A Novel Flexible Inertia Weight Particle Swarm Optimization Algorithm.

Particle swarm optimization (PSO) is an evolutionary computing method based on intelligent collective behavior of some animals. It is easy to implement and there are few parameters to adjust. The performance of PSO algorithm depends greatly on the appropriate parameter selection strategies for fine tuning its parameters. Inertia weight (IW) is one of PSO's parameters used to bring about a balance between the exploration and exploitation characteristics of PSO. This paper proposes a new nonlinear strategy for selecting inertia weight which is named Flexible Exponential Inertia Weight (FEIW) strategy because according to each problem we can construct an increasing or decreasing inertia weight strategy with suitable parameters selection. The efficacy and efficiency of PSO algorithm with FEIW strategy (FEPSO) is validated on a suite of benchmark problems with different dimensions. Also FEIW is compared with best time-varying, adaptive, constant and random inertia weights. Experimental results and statistical analysis prove that FEIW improves the search performance in terms of solution quality as well as convergence rate

Wilcoxon-ranks and p-value on the average and minimum number of iterations of successful runs according to Table 5.

<p>Wilcoxon-ranks and p-value on the average and minimum number of iterations of successful runs according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t005" target="_blank">Table 5</a>.</p

Wilcoxon-ranks and p-value on the average and minimum error according to Tables 7 and 8.

<p>Wilcoxon-ranks and p-value on the average and minimum error according to Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t007" target="_blank">7</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t008" target="_blank">8</a>.</p

Convergence graph for some PSO variants.

<p>(A) Sphere Function with <i>ε</i> = 10⁻²⁰. (B) Griewank Function with <i>ε</i> = 10⁻¹. (C) Ackley Function with <i>ε</i> = 10⁻¹⁵. (D) Zakharov Function with <i>ε</i> = 10⁻²⁰⁰. (E) Schwefel's Problem 2.22 with <i>ε</i> = 10⁻²⁰. (F) Weierstrass Function with <i>ε</i> = 10⁻³⁰.</p

Friedman test based on Tables 7 and 8.

<p>Friedman test based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t007" target="_blank">7</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t008" target="_blank">8</a>.</p

Wilcoxon-ranks and p-value on the average and minimum error according to Tables 13 and 14.

<p>Wilcoxon-ranks and p-value on the average and minimum error according to Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t013" target="_blank">13</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t014" target="_blank">14</a>.</p

Friedman test based on Tables 13 and 14.

<p>Friedman test based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t013" target="_blank">13</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t014" target="_blank">14</a>.</p

Bonferroni-Dunn bar chart.

<p>(A) Average error based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t011" target="_blank">11</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t012" target="_blank">12</a>. (B) Average error based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t013" target="_blank">13</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t014" target="_blank">14</a>. (C) Minimum error based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t011" target="_blank">11</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t012" target="_blank">12</a>. (D) Minimum error based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t013" target="_blank">13</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t014" target="_blank">14</a>. (E) Standard deviation of error based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t011" target="_blank">11</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t012" target="_blank">12</a>. (F) Standard deviation of error based on Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t013" target="_blank">13</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t014" target="_blank">14</a>.</p

Summary of results of Tables 15–22.

<p>Summary of results of Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t015" target="_blank">15</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t002" target="_blank">2</a><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t002" target="_blank">2</a>.</p

Best and worst IW strategies for each benchmark function in terms of average, minimum and standard deviation of error according to Tables 13 and 14.

<p>Best and worst IW strategies for each benchmark function in terms of average, minimum and standard deviation of error according to Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t013" target="_blank">13</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161558#pone.0161558.t014" target="_blank">14</a>.</p