Introduction
Preliminaries
Clustering
Whale optimization algorithm (WOA)
Exploitation phase
Exploration phase
Proposed method
Tournament empowered WOA
MR-TWOA-based recommendation method
Time complexity
Experimental results
Performance of TWOA on benchmark problems
Function |
\(V_{no}\)
| Range |
\(f_{min}\)
|
---|---|---|---|
\(F_1(x)=\sum _{i=1}^nx_i^2\)
| 30 | [− 100, 100] | 0 |
\(F_2(x)=\sum _{i=1}^n \mid x_i \mid + \prod _{i=1}^n\mid x_i\)
| 30 | [− 10, 10] | 0 |
\( F_3(x) = \sum _{i=1}^{d} \sum _{j=1}^{i} x_j^2 \)
| 30 | [− 100, 100] | 0 |
\(F_4(x)=\max _i \left\{ \mid x \mid , 1\le i \le n\right\} \)
| 30 | [− 100, 100] | 0 |
\(F_5(x)=\sum _{i=1}^{n-1}\left[ 100(x_{i+1}- x_i^2)^2 +(x_i-1)^2\right] \)
| 30 | [− 30, 30] | 0 |
\(F_6(x)=\sum _{i=1}^{n}(\left[ x_i+0.5\right] )^2\)
| 30 | [− 100, 100] | 0 |
\(F_7(x)=\sum _{i=1}^{n} ix_i^4 + random [0,1)\)
| 30 | [− 1.28, 1.28] | 0 |
Function |
\(V_\mathrm{no}\)
| Range |
\(f_\mathrm{min}\)
|
---|---|---|---|
\( F_{8}(x) = \sum _{i=1}^n - x_i \sin \sqrt{|x_i|} \)
| 30 | [− 500, 500] | 0 |
\( F_{9}(x) = \sum _{i=1}^n [x_i^2 - 10\cos (2\pi x_i)+10]\)
| 30 | [− 5.12, 5.12] | 0 |
\( F_{10}(x) = - 20 \exp ^{-0.02 \sqrt{n^-1\sum _{i=1}^n x_i^2}} - e^{n^-1 \sum _{i=1}^n cos(2\pi x_i)} + 20 + e\)
| 30 | [− 32, 32] | 0 |
\(F_{11}(x) = \frac{1}{4000} \sum _{i=1}^n x_i^2 -\prod _{i=1}^n \cos (\frac{x_i}{\sqrt{i}}) + 1\)
| 30 | [− 600, 600] | 0 |
\(F_{12}(x) = \frac{\pi }{n} \left\{ 10 \sin (\pi y_1)+\sum _{i=1}^{n-1} (y_i-1)^2 \left[ 1+ 10 \sin ^2 (\pi y_1)\right] +(y_n-1)^2\right\} \)
| |||
+ \(\sum _{i=1}^n u(x_i, 10, 100, 4),\)\( y_i = 1+\frac{x_i +1}{4}\), | |||
\(u(x_i, a, k, m)= {\left\{ \begin{array}{ll}k(x_i-a)^m &{} x_i > 0\\ 0 &{} -a<x_i < 1\\ k(-x_i-a)^m &{} -x_i-a \end{array}\right. }\)
| 30 | [− 50, 50] | 0 |
\(F_{13}(x) = 0.1\{\sin ^2(3\pi x_1)+\sum _{i=1}^n (x_i-1)\left[ 1+\sin ^2(3\pi x_i+1)\right] \)
| |||
\(\qquad +(x_n-1)^2\left[ 1+\sin ^2(2\pi x_n)\right] \}+\sum _{i=1}^n u(x_i, 5, 100, 4)\)
| 30 | [− 50, 50] | 0 |
Function |
\(V_\mathrm{no}\)
| Range |
\(f_\mathrm{min}\)
|
---|---|---|---|
\(F_{14}(x) = \left( \frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^2(x_i-a_{ij})^6}\right) ^{-1}\)
