1 Introduction
2 Related works
3 Original WOA and its variants
3.1 Original WOA
3.2 Compact WOA
3.3 Parallel techniques
3.4 Time and space complexity of PCWOA
Algorithm | WOA | CWOA | PCWOA |
---|---|---|---|
Time complexity | O(\(I*P*D\)) | O(\(I*D\)) | O(\(I*D*G\)) |
Space complexity | O(P) | O(2) | O(G) |
4 Experimental results and analysis
4.1 Benchmark functions and algorithm parameters
Algorithm | Parameters |
---|---|
WOA | Virtual \(Np=30, ub=100, lb=-100, c=1;\) |
CWOA | Virtual \(Np=30, ub=100, lb=-100, c=1, \lambda =10;\) |
PWOA | Virtual \(Np=30, ub=100, lb=-100, c=1, groups=3;\) |
PCWOA | Virtual \(Np=30, ub=100, lb=-100, c=1, \lambda =10, groups=3;\) |
MCSCA | Virtual \(Np=30, ub=100, lb=-100, \lambda =10, groups=10, F_{1,2,3,4}=0.5;\) |
CBA | Virtual \(Np=30, ub=100, lb=-100,\lambda =10, A=0.5, r=0.5;\) |
CSCA | Virtual \(Np=30, ub=100, lb=-100, \lambda =10;\) |
SCA | Virtual \(Np=30, ub=100, lb=-100;\) |
MVO | Virtual \(Np=30, ub=100, lb=-100, P=6, WEP_{Min}=0.2, WEP_{Max}=1;\) |
AOA | Virtual \(Np=30, ub=100, lb=-100, \alpha =5, \mu =0.5;\) |
SSA | Virtual \(Np=30, ub=100, lb=-100;\) |
PSO | Virtual \(Np=30, ub=100, lb=-100, c1=c2=2,w\in [0.4,0.9];\) |
4.2 Compared with WOA and its variants
F | WOA | CWOA | PCWOA |
---|---|---|---|
F1 | −4.24E+02 | 2.14E+04 | −1.35E+03 |
F2 | 1.17E+08 | 2.08E+08 | 6.45E+07 |
F3 | 6.39E+10 | 1.18E+11 | 3.51E+10 |
F4 | 1.03E+05 | 7.72E+04 | 7.45E+04 |
F5 | 8.99E+00 | 4.17E+03 | −8.34E+02 |
F6 | −6.04E+02 | 9.12E+02 | −7.46E+02 |
F7 | 4.90E+03 | −5.20E+02 | 3.30E+02 |
F8 | −6.79E+02 | −6.79E+02 | −6.79E+02 |
F9 | −5.60E+02 | −5.65E+02 | −5.64E+02 |
F10 | 1.88E+02 | 2.21E+03 | −3.09E+02 |
F11 | 1.56E+02 | 1.65E+02 | 1.55E+02 |
F12 | 3.12E+02 | 2.45E+02 | 2.41E+02 |
F13 | 4.02E+02 | 3.24E+02 | 3.14E+02 |
F14 | 6.06E+03 | 6.97E+03 | 5.65E+03 |
F15 | 6.74E+03 | 7.06E+03 | 5.88E+03 |
F16 | 2.02E+02 | 2.03E+02 | 2.02E+02 |
F17 | 1.02E+03 | 1.03E+03 | 9.75E+02 |
F18 | 1.05E+03 | 1.11E+03 | 1.05E+03 |
F19 | 7.16E+02 | 2.76E+04 | 5.53E+02 |
F20 | 7.16E+02 | 6.15E+02 | 6.14E+02 |
F21 | 1.73E+03 | 2.90E+03 | 1.20E+03 |
F22 | 7.82E+03 | 8.40E+03 | 7.42E+03 |
F23 | 8.39E+03 | 8.39E+03 | 7.16E+03 |
