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moving back to feasible best positions facilitates better exploration If the feasible space consists of disjointed feasible regions, the chances are increased for particles to explore di erent feasible regions El-Gallad et al [234] replaced infeasible particles with their feasible personal best positions This approach assumes feasible initial particles and that personal best positions are replaced only with feasible solutions The approach is very similar to that of Hu and Eberhart [388], but with less diversity Replacement of an infeasible particle with its feasible personal best, forces an immediate repair The particle is immediately brought back into feasible space The approach of Hu and Eberhart allow an infeasible particle to explore more by pulling it back into feasible space over time But, keep in mind that during this exploration, the infeasible particle will have no in uence on the rest of the swarm while it moves within infeasible space Venter and Sobieszczanski-Sobieski [874, 875] proposed repair of infeasible solutions by setting vi (t) = 0 vi (t + 1) = c1 r1 (t)(yi (t) xi (t)) + c2 r2 (t)( (t) xi (t)) y (1691) (1692)
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for all infeasible particles, i In other words, the memory of previous velocity (direction of movement) is deleted for infeasible particles, and the new velocity depends only on the cognitive and social components Removal of the momentum has the e ect that infeasible particles are pulled back towards feasible space (assuming that the personal best positions are only updated if no constraints are violated)
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A great number of PSO variations can be found for solving MOPs This section describes only a few of these but provides references to other approaches The dynamic neighborhood MOPSO, developed by Hu and Eberhart [387], dynamically determines a new neighborhood for each particle in each iteration, based on distance in objective space Neighborhoods are determined on the basis of the simplest objective Let f1 (x) be the simplest objective function, and let f2 (x) be the second objective The neighbors of a particle are determined as those particles closest to the particle with respect to the tness values for objective f1 (x) The neighborhood best particle is selected as the particle in the neighborhood with the best tness according to the second objective, f1 (x) Personal best positions are replaced only if a particle s new position dominates its current personal best solution The dynamic neighborhood MOPSO has a few disadvantages: It is not easily scalable to more than two objectives, and its usability is therefore restricted to MOPs with two objectives It assumes prior knowledge about the objectives to decide which is the most simple for determination of neighborhoods
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16 Particle Swarm Optimization
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It is sensitive to the ordering of objectives, since optimization is biased towards improving the second objective Parsopoulos and Vrahatis [659, 660, 664, 665] developed the vector evaluated PSO (VEPSO), on the basis of the vector-evaluated genetic algorithm (VEGA) developed by Scha er [761] (also refer to Section 963) VEPSO uses two sub-swarms, where each sub-swarm optimizes a single objective This algorithm is therefore applicable to MOPs with only two objectives VEPSO follows a kind of coevolutionary approach The global best particle of the rst swarm is used in the velocity equation of the second swarm, while the second swarm s global best particle is used in the velocity update of the rst swarm That is, S1 vij (t + 1) S2 vij (t + 1) = + = + wS1 vij (t) + c1 r1j (t)(S1 yij (t) S1 xij (t)) c2 r2j (t)(S2 i (t) S1 xij (t)) y wS2 vij (t) + c1 r1j (t)(S2 yij (t) S2 xij (t)) c2 rij (t)(S1 j (t) Sx2j (t)) y
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(1693) (1694)
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where sub-swarm S1 evaluates individuals on the basis of objective f1 (x), and subswarm S2 uses objective f2 (x) The MOPSO algorithm developed by Coello Coello and Lechuga is one of the rst PSObased MOO algorithms that extensively uses an archive [147, 148] This algorithm is based on the Pareto archive ES (refer to Section 1262), where the objective function space is separated into a number of hypercubes A truncated archive is used to store non-dominated solutions During each iteration, if the archive is not yet full, a new particle position is added to the archive if the particle represents a non-dominated solution However, because of the size limit of the archive, priority is given to new non-dominated solutions located in less populated areas, thereby ensuring that diversity is maintained In the case that members of the archive have to be deleted, those members in densely populated areas have the highest probability of deletion Deletion of particles is done during the process of separating the objective function space into hypercubes Densely populated hypercubes are truncated if the archive exceeds its size limit After each iteration, the number of members of the archive can be reduced further by eliminating from the archive all those solutions that are now dominated by another archive member For each particle, a global guide is selected to guide the particle toward less dense areas of the Pareto front To select a guide, a hypercube is rst selected Each hypercube is assigned a selective tness value, fsel (Hh ) = fdel (Hh ) (1695)
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where fdel (Hh ) = Hh ns is the deletion tness value of hypercube Hh ; = 10 and Hh ns represents the number of nondominated solutions in hypercube Hh More densely populated hypercubes will have a lower score Roulette wheel selection is then used to select a hypercube, Hh , based on the selection tness values The global guide for particle i is selected randomly from among the members of hypercube Hh
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