| 2 | [− 65, 65] | 1 |
\(F_{15}(x) = \sum _{i=1}^{11}\left[ a_i-\frac{x_1(b_i^2+b_ix_2)}{b_i^2+b_ix_3+x_4}\right] ^2\)
| 4 | [− 5,5] | 0.0003 |
\(F_{16}(x) = 4x_1^2-2.1x_1^4+\frac{1}{3}x_1^6+x_1x_2+4x_2^2+4x_2^4\)
| 2 | [− 5,5] | − 1.0316 |
\(F_{17}(x) = (x_2-\frac{5.1}{4\pi ^2}x_1^2+\frac{5}{\pi }x_1-6)^2+10(1-\frac{1}{8\pi })\cos x_1+10\)
| 2 | [− 5, 5] | 0.398 |
\(F_{18}(x) = \left[ 1+(x_1+x_2+1)^2(19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2)\right] ~~~~\times \left[ 30+(2x_1+3x_2+1)^2(18-32x_1+12x_1^2-48x_2+36x_1x_2+27x_2^2)\right] \)
| 2 | [−2, 2] | 3 |
\(F_{19}(x) =-\,\sum _{i=1}^4 c_i\exp (-\sum _{j=1}^3a_{ij}(x_j-p_{ij})^2)\)
| 3 | [1, 3] | − 3.86 |
\(F_{20}(x) =-\,\sum _{i=1}^4 c_i\exp (-\sum _{j=1}^6a_{ij}(x_j-p_{ij})^2)\)
| 6 | [0, 1] | − 3.32 |
\(F_{21}(x) =-\,\,\sum _{i=1}^5 \left[ (X-a_i)(X-a_i)^T+c_i \right] ^{-1}\)
| 4 | [0, 10] | − 10.1532 |
\(F_{22}(x) =-\,\,\sum _{i=1}^7 \left[ (X-a_i)(X-a_i)^T+c_i \right] ^{-1}\)
| 4 | [0, 10] | − 10.4028 |
\(F_{23}(x) =-\,\,\sum _{i=1}^10 [(X-a_i)(X-a_i)^T+c_i]^{-1}\)
| 4 | [0, 10] | − 10.5363 |
Parameter name | SCA | ICS | EGWO | WOA | TWOA |
---|---|---|---|---|---|
Population size (pop) | 30 | 30 | 30 | 30 | 30 |
Number of iterations (itr) | 500 | 500 | 500 | 500 | 500 |
a
| 2 | 2 | 2 | 2 | 2 |
Probability (Pa) | – | .25 | – | – | – |
Step scaling factor | – | .01 | – | – | – |
Crossover rate (C) | – | 0.1 | – | – | – |
Mutation rate (C) | – | – | 0.1 | – | – |
Fn. | TWOA | WOA | ICS | EGWO | SCA | |||||
---|---|---|---|---|---|---|---|---|---|---|
MEAN | STD | MEAN | STD | MEAN | STD | MEAN | STD | MEAN | STD | |
F1 | 7.04E−73 | 2.71E−72 | 8.71E−73 | 4.40E−72 | 4.56E+01 | 2.38E+01 | 1.64E+01 | 3.09E+01 | 2.32E−03 | 3.02E−03 |
F2 | 4.14E−51 | 7.47E−51 | 1.04E−50 | 1.21E−50 | 2.50E+01 | 3.11E+01 | 3.00E−02 | 4.51E−02 | 5.09E+01 | 5.63E+01 |
F3 | 9.19E+03 | 5.58E+03 | 6.22E+04 | 9.06E+03 | 4.45E+03 | 2.49E+03 | 1.11E+04 | 6.66E+03 | 6.10E+03 | 3.73E+03 |
F4 | 3.62E+01 | 1.97E+01 | 6.39E+01 | 3.19E+01 | 1.88E+01 | 1.60E+01 | 4.39E+01 | 4.64E+00 | 2.07E+01 | 4.92E+00 |
F5 | 2.76E+01 | 3.08E−01 | 3.45E+01 | 5.64E−01 | 9.51E+03 | 2.02E+04 | 1.78E+05 | 7.77E+05 | 3.04E+02 | 4.72E+02 |