F24 | 1.32E+03 | 1.30E+03 | 1.29E+03 |
F25 | 1.43E+03 | 1.39E+03 | 1.40E+03 |
F26 | 1.57E+03 | 1.43E+03 | 1.49E+03 |
F27 | 2.68E+03 | 2.54E+03 | 2.53E+03 |
F28 | 2.68E+03 | 5.88E+03 | 5.67E+03 |
Win | 2 | 3 | 23 |
4.3 Compared with other compact algorithms
F | PCWOA | CBA | CSCA | MCSCA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Ave | Std | Rank | Ave | Std | Rank | Ave | Std | Rank | Ave | Std | Rank | |
F1 | \(-\)1.35E+03 | 1.67E+01 | 1 | 5.87E+04 | 8.33E+02 | 3 | 1.32E+05 | 1.55E+04 | 4 | 1.60E+04 | 2.89E+03 | 2 |
F2 | 5.99E+07 | 2.33E+07 | 1 | 4.68E+09 | 1.98E+08 | 4 | 2.39E+09 | 4.36E+08 | 3 | 2.48E+08 | 6.83E+07 | 2 |
F3 | 3.44E+10 | 1.79E+10 | 1 | 4.23E+21 | 1.41E+21 | 4 | 6.62E+18 | 1.58E+19 | 3 | 4.33E+13 | 1.35E+14 | 2 |
F4 | 7.26E+04 | 2.36E+04 | 3 | 6.84E+04 | 2.70E+02 | 2 | 2.79E+05 | 9.95E+04 | 4 | 5.25E+04 | 4.69E+03 | 1 |
F5 | \(-\)8.23E+02 | 3.49E+01 | 1 | 5.43E+04 | 1.61E+03 | 4 | 4.94E+04 | 9.71E+03 | 3 | 4.21E+03 | 1.40E+03 | 2 |
F6 | \(-\)7.45E+02 | 5.58E+01 | 1 | 1.86E+04 | 4.42E+02 | 3 | 3.16E+04 | 4.03E+03 | 4 | 2.07E+03 | 1.13E+03 | 2 |
F7 | 2.20E+03 | 7.75E+05 | 1 | 2.81E+08 | 2.54E+07 | 4 | 1.36E+06 | 1.86E+06 | 3 | 1.35E+03 | 3.11E+03 | 2 |
F8 | \(-\)6.79E+02 | 6.20E\(-\)02 | 2 | \(-\)6.79E+02 | 6.00E\(-\)02 | 1 | \(-\)6.79E+02 | 6.90E\(-\)02 | 3 | \(-\)6.78E+02 | 3.10E\(-\)01 | 4 |
F9 | \(-\)5.65E+02 | 2.66E+00 | 1 | \(-\)5.42E+02 | 1.25E+00 | 4 | \(-\)5.55E+02 | 1.52E+00 | 2 | \(-\)5.45E+02 | 9.11E+00 | 3 |
F10 | \(-\)2.96E+02 | 6.82E+01 | 1 | 1.16E+04 | 2.46+02 | 4 | 1.73E+04 | 2.35E+03 | 3 | 2.27E+03 | 4.62E+02 | 2 |
F11 | 1.37E+02 | 1.05E+02 | 2 | 6.14E+02 | 3.15E+01 | 3 | 1.17E+03 | 1.55E+02 | 4 | 1.07E+02 | 6.74E+01 | 1 |
F12 | 2.66E+02 | 1.19E+02 | 2 | 6.80E+02 | 1.76E+01 | 3 | 1.57E+03 | 1.97E+02 | 4 | 2.32E+02 | 8.56E+01 | 1 |
F13 | 3.75E+02 | 1.08E+02 | 2 | 9.37E+02 | 4.10E+01 | 3 | 1.68E+03 | 1.95E+02 | 4 | 3.28E+02 | 5.40E+01 | 1 |
F14 | 5.71E+03 | 7.89E+02 | 1 | 6.60E+03 | 2.84E+02 | 2 | 9.24E+03 | 3.93E+02 | 4 | 6.96E+03 | 5.52E+02 | 3 |
F15 | 5.90E+03 | 7.51E+02 | 1 | 6.07E+03 | 4.60E+02 | 2 | 9.32E+03 | 3.52E+02 | 4 | 7.51E+03 | 4.20E+02 | 3 |
