4. Results

The proposed BAM-PSO algorithm is compared with five different biologically inspired algorithms: PSO with inertial vector and boundaries [27], ant colony system (ACS) [6], differential evolution (DE) [31], simplified swarm optimization (SSO) [20] and particle swarm optimization with aging leader and challengers (ALC-PSO) [24]. These algorithms are selected because of several factors: first, PSO is the base algorithm for BAM-PSO, so it is natural to compare performance with the original optimizer, SSO and ALC-PSO are other well-known variants of PSO that in some way, claim to alleviate the premature convergence problem and, specifically, ALC-PSO is related in many ways to BAM-PSO. Finally, while ACS and DE are not related closely to BAM-PSO, they are swarm-based and evolution-based optimization algorithms respectively and thus, were considered as good candidates for performance comparison.

To test optimization performance of these algorithms, well-established benchmark functions are selected in low and high dimensionality [33]. These selected functions help evaluate algorithm's performance over a broad type of problems, because they possess multiple local minima, complex non-linear structure, or have bowl-shaped/plate-shaped structure [36, 37]; even some of them have a steep ridge and drops structure. From the literature, a list of 18 functions was considered relevant enough to test BAM-PSO performance. The selected benchmark functions are shown in Table 1.

where <sup>f</sup> <sup>∗</sup>ð Þ represents the objective function value for the best candidate solution, <sup>θ</sup> the remaining leadership's term, Θ represents the maximum leadership term, δ<sup>g</sup> Best defines the entire swarm improvement factor, δ<sup>l</sup> Best represents the individual particle improvement factor, leader represents the particle within the swarm that is the acting leader (not necessarily pgjð Þt ) and whose all particles will follow according to Eqs. (5) and (6); finally, δLeader represents

Eqs. (7), (8) and (9) indicate the leading performance of the leader. The lifespan controller utilizes these performance evaluations to adjust the leading term of leader according to the

When the leading term of leader reaches θ ¼ 0 the leader is considered exhausted and replaced

Step 4: Adjust lifespan of all particles within the swarm according to Eqs. (2)–(4) and replace

Step 5: Terminal condition check. If the number of iterations is larger than the predefined or the error has reached a minimum expected value, the algorithm terminates. Otherwise go to

The proposed BAM-PSO algorithm is compared with five different biologically inspired algorithms: PSO with inertial vector and boundaries [27], ant colony system (ACS) [6], differential evolution (DE) [31], simplified swarm optimization (SSO) [20] and particle swarm optimization with aging leader and challengers (ALC-PSO) [24]. These algorithms are selected because of several factors: first, PSO is the base algorithm for BAM-PSO, so it is natural to compare performance with the original optimizer, SSO and ALC-PSO are other well-known variants of PSO that in some way, claim to alleviate the premature convergence problem and, specifically, ALC-PSO is related in many ways to BAM-PSO. Finally, while ACS and DE are not related closely to BAM-PSO, they are swarm-based and evolution-based optimization algorithms respectively and thus, were considered as good candidates for performance comparison.

To test optimization performance of these algorithms, well-established benchmark functions are selected in low and high dimensionality [33]. These selected functions help evaluate algorithm's performance over a broad type of problems, because they possess multiple local

θ ¼ θ � 1 leader term<sup>0</sup> ð Þ s reduction :

the leader's individual improvement factor.

16 Particle Swarm Optimization with Applications

if δgBest < 0: θ ¼ θ þ 2 ð Þ up to Θ , else :

if δlBest < 0: θ ¼ θ þ 1 ð Þ up to Θ :else : if δLeaderθ < 0: θ ¼ θ ð Þ no increase , else :

Step 2 for a new round of iteration.

4. Results

by newly generated challengers as described in [24].

particles with random ones for every depleted lifespan.

