Table 3: Comparison of mean/standard deviation of solutions obtained by using hybrid HPSO-SA algorithm and approaches described in [2].

𝐹 𝑢 𝑛 𝑐 𝑡 𝑖 𝑜 𝑛 𝑃 𝑆 𝑂 𝑄 𝐼 𝑃 𝑆 𝑂 𝐴 𝑇 𝑅 𝐸 𝑃 𝑆 𝑂 𝐺 𝑀 𝑃 𝑆 𝑂 HPSO-SA

Rastrigin 2 2 . 3 3 9 1 5 8 1 1 . 9 4 6 8 8 8 1 9 . 4 2 5 9 7 9 2 0 . 0 7 9 1 8 5 𝟎
15.932042 9 . 1 6 1 5 2 6 1 4 . 3 4 9 0 4 6 1 3 . 7 0 0 2 0 2 𝟎

Sphere 1 . 1 6 7 7 4 9 𝑒 4 5 0 . 0 0 0 0 0 0 4 , 0 0 0 2 8 9 𝑒 1 7 7 , 2 6 3 5 7 9 𝑒 1 7 𝟓 . 𝟑 𝟔 𝟓 𝟔 𝐞 𝟑 𝟐
5 . 2 2 2 3 3 1 𝑒 4 6 0 . 0 0 0 0 0 0 0 . 0 0 0 2 4 6 6 . 1 8 8 8 5 4 𝑒 1 7 𝟐 . 𝟗 𝟖 𝟒 𝟗 𝟐 𝐞 𝟑 𝟏

Griewank 0 . 0 3 1 6 4 6 0 . 0 1 1 5 8 0 0 . 0 2 5 1 5 8 0 . 0 2 4 4 6 2 𝟑 , 𝟑 𝟐 𝟐 𝟓 𝟓 𝐞 𝟐 𝟎
0 . 0 2 5 3 2 2 0 . 0 1 2 8 5 0 0 . 0 2 8 1 4 0 0 . 0 3 9 3 0 4 𝟐 . 𝟔 𝟖 𝟒 𝟏 𝟓 𝐞 𝟐 𝟎

Rosenbrock 2 2 . 1 9 1 7 2 5 8 . 9 3 9 0 1 1 1 9 . 4 9 0 8 2 0 1 4 . 1 5 9 5 4 7 𝟎 , 𝟐 𝟐 𝟕 𝟎 𝟒 𝟖 𝟏 𝟖 𝟖
1 . 6 1 5 5 4 4 𝑒 + 0 4 3 . 1 0 6 3 5 9 3 . 9 6 4 3 3 5 𝑒 + 0 4 4 . 3 3 5 4 3 9 𝑒 + 0 4 𝟎 . 𝟐 𝟒 𝟑 𝟗 𝟕 𝟖 𝟎 𝟓 𝟕

Noisy 8 . 6 8 1 6 0 2 0 . 4 5 1 1 0 9 8 . 0 4 6 6 1 7 7 . 1 6 0 6 7 5 𝟎 , 𝟎 𝟎 𝟐 𝟎 𝟏 𝟗 𝟗 𝟗 𝟖
9 . 0 0 1 5 3 4 0 . 3 2 8 6 2 3 8 . 8 6 2 3 8 5 7 . 6 6 5 8 0 2 𝟎 . 𝟎 𝟎 𝟎 𝟔 𝟓 𝟎 𝟑 𝟒 𝟕

Schwefel 6 1 7 8 . 5 5 9 8 9 6 6 3 5 5 . 5 8 6 6 4 0 6 1 8 3 . 6 7 7 6 0 0 6 0 4 7 . 6 7 0 8 9 8 𝟖 𝟑 𝟕 𝟗 . 𝟔 𝟔
4 . 8 9 3 3 2 9 𝑒 + 0 2 4 7 7 . 5 3 2 5 8 4 4 6 9 . 6 1 1 1 0 4 4 8 2 . 9 2 6 7 3 8 𝟐 . 𝟐 𝟎 𝟒 𝟐 𝟓 𝐞 𝟏 𝟗

Ackley 3 . 4 8 3 9 0 3 𝑒 1 8 2 . 4 6 1 8 1 1 𝑒 2 4 0 . 0 1 8 4 9 3 1 . 4 7 4 9 3 3 𝑒 1 8 𝟕 . 𝟒 𝟑 𝟓 𝟒 𝟔 𝐞 𝟏 𝟔
8 . 3 5 9 5 3 5 𝑒 1 9 0 . 0 1 4 4 2 5 0 . 0 1 4 7 4 7 1 . 1 5 3 7 0 9 𝑒 0 8 𝟏 . 𝟎 𝟗 𝟑 𝟖 𝟐 𝐞 𝟏 𝟓

Michalewizc 1 8 . 1 5 9 4 0 0 1 8 . 4 6 9 6 0 0 1 8 . 9 8 2 9 0 0 1 8 . 3 9 9 8 0 0 𝟏 𝟗 , 𝟔 𝟐 𝟓 𝟓 𝟓 𝟖 𝟎 𝟔
1 . 0 5 1 0 5 0 0 . 0 9 2 9 6 6 0 . 2 7 2 5 7 9 0 . 4 0 3 7 2 2 𝟎 . 𝟎 𝟎 𝟗 𝟐 𝟔 𝟗 𝟒 𝟒

Himmelblau 3 . 3 3 1 4 8 8 3 . 7 8 3 9 6 1 3 . 7 5 1 4 5 8 3 . 4 6 0 2 3 3 𝟑 , 𝟕 𝟖 𝟑 𝟗 𝟔
1 . 2 4 3 2 9 0 0 . 1 9 0 3 9 4 0 . 1 7 4 4 6 0 0 . 4 5 7 8 2 0 𝟑 . 𝟏 𝟔 𝟎 𝟎 𝟏 𝐞 𝟏 𝟓

Shubert 1 8 6 . 7 3 0 9 4 1 1 8 6 . 7 3 0 9 4 2 1 8 6 . 7 3 0 9 4 1 1 8 6 . 7 3 0 9 4 2 𝟏 𝟖 𝟔 , 𝟕 𝟑 𝟎 𝟗 𝟒 𝟐
1 . 4 2 4 1 5 4 𝑒 0 5 0 . 0 0 0 0 0 0 1 . 4 2 4 1 5 4 𝑒 0 5 1 . 5 2 5 8 7 9 𝑒 0 5 𝟖 . 𝟔 𝟔 𝟕 𝟒 𝟔 𝐞 𝟏 𝟒