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mq (t)pm (xi ) be used: if pm (xi ) < 0001 if pm (xi ) 01 if pm (xi ) 10 otherwise
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As an example, the following penalties can 10 t t 20 mq (t) = 100 t 300 t
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The multi-level function approach has the weakness that the number of parameters that has to be maintained increases signi cantly with increase in the number of levels, nq , and the number of constraints, ng + nh Joines and Houck [425] proposed dynamic penalties, where
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where , and are constants The longer the search continues, the higher the penalty for constraint violations This allows for better exploration Other penalty methods can be found in [587, 588, 691] Often referred to as the death penalty method, unfeasible solutions can be rejected However, Michalewicz [585] shows that the method performs badly when the feasible region is small compared to the entire search space The interested reader is referred to [584] for a more complete survey of constraint handling methods
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Extensive research has been done to solve multi-objective optimization problems (MOP) as de ned in De nition A10 [149, 195] This section summarizes only a few of these GA approaches to multi-objective optimization (MOO) GA approaches for solving MOPs can be grouped into three main categories [421]: Weighted aggregation approaches where the objective is de ned as a weighted sum of sub-objectives Population-based non-Pareto approaches, which do not make use of the dominance relation as de ned in Section A8 Pareto-based approaches, which apply the dominance relation to nd an approximation of the Pareto front Examples from the rst and last classes are considered below
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One of the simplest approaches to deal with MOPs is to de ne an aggregate objective function as a weighted sum of sub-objectives:
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where nk 2 is the total number of sub-objectives, and k [0, 1], k = 1, , nk with nk k=1 k = 1 While the aggregation approach above is very simple to implement and computationally e cient, it su ers from the following problems: It is di cult to get the best values for the weights, k , since these are problem dependent These methods have to be re-applied to nd more than one solution, since only one solution can be obtained with a single run of an aggregation algorithm However, even for repeated applications, there is no guarantee that di erent solutions will be found The conventional weighted aggregation as given above cannot solve MOPs with a concave Pareto front [174] To address these problems, Jin et al [421, 422], proposed aggregation methods with dynamically changing weights (for nk = 2) and an approach to maintain an archive of nondominated solutions The following approaches have been used to dynamically adapt weights: Random distribution of weights, where for each individual, 1,i (t) = U (0, ns )/ns 2,i (t) = 1 1,i (t) (940) (941)
96 Advanced Topics Bang-bang weighted aggregation, where 1 (t) = 2 (t) = sign(sin(2 t/ )) 1 1 (t)
(942) (943)
where is the weights change frequency Weights change abruptly from 0 to 1 each generation Dynamic weighted aggregation, where 1 (t) = 2 (t) = | sin(2 t/ )| 1 1 (t) (944) (945)
With this approach, weights change more gradually Jin et al [421, 422] used Algorithm 910 to produce an archive of nondominated solutions This algorithm is called after the reproduction (crossover and mutation) step Algorithm 910 Algorithm to Maintain an Archive of Nondominated Solutions for each offspring, xi (t) do if xi (t) dominates an individual in the current population, C(t), and xi (t) is not dominated by any solutions in the archive and xi (t) is not similar to any solutions in the archive then if archive is not full then Add xi (t) to the archive; else if xi (t) dominates any solution xa in the archive then Replace xa with xi (t); else if any xa1 in the archive dominates another xa2 in the archive then Replace xa2 with xi (t); else Discard xi (t); end end else Discard xi (t); end for each solution xa1 in the archive do if xa1 dominates xa2 in the archive then Remove xa2 from the archive; end end end