Figure 2 shows the flow chart for BAM-PSO algorithm.

following decision tree:

For comparison purposes, all algorithms were configured in similar vein when it was possible, e.g. the ACS algorithm [6] uses ant-type vectors which can be considered particles in a swarm like those found in the PSO [8, 21], ALC-PSO [24] and the proposed BAM-PSO algorithms. Nevertheless, the behavior and setting are very different, since ant-type vectors behavior is determined by a mathematical model that simulates the pheromone attraction between biological ants. The DE algorithm [34] does not have a swarm-based mathematical model for the dynamics of particles, but instead the mathematical model used to simulate evolution is based on vectors and mutation factors. Finally, SSO algorithm [20] does not consider linear equations


Table 1. Benchmark functions used in algorithm performance comparison for BAM-PSO.

to update the information of the particles, instead a probability function is considered to decide the next particle position based on previously defined settings.

## 4.1. Evaluating the algorithms in low dimensional settings

The swarm size S for every algorithm is set to 20, dimension D for every function is set to 2, and total iterations are set to 10,000 for each objective function. Table 2 reveals the performance for the different selected algorithms in a low dimension, the results show the best possible solution offered by the algorithm after terminal condition was reached. As we can see, both BAM-PSO and ALC-PSO algorithms show improved performance in comparison to the other algorithms. Meaning that BAM-PSO provides good results in low dimensional problems for all the benchmark functions, outperforming most of the other tested algorithms. It is important to note, that results marked in Bold are the best solution obtained for each case.

## 4.2. Evaluating the algorithms in high dimensional settings

Our second simulation scenario consists in evaluating the performance of the BAM-PSO with high dimensional problems. In this case, the total of 18 benchmark functions from Table 1 was considered and the function dimension D was configured to 30.

Based on the previous results, ALC-PSO, SSO, and PSO algorithm were selected to compare results with the BAM-PSO because of their shared origin. However, ACS was also included due to its swarm nature.

At first glance, the results shown in Table 3 suggest that the BAM-PSO provides the best performance of all compared algorithms in highly-dimensional problems for several benchmark functions. It is important to note, that results marked in Bold are the best solution obtained for each case.

In Table 4, we can observe that BAM-PSO outperforms the other algorithms with an accepted level of significance using this procedure. However, this test is a simple first-line procedure and to uncover more evidence over the results, we rely on a more robust and sensitive

Dimension = 30 BAM-PSO ALC\_PSO PSO SSO ACS

f <sup>1</sup> 3.420000E-01 4.810000E-02 1.664995E-02 1.017972E + 00 1.009513E + 00 f <sup>2</sup> 1.710000E + 00 2.950000E + 00 6.179332E-01 1.936833E + 01 1.575243E + 01 f <sup>3</sup> 9.720000E-04 1.060000E + 00 1.054778E-05 3.628362E + 02 2.766939E + 02 f <sup>4</sup> 3.830000E + 00 1.420000E + 03 2.003442E + 02 1.421872E + 06 4.577661E + 05 f <sup>5</sup> 2.701825E-07 1.943493E-01 1.314056E + 01 6.189616E + 02 5.320685E + 05 f <sup>6</sup> 1.000012E + 00 1.347625E + 04 5.670018E + 03 2.350724E + 06 3.796814E + 06 f <sup>7</sup> 1.838826E-25 1.846162E-01 8.309639E-01 5.267949E + 03 7.131057E + 03 f <sup>8</sup> 1.693373E-15 5.924127E-08 2.569830E-10 5.672124E-02 2.042905E-01 f <sup>9</sup> 1.591141E + 00 4.989256E-03 1.084663E + 03 6.528900E + 02 4.572974E + 02 f <sup>10</sup> 4.265668E-03 2.575325E-01 6.748519E + 01 2.362575E + 01 6.768979E + 01 f <sup>11</sup> 1.000007E + 00 0.000000E + 00 8.449218E + 120 3.61590E + 115 1.11140E + 121 f <sup>12</sup> 1.00008E + 119 9.432255E + 119 2.858556E + 125 1.96354E + 124 3.06278E + 125 f <sup>13</sup> 1.000053E + 00 9.837181E + 00 5.033260E + 01 3.015314E + 03 3.832691E + 03 f <sup>14</sup> 1.005478E + 00 2.327770E + 03 9.314781E + 03 8.149385E + 03 1.062061E + 04 f <sup>15</sup> 2.039775E-23 0.000000E + 00 8.520128E-06 4.815499E-04 9.508960E + 02 f <sup>16</sup> 9.177340E + 00 1.276359E + 01 1.683612E + 02 1.086427E + 02 4.458898E + 01 f <sup>17</sup> 1.000155E + 00 6.796335E + 00 1.001832E + 00 1.026017E + 02 1.676198E + 02 f <sup>18</sup> 9.112266E-01 2.056725E + 02 4.960351E + 00 7.179904E + 05 8.280520E + 05