F6 | 1.06E+00 | 2.93E−01 | 5.22E−01 | 3.35E−01 | 4.66E+01 | 2.58E+01 | 2.21E+01 | 2.82E+01 | 1.72E−03 | 1.73E−03 |
F7 | 1.10E−04 | 7.02E−05 | 2.82E−03 | 3.05E−03 | 4.71E−02 | 1.77E−02 | 1.05E−01 | 1.17E−01 | 3.22E−01 | 1.14E−01 |
F8 | − 1.32E+04 | 1.28E+02 | − 1.09E+04 | 1.92E+03 | − 9.06E+03 | 7.45E+02 | − 4.53E+03 | 3.70E+02 | − 7.08E+03 | 9.09E+02 |
F9 | 1.07E+02 | 2.13E+01 | 2.31E−15 | 1.19E−14 | 1.36E+02 | 3.76E+01 | 4.25E+01 | 3.97E+01 | 1.05E+02 | 3.20E+01 |
F10 | 8.75E−16 | 8.11E−01 | 5.42E−15 | 3.38E+01 | 6.74E+00 | 1.43E+00 | 1.96E+01 | 1.03E+01 | 6.23E+00 | 3.16E+00 |
F11 | 6.77E−02 | 3.43E−02 | 1.83E−01 | 8.79E+00 | 1.45E+00 | 1.87E−01 | 1.18E+00 | 3.36E−01 | 8.90E−02 | 1.06E−01 |
F12 | 1.67E−02 | 7.68E−03 | 6.16E−02 | 2.86E−01 | 1.28E+01 | 6.24E+00 | 1.82E+05 | 1.05E+06 | 1.71E+01 | 6.76E+00 |
F13 | 5.70E−01 | 2.11E−01 | 7.66E−01 | 4.25E−01 | 6.76E+01 | 1.18E+02 | 9.36E+04 | 2.60E+05 | 3.50E+01 | 2.61E+01 |
F14 | 3.75E+00 | 2.92E+00 | 3.73E+00 | 3.84E+00 | 1.25E+00 | 4.87E−16 | 2.38E+00 | 2.57E+00 | 3.04E+00 | 2.15E+00 |
F15 | 7.17E−04 | 2.25E-04 | 1.05E−03 | 1.29E−03 | 8.14E−03 | 1.92E−02 | 1.29E−03 | 4.65E−04 | 4.62E−03 | 8.64E−03 |
F16 | − 1.33E+00 | 7.76E−14 | − 1.26E+00 | 1.55E−09 | − 1.29E+00 | 5.18E−13 | − 1.23E+00 | 7.01E−05 | − 1.02E+00 | 5.14E−05 |
F17 | 3.91E−01 | 2.43E−09 | 4.86E−01 | 3.09E−05 | 4.98E−01 | 3.43E−08 | 4.76E−01 | 1.33E−08 | 5.15E−01 | 1.93E−07 |
F18 | 2.96E+00 | 5.53E−13 | 3.66E+00 | 1.58E−04 | 7.13E+00 | 1.84E+01 | 3.59E+00 | 1.55E−04 | 3.89E+00 | 4.48E−03 |
F19 | − 4.99E+00 | 2.85E−13 | − 4.70E+00 | 1.35E−02 | − 4.73E+00 | 2.47E−01 | − 4.60E+00 | 2.92E−03 | − 3.80E+00 | 2.06E−02 |
F20 | −4.18E+00 | 4.03E−02 | − 3.89E+00 | 2.60E−01 | − 4.06E+00 | 8.06E−02 | − 3.59E+00 | 1.96E−01 | − 3.23E+00 | 1.41E−01 |
F21 | − 7.53E+00 | 4.36E+00 | − 3.14E+00 | 1.57E+00 | − 7.59E+00 | 3.33E+00 | − 8.97E+00 | 2.45E+00 | − 7.26E+00 | 3.21E+00 |
F22 | − 9.65E+00 | 1.89E+00 | − 9.02E+00 | 3.61E+00 | − 6.48E+00 | 4.41E+00 | − 3.11E+00 | 2.38E+00 | − 7.38E+00 | 4.19E+00 |
F23 | − 8.56E+00 | 2.19E+00 | − 8.54E+00 | 3.69E+00 | − 7.30E+00 | 4.66E+00 | − 4.46E+00 | 2.27E+00 | − 7.54E+00 | 4.60E+00 |
Performance analysis of MR-TWOA
Name | Cluster | Dimension | Data objects |
