Engineering Optimization: Theory and Practice, Fourth Edition

13.3 Simulated Annealing 703 13.3.2 Procedure The simulated annealing method simulates the process of slow cooling of molten met ...

704 Modern Methods of Optimization is not same in all situations. As can be seen from Eq. (13.18), this probability depends on t ...

13.3 Simulated Annealing 705 computational effort. A smaller value ofn, on the other hand, might result either in a premature co ...

706 Modern Methods of Optimization Start with initial vector, X 1 , initial temperature and other parameters (T,n,c) Findf 1 = f ...

13.3 Simulated Annealing 707 Step 1: Choose the parameters of the SA method. The initial temperature is taken as the average val ...

708 Modern Methods of Optimization Step 3: Generate a new design point in the vicinity of the current design point{ X 2 = 1 2. 7 ...

13.4 Particle Swarm Optimization 709 As an example, consider the behavior of birds in a flock. Although each bird has a limited ...

710 Modern Methods of Optimization (similar to chromosomes in genetic algorithms). Evaluate the objective function values corres ...

13.4 Particle Swarm Optimization 711 Vj(i)=θVj(i− 1 )+c 1 r 1 [Pbest ,j−Xj(i− 1 )] +c 2 r 2 [Gbest−Xj(i− 1 )];j= 1 , 2 ,... , N ...

712 Modern Methods of Optimization C(i)=(ci)α (13.27) H(X)= ∑m j= 1 { φ[gj( [X)]qj(X)]γ[qi(X)] } (13.28) φ[qj(X)]=a ( 1 − 1 eqj( ...

13.4 Particle Swarm Optimization 713 so that v 1 ( 1 )= 0 + 0. 3294 (− 1. 5 + 1. 5 )+ 0. 9542 ( 1. 25 + 1. 5 )= 2. 6241 v 2 ( 1 ...

714 Modern Methods of Optimization 6.Find the objective function values at the currentxj(i): f[x 1 ( 2 )]= 4. 4480 , f[x 2 ( 2 ) ...

13.5 Ant Colony Optimization 715 layers is equal to the number of design variables and the number of nodes in a par- ticular lay ...

716 Modern Methods of Optimization is updated as follows: τij←τij+ τ(k) (13.33) Because of the increase in the pheromone, the p ...

13.5 Ant Colony Optimization 717 13.5.5 Algorithm The step-by-step procedure of ACO algorithm for solving a minimization problem ...

718 Modern Methods of Optimization Step 4: Test for the convergence of the process. The process is assumed to have con- verged i ...

13.5 Ant Colony Optimization 719 x 11 = 0·0 x 12 = 0·5 x 13 = 1·0 x 14 = 1·5 x 15 = 2·0 x 16 = 2·5 x 17 = 3·0 Home Food (Destina ...

720 Modern Methods of Optimization Step 4: Assuming that the ants return home and start again in search of food, we set the iter ...

13.5 Ant Colony Optimization 721 value assumed (or the path selected in Fig. 13.4) by different ants can be seen to be ant 1 :x ...

722 Modern Methods of Optimization Step 2: For any antk, the probability of selecting pathx 1 jin Fig. 13.4 is given by p 1 j= τ ...