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19

The Wilcoxon test results shown in Table 5 shed light over the fact that BAM-PSO algorithm can go beyond the results provided by the PSO, SSO and ACS with great statistical significance (P < 0.001), but the procedure finds not enough evidence to conclude that the BAM-PSO can outperform the ALC-PSO at this level of statistical significance; however, the results are good enough to show that BAM-PSO can outperform ALC-PSO with good statistical significance (P < 0.01), and considering the literature claim that: if the resulting P-value is small enough (P < 0.05), then it can be accepted that the median of the differences between the paired

BAM-PSO vs. ALC\_PSO PSO SSO ACS Positive results 14 15 18 18 Negative results 4 3 0 0 Significant difference? (P < 0.05) Yes Yes Yes Yes

procedure, which is the Wilcoxon Test.

Table 3. Optimization results comparison for D = 30.

Table 4. Non-parametrical sign test for benchmark results at D = 30.

## 4.3. Non-parametrical statistical analysis of BAM-PSO performance results

In order to conclude whether or not BAM-PSO outperforms the other selected algorithms, more accurate means of comparison other than simple observation of benchmark results are required; for this reason, some of the most popular non-parametric statistical tests were employed. This type of analysis is widely accepted as a metric of performance comparison between algorithms in a pair-wise configuration [43]. To this end, using the statistical procedures defined by [38–40], the Signed Test and the Wilcoxon Test statistical analysis were selected.


Table 2. Optimization results comparison for D = 2.

Particle Swarm Optimization Algorithm with a Bio-Inspired Aging Model http://dx.doi.org/10.5772/intechopen.71791 19


Table 3. Optimization results comparison for D = 30.

to update the information of the particles, instead a probability function is considered to

The swarm size S for every algorithm is set to 20, dimension D for every function is set to 2, and total iterations are set to 10,000 for each objective function. Table 2 reveals the performance for the different selected algorithms in a low dimension, the results show the best possible solution offered by the algorithm after terminal condition was reached. As we can see, both BAM-PSO and ALC-PSO algorithms show improved performance in comparison to the other algorithms. Meaning that BAM-PSO provides good results in low dimensional problems for all the benchmark functions, outperforming most of the other tested algorithms. It is important to note, that results marked in Bold are the best solution obtained for each case.

Our second simulation scenario consists in evaluating the performance of the BAM-PSO with high dimensional problems. In this case, the total of 18 benchmark functions from Table 1 was

Based on the previous results, ALC-PSO, SSO, and PSO algorithm were selected to compare results with the BAM-PSO because of their shared origin. However, ACS was also included

At first glance, the results shown in Table 3 suggest that the BAM-PSO provides the best performance of all compared algorithms in highly-dimensional problems for several benchmark functions. It is important to note, that results marked in Bold are the best solution

In order to conclude whether or not BAM-PSO outperforms the other selected algorithms, more accurate means of comparison other than simple observation of benchmark results are required; for this reason, some of the most popular non-parametric statistical tests were employed. This type of analysis is widely accepted as a metric of performance comparison between algorithms in a pair-wise configuration [43]. To this end, using the statistical procedures defined by

4.3. Non-parametrical statistical analysis of BAM-PSO performance results

[38–40], the Signed Test and the Wilcoxon Test statistical analysis were selected.

Dimension = 2 BAM-PSO ALC-PSO DE SSO ACS PSO

f <sup>1</sup> 0.000000E + 00 0.000000E + 00 7.800000E-03 4.250000E-02 4.880000E-09 0.000000E + 00 f <sup>2</sup> 8.880000E-16 8.880000E-16 2.120000E + 00 4.300000E-02 4.160000E-02 8.880000E-16 f <sup>3</sup> 0.000000E + 00 0.000000E + 00 2.800000E-01 5.460000E-06 5.460000E-08 3.550000E-43 f <sup>4</sup> 0.000000E + 00 0.000000E + 00 2.610000E + 00 4.300000E-03 7.300000E-01 0.000000E + 00

decide the next particle position based on previously defined settings.

