mateo, CA, July 1993. Tamaki, H., H. Kita and S. Kobayashi. 1996. Multi-Objective Optimization by Genetic Algorithms: A Review. Proceedings of the IEEE Conference on Evolutionary Computation, ICEC 1996, pp 517-522. Younes, A., H. Ghenniwa and S. Areibi. 2002. An Adaptive Genetic Algorithm for Multi-Objective Flexible Manufacturing Systems. GECCO, New York, July 2002.

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