F16 | 2.02E+02 | 4.10E\(-\)01 | 1 | 2.03E+02 | 6.50E\(-\)01 | 3 | 2.04E+02 | 6.10E\(-\)01 | 4 | 2.03E+02 | 3.60E\(-\)01 | 2 |
F17 | 9.39E+02 | 1.31E+02 | 1 | 1.21E+03 | 1.58E+01 | 3 | 4.67E+03 | 4.06E+02 | 4 | 9.40E+02 | 1.06E+02 | 2 |
F18 | 1.00E+03 | 1.26E+02 | 2 | 1.30E+03 | 1.57E+01 | 3 | 4.79E+03 | 3.45E+02 | 4 | 9.99E+02 | 7.15E+01 | 1 |
F19 | 5.46E+02 | 1.68E+01 | 1 | 7.53E+05 | 4.87E+04 | 3 | 9.31E+06 | 3.62E+06 | 4 | 9.66E+03 | 3.52E+03 | 2 |
F20 | 6.14E+02 | 3.70E\(-\)01 | 1 | 6.15E+02 | 0.00E+00 | 3 | 6.15E+02 | 6.51E\(-\)13 | 2 | 6.15E+02 | 2.40E\(-\)01 | 4 |
F21 | 1.13E+03 | 5.84E+01 | 1 | 3.21E+03 | 2.35E+01 | 3 | 1.12E+04 | 1.14E+03 | 4 | 2.53E+03 | 2.55E+02 | 2 |
F22 | 7.27E+03 | 7.18E+02 | 1 | 1.11E+04 | 2.22E+02 | 3 | 1.13E+04 | 2.46E+02 | 4 | 8.66E+03 | 6.39E+02 | 2 |
F23 | 7.26E+03 | 7.71E+02 | 1 | 9.69E+03 | 4.36E+02 | 3 | 1.04E+04 | 3.71E+02 | 4 | 9.11E+03 | 3.83E+02 | 2 |
F24 | 1.30E+03 | 5.69E+00 | 2 | 1.81E+03 | 2.38E+01 | 4 | 1.40E+03 | 3.07E+01 | 3 | 1.30E+03 | 5.39E+00 | 1 |
F25 | 1.39E+03 | 5.28E+00 | 1 | 1.56E+03 | 5.68E+00 | 4 | 1.49E+03 | 1.66E+01 | 3 | 1.40E+03 | 3.86E+00 | 2 |
F26 | 1.46E+03 | 8.88E+01 | 1 | 2.30E+03 | 8.16E+01 | 4 | 1.63E+03 | 9.56E+00 | 3 | 1.55E+03 | 7.54E+01 | 2 |
F27 | 2.52E+03 | 5.48E+01 | 1 | 4.39E+03 | 5.37E+01 | 4 | 2.90E+03 | 4.65E+01 | 3 | 2.65E+03 | 3.78E+01 | 2 |
F28 | 5.91E+03 | 9.22E+02 | 2 | 1.07E+04 | 2.21E+02 | 3 | 1.51E+04 | 1.62E+03 | 4 | 5.66E+03 | 5.57E+02 | 1 |
Mean | 1.14 | 3.18 | 3.5 | 1.82 | ||||||||
Rank | 1 | 3 | 4 | 2 |
4.4 Compared with popular algorithms
F | PCWOA | SCA | ALO | AOA | SSA | |||||
---|---|---|---|---|---|---|---|---|---|---|
Ave | Best | Ave | Best | Ave | Best | Ave | Best | Ave | Best | |
F1 | \(-\)1.35E+03 | \(-\)1.38E+03 | 1.37E+04 | 7.20E+03 | 2.33E+04 | 8.75E+03 | 6.50E+04 | 4.84E+04 | \(-\)1.40E+03 | \(-\)1.40E+03 |
F2 | 7.49E+07 | 3.35E+07 | 2.13E+08 | 5.03E+07 | 2.91E+08 | 1.68E+08 | 1.59E+09 | 7.09E+08 | 1.29E+08 | 3.69E+07 |
F3 | 3.27E+10 | 6.14E+09 | 6.71E+10 | 2.56E+10 | 3.32E+13 | 3.56E+10 | 1.97E+20 | 3.82E+14 | 6.86E+10 | 4.50E+10 |
F4 | 7.09E+04 | 3.42E+04 | 4.56E+04 | 3.00E+04 | 1.58E+05 | 1.12E+05 | 6.80E+04 | 6.11E+04 | 5.52E+04 | 3.07E+04 |