4.1. Evaluating the algorithms in low dimensional settings

18 Particle Swarm Optimization with Applications

4.2. Evaluating the algorithms in high dimensional settings

considered and the function dimension D was configured to 30.

due to its swarm nature.

obtained for each case.

Table 2. Optimization results comparison for D = 2.

In Table 4, we can observe that BAM-PSO outperforms the other algorithms with an accepted level of significance using this procedure. However, this test is a simple first-line procedure and to uncover more evidence over the results, we rely on a more robust and sensitive procedure, which is the Wilcoxon Test.

The Wilcoxon test results shown in Table 5 shed light over the fact that BAM-PSO algorithm can go beyond the results provided by the PSO, SSO and ACS with great statistical significance (P < 0.001), but the procedure finds not enough evidence to conclude that the BAM-PSO can outperform the ALC-PSO at this level of statistical significance; however, the results are good enough to show that BAM-PSO can outperform ALC-PSO with good statistical significance (P < 0.01), and considering the literature claim that: if the resulting P-value is small enough (P < 0.05), then it can be accepted that the median of the differences between the paired


Table 4. Non-parametrical sign test for benchmark results at D = 30.


aging mechanism implemented in BAM-PSO allows the algorithm to provide better results

The authors thank the support of CONACYT Mexico, through Projects CB256769 and 340

, Nancy Arana-Daniel<sup>1</sup>

Particle Swarm Optimization Algorithm with a Bio-Inspired Aging Model

http://dx.doi.org/10.5772/intechopen.71791

21

,

CB258068 (Project supported by Fondo Sectorial de Investigación para la Educación).

, Esteban A. Hernandez-Vargas<sup>2</sup>

\*

[1] Von Neumann J. Zur theorie der gesellschaftsspiele. Mathematische Annalen. 1928;100

[2] Maynard J, Price R. The logic of animal conflict. Nature. 1973;5427(246):15-18. DOI: 10.10

[3] Henson S, Hayward J. The mathematics of animal behavior: An interdisciplinary dia-

[4] Barricelli N. Esempi numerici di processi di evoluzione. Methodos. 1954;6(21–22):45-68 [5] Goss S, Aaron S, Deneubourg J, Pasteels J. Self-organized shortcuts in the angentine ant.

[6] Colorni A, Dorigo M, Maniezzo V. Distributed optimization by ant colonies. Proceedings

[7] Socha K, Dorigo M. Ant colony optimization for continuous domains. European Journal

[8] Kennedy J, Eberhart R. Particle swarm optimization. Proceedings of the IEEE International Joint Conference on Neural Networks. 1995;4(1):1942-1948. DOI: 10.1109/ICNN.

of Operational Research. 2008;1(1):1155-1173. DOI: 10.1016/j.ejor.2006.06.046

Naturwissenschaften. 1989;76(12):579-581. DOI: 10.1007/BF00462870

than some other popular optimizers that does not implement aging.

Acknowledgements

Author details

References

38/246015a0

1995.488968

Eduardo Rangel-Carrillo<sup>1</sup>

Carlos Lopez-Franco<sup>1</sup> and Alma Y. Alanis<sup>1</sup>

(1):295-320. DOI: 10.1007/BF01448847

logue. Notices of the AMS. 2010;57(10):1248-1258

of ECAL91. 1991;1(1):134-142. DOI: 10.1007/BF00462870

\*Address all correspondence to: almayalanis@gmail.com

1 CUCEI, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico

2 Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany

Table 5. Non-parametrical Wilcoxon test for benchmark results at D = 30.

observations is statistically significantly different from 0 [44]. We can conclude then, that BAM-PSO has a greater performance over a broad set of benchmark functions over all other selected algorithms with statistical relevance, including ALC-PSO.

The performance of BAM-PSO can be explained by its senescence mechanism: after particles falls into local minimum, they offer less improvement; then, the senescence mechanism starts acting by producing senescence on the swarm; then, exhausted particles are replaced with random ones through the search space. This favors exploration after premature convergence without completely eliminating exploitation of search space near the local minimum, which in the end provides better optimization results than other PSO variants.
