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contribution to particle velocity is proportional to the distance between a particle and the best position found by the neighborhood of particles The velocity is calculated as vij (t + 1) = vij (t) + c1 r1j (t)[yij (t) xij (t)] + c2 r2j (t)[ ij (t) xij (t)] y (166)

where yij is the best position, found by the neighborhood of particle i in dimension j The local best particle position, yi , ie the best position found in the neighborhood Ni , is de ned as yi (t + 1) {Ni |f ( i (t + 1)) = min{f (x)}, y with the neighborhood de ned as Ni = {yi nNi (t), yi nNi +1 (t), , yi 1 (t), yi (t), yi+1 (t), , yi+nNi (t)} (168) x Ni } (167)

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Barlowe, Cindy 276 4 8 4 725 15 5 8 2 4 207 207 207 276 207 207 207 276 207 207 207 207 233 440 80 160 234 125 167 127 425 Total 125 3334 25 8 2 180 40 80 394 305 207 207 550 187 98 00 00 343 207 207 207 207 207 360 8 8 205 412 700 288 24 100 307 100 207 2 7 4 8 92 299 160 139 235 511 200 311 208 415 40 375 8 12 16 8 369 576 160 416 69 276 70 206 94 370 450 80 52 259 145 114 00 207 110 97 386 593 190 403 00 276 120 156 8

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for neighborhoods of size nNi The local best position will also be referred to as the neighborhood best position It is important to note that for the basic PSO, particles within a neighborhood have no relationship to each other Selection of neighborhoods is done based on particle indices However, strategies have been developed where neighborhoods are formed based on spatial similarity (refer to Section 162) There are mainly two reasons why neighborhoods based on particle indices are preferred: 1 It is computationally inexpensive, since spatial ordering of particles is not required For approaches where the distance between particles is used to form neighborhoods, it is necessary to calculate the Euclidean distance between all pairs of particles, which is of O(n2 ) complexity s 2 It helps to promote the spread of information regarding good solutions to all particles, irrespective of their current location in the search space It should also be noted that neighborhoods overlap A particle takes part as a member of a number of neighborhoods This interconnection of neighborhoods also facilitates the sharing of information among neighborhoods, and ensures that the swarm converges on a single point, namely the global best particle The gbest PSO is a special case of the lbest PSO with nNi = ns Algorithm 162 summarizes the lbest PSO

The two versions of PSO discussed above are similar in the sense that the social component of the velocity updates causes both to move towards the global best particle This is possible for the lbest PSO due to the overlapping neighborhoods There are two main di erences between the two approaches with respect to their convergence characteristics [229, 489]:

5/14/2001

161 Basic Particle Swarm Optimization Algorithm 162 lbest PSO Create and initialize an nx -dimensional swarm; repeat for each particle i = 1, , ns do //set the personal best position if f (xi ) < f (yi ) then yi = xi ; end //set the neighborhood best position y if f (yi ) < f ( i ) then y = yi ; end end for each particle i = 1, , ns do update the velocity using equation (166); update the position using equation (161); end until stopping condition is true;

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