F5 | \(-\)8.45E+02 | \(-\)8.87E+02 | 2.07E+03 | 1.07E+03 | 5.43E+03 | 1.25E+03 | 2.88E+04 | 8.97E+03 | 1.25E+03 | \(-\)2.25E+02 |
F6 | \(-\)7.46E+02 | \(-\)8.60E+02 | 2.03E+02 | \(-\)4.68E+02 | 1.15E+03 | 1.94E+01 | 1.60E+04 | 8.73E+03 | \(-\)4.96E+02 | \(-\)6.87E+02 |
F7 | 2.63E+00 | \(-\)6.39E+02 | \(-\)5.84E+02 | \(-\)6.61E+02 | 6.74E+02 | \(-\)6.67E+02 | 1.37E+07 | 5.33E+01 | \(-\)6.65E+02 | \(-\)7.17E+02 |
F8 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 | \(-\)6.79E+02 |
F9 | \(-\)5.64E+02 | \(-\)5.71E+02 | \(-\)5.58E+02 | \(-\)5.62E+02 | \(-\)5.62E+02 | \(-\)5.66E+02 | \(-\)5.56E+02 | \(-\)5.62E+02 | \(-\)5.72E+02 | \(-\)5.80E+02 |
F10 | \(-\)3.14E+02 | \(-\)4.16E+02 | 1.62E+03 | 9.41E+02 | 2.27E+03 | 1.61E+03 | 1.03E+04 | 6.18E+03 | 6.20E+02 | 1.96E+02 |
F11 | 1.25E+02 | \(-\)1.86E+02 | 8.97E+02 | \(-\)8.15E+01 | 2.78E+01 | \(-\)8.47E+01 | 5.43E+02 | 1.92E+02 | \(-\)2.25E+02 | \(-\)2.97E+02 |
F12 | 2.47E+02 | 6.70E+01 | 1.16E+02 | 3.93E+01 | 2.55E+02 | 1.16E+02 | 6.57E+02 | 3.90E+02 | 1.10E+02 | 1.15E+01 |
F13 | 3.01E+02 | 1.76E+02 | 3.32E+02 | 1.87E+02 | 3.78E+02 | 2.34E+02 | 7.48E+02 | 5.50E+02 | 2.63E+02 | 1.90E+02 |
F14 | 5.82E+03 | 4.52E+03 | 7.30E+03 | 6.91E+03 | 5.83E+03 | 5.19E+03 | 4.55E+03 | 3.47E+03 | 5.48E+03 | 4.64E+03 |
F15 | 5.83E+03 | 4.27E+03 | 7.87E+03 | 6.91E+03 | 5.77E+03 | 4.22E+03 | 7.70E+03 | 6.51E+03 | 6.15E+03 | 5.39E+03 |
F16 | 2.02E+02 | 2.01E+02 | 2.02E+02 | 2.02E+02 | 2.02E+02 | 2.01E+02 | 2.02E+02 | 2.01E+02 | 2.02E+02 | 2.01E+02 |
F17 | 9.35E+02 | 7.22E+02 | 8.43E+02 | 7.37E+02 | 1.19E+03 | 9.50E+02 | 1.03E+03 | 9.25E+02 | 8.68E+02 | 6.03E+02 |
F18 | 9.65E+02 | 7.16E+02 | 9.53E+02 | 8.69E+02 | 1.16E+03 | 9.50E+02 | 1.26E+03 | 1.23E+03 | 1.01E+03 | 7.58E+02 |
F19 | 5.53E+02 | 5.24E+02 | 8.35E+03 | 2.38E+03 | 3.03E+04 | 5.57E+03 | 8.43E+05 | 5.76E+05 | 4.56E+03 | 5.44E+02 |
F20 | 6.15E+02 | 6.14+02 | 6.15E+02 | 6.14E+02 | 6.15E+02 | 6.15E+02 | 6.15E+02 | 6.15E+02 | 6.15E+02 | 6.15E+02 |
F21 | 1.12E+03 | 1.03E+03 | 2.76E+03 | 2.50E+03 | 2.99E+03 | 2.68E+02 | 3.39E+03 | 3.29E+03 | 2.60E+03 | 2.28E+03 |
F22 | 7.34E+03 | 6.15E+03 | 8.74E+03 | 7.75E+03 | 7.17E+03 | 6.40E+03 | 7.49E+03 | 6.34E+03 | 7.18E+03 | 6.08E+03 |