---|---|---|---|
Iris (Replicated) | 3 | 7 | 10,000,050 |
CMC (Replicated) | 3 | 9 | 10,000,197 |
Wine (Replicated) | 2 | 18 | 5,000,000 |
Vovel (Replicated) | 10 | 10 | 1,025,010 |
S. no | Dataset | Criteria | MR-Kmeans | MR-KPSO | MR-ABC | MR-BAT | MR-WOA | MR-TWOA |
---|---|---|---|---|---|---|---|---|
1 | Repriduced Iris | Fm | 0.636 | 0.767 | 0.833 | 0.781 | 0.801 | 0.848 |
CT |
7.95E+04
| 10.25E+04 | 10.27E+04 | 10.39E+04 | 10.20E+04 | 9.20E+04 | ||
2 | Reproduced CMC | Fm | 0.290 | 0.320 | 0.380 | 0.381 | 0.297 | 0.392 |
CT |
7.80E+E04
| 11.40E+E04 | 11.46E+04 | 10.41E+E04 | 11.52E+E04 | 10.51E+E04 | ||
3 | Reproduced Wine | Fm | 0.45 | 0.510 | 0.730 | 0.718 | 0.750 | 0.790 |
CT |
10.12E+04
| 17.19E+04 | 17.28E+04 | 20.29E+04 | 17.14E+04 | 16.15E+04 | ||
4 | Reproduced Vovel | Fm | 0.555 | 0.630 | 0.635 | 0.621 | 0.610 | 0.650 |
CT |
11.65E+04
| 15.32E+04 | 14.32E+04 | 14.20E+04 | 14.22E+04 | 13.26E+04 |
Clusters | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | |
---|---|---|---|---|---|---|---|---|---|
MR-TWOA | MAE | 0.741 | 0.690 | 0.681 | 0.690 | 0.689 | 0.680 | 0.686 | 0.687 |
Precison | 0.410 | 0.420 | 0.430 | 0.430 | 0.420 | 0.440 | 0.440 | 0.450 | |
Recall | 0.130 | 0.120 | 0.210 | 0.370 | 0.450 | 0.550 | 0.670 | 0.690 | |
MR-WOA | MAE | 0.790 | 0.770 | 0.770 | 0.770 | 0.781 | 0.785 | 0.786 | 0.788 |
Precision | 0.410 | 0.370 | 0.390 | 0.360 | 0.350 | 0.390 | 0.360 | 0.350 | |
Recall | 0.120 | 0.120 | 0.210 | 0.310 | 0.420 | 0.530 | 0.610 | 0.690 | |
MR-BAT | MAE | 0.820 | 0.790 | 0.790 | 0.790 | 0.800 | 0.805 | 0.806 | 0.807 |
Precision | 0.370 | 0.370 | 0.390 | 0.370 | 0.350 | 0.350 | 0.340 | 0.330 | |
Recall | 0.130 | 0.110 | 0.260 | 0.270 | 0.300 | 0.380 | 0.440 | 0.700 | |
MR-ABC | MAE | 0.819 | 0.810 | 0.810 | 0.810 | 0.810 | 0.805 | 0.810 | 0.810 |
Precision | 0.350 | 0.320 | 0.330 | 0.320 | 0.320 | 0.320 | 0.310 | 0.320 | |
Recall | 0.120 | 0.160 | 0.220 | 0.260 | 0.360 | 0.420 | 0.440 | 0.490 | |
MR-PSO | MAE | 0.825 | 0.825 | 0.824 | 0.828 | 0.824 | 0.824 | 0.825 | 0.825 |
Precision | 0.320 | 0.310 | 0.280 | 0.300 | 0.290 | 0.290 | 0.290 | 0.260 | |
Recall | 0.100 | 0.120 | 0.160 | 0.230 | 0.340 | 0.440 | 0.460 | 0.500 |