F23 | 7.03E+03 | 5.03E+03 | 9.21E+03 | 8.44E+03 | 7.66E+03 | 6.76E+03 | 9.64E+03 | 7.86E+03 | 7.73E+03 | 6.00E+03 |
F24 | 1.30E+03 | 1.28E+03 | 1.32E+03 | 1.31E+03 | 1.33E+03 | 1.32E+03 | 1.57E+03 | 1.38E+03 | 1.31E+03 | 1.29E+03 |
F25 | 1.39E+03 | 1.37E+03 | 1.43E+03 | 1.43E+03 | 1.47E+03 | 1.44E+03 | 1.54E+03 | 1.48E+03 | 1.42E+03 | 1.40E+03 |
F26 | 1.47E+03 | 1.40E+03 | 1.53E+03 | 1.41E+03 | 1.59E+03 | 1.59E+03 | 1.61E+03 | 1.46E+03 | 1.57E+03 | 1.41E+03 |
F27 | 2.53E+03 | 2.38E+03 | 2.71E+03 | 2.63E+03 | 2.66E+03 | 2.49E+03 | 3.10E+03 | 2.75E+03 | 2.55E+03 | 2.43E+03 |
F28 | 5.96E+03 | 4.40E+03 | 4.30E+03 | 3.86E+03 | 6.55E+03 | 5.32E+03 | 7.96E+03 | 7.13E+03 | 4.01E+03 | 3.23E+03 |
5 DV-Hop localization algorithm based on PCWOA
5.1 DV-Hop localization algorithm
5.2 Apply PCWOA in DV-Hop localization algorithm
6 Simulation research
6.1 Influences of the number of anchor nodes
Anchor node | DV-Hop | PSO | WOA | PWOA | PCWOA |
---|---|---|---|---|---|
5 | 0.4988 | 0.4397 | 0.4031 | 0.3812 | 0.3956 |
10 | 0.3939 | 0.2804 | 0.2644 | 0.2584 | 0.2830 |
15 | 0.3577 | 0.2540 | 0.2550 | 0.2539 | 0.2376 |
20 | 0.3402 | 0.2261 | 0.2228 | 0.2215 | 0.2124 |
25 | 0.3300 | 0.2003 | 0.2005 | 0.2003 | 0.1966 |
30 | 0.3425 | 0.1979 | 0.1989 | 0.1965 | 0.1907 |
35 | 0.3137 | 0.1913 | 0.1918 | 0.1910 | 0.1862 |
40 | 0.3027 | 0.1839 | 0.1851 | 0.1838 | 0.1799 |
6.2 Influences of communication radius
Communication radius | DV-Hop | PSO | WOA | PWOA | PCWOA |
---|---|---|---|---|---|
20 | 0.3402 | 0.2663 | 0.2235 | 0.2214 | 0.2124 |
25 | 0.3330 | 0.2066 | 0.2059 | 0.2057 | 0.2023 |
30 | 0.3095 | 0.1939 | 0.1916 | 0.1906 | 0.1886 |
35 | 0.3158 | 0.2036 | 0.2044 | 0.2034 | 0.2027 |
40 | 0.3125 | 0.1940 | 0.1944 | 0.1939 | 0.1921 |
6.3 Influences of the total number of nodes
Total nodes | DV-Hop | PSO | WOA | PWOA | PCWOA |
---|---|---|---|---|---|
100 | 0.3825 | 0.2285 | 0.2272 | 0.2233 | 0.2211 |
150 | 0.3502 | 0.1962 | 0.1969 | 0.1953 | 0.1936 |
200 | 0.3095 | 0.1947 | 0.1912 | 0.1906 | 0.1890 |
250 | 0.3413 | 0.2209 | 0.2180 | 0.2171 | 0.2145 |
300 | 0.3193 | 0.2178 | 0.2156 | 0.2145 | 0.2144 |
350 | 0.3405 | 0.2268 | 0.2258 | 0.2264 | 0.2244 |
400 | 0.3318 | 0.2267 | 0.2258 | 0.2203 | 0.2199 |