Mathematical Problems in Engineering

Research Article

A Novel Ranking Method Based on Subjective Probability Theory for Evolutionary Multiobjective Optimization

Table 3

Comparison of the new GP algorithm with the proposed ranking approach and NRGAII-GP for a classical nonlinear system design problem: (a) model results of the new MOGP algorithm with the proposed ranking approach for 10 trials; (b) model results of NRGAII-GP for 10 trials.
(a)
TrialsResults of the MOGP algorithm with the proposed ranking approach
StructureNumber of featuresMSE
1 𝐱 3 + 𝐱 3 𝐱 4 21.8479e-4
2 𝐱 3 + 𝐱 4 𝐱 3 21.8265e-4
3 𝐱 3 + 𝐱 4 𝐱 3 21.8143e-4
4 𝐱 4 𝐱 3 + 𝐱 3 21.7979e-4
5 𝐱 3 + 𝐱 3 𝐱 4 21.7688e-4
6 𝐱 3 + 𝐱 4 𝐱 3 21.7720e-4
7 𝐱 4 𝐱 3 + 𝐱 3 21.8062e-4
8 𝐱 4 𝐱 3 + 𝐱 3 21.7902e-4
9 𝐱 3 + 𝐱 3 𝐱 4 21.8638e-4
10 𝐱 3 + 𝐱 4 𝐱 3 21.7611e-4
(b)
TrialsResults of the MOGP algorithm with NSGA-II
StructureNumber of featuresMSE
1 ( 𝐱 3 + 𝐱 2 ) + 𝐱 1 + ( 𝐱 1 𝐱 2 𝐱 3 𝐱 2 ) + ( 𝐱 4 + 𝐱 3 ) ( 𝐱 3 + 𝐱 4 ) + 𝐱 1 𝐱 1 + 𝐱 3 𝐱 4 41.8479e-4
2 ( ( 𝐱 4 + 𝐱 1 ) + 𝐱 3 𝐱 4 ) + 𝐱 1 𝐱 1 𝐱 1 + ( 𝐱 4 𝐱 4 ) ( 𝐱 3 𝐱 4 ) 31.8265e-4
3 ( 𝐱 4 𝐱 3 ) ( 𝐱 4 𝐱 2 ) ( 𝐱 4 + 𝐱 2 ) 𝐱 3 𝐱 2 + ( 𝐱 4 + 𝐱 3 ) + ( 𝐱 3 𝐱 4 ) 31.8143e-4
4 𝐱 2 𝐱 3 + 𝐱 4 𝐱 3 + ( 𝐱 2 + 𝐱 3 ) + 𝐱 4 + ( 𝐱 4 + 𝐱 3 ) 𝐱 4 𝐱 4 ( 𝐱 3 𝐱 3 ) ( 𝐱 2 𝐱 2 ) 31.7979e-4
5 ( 𝐱 2 𝐱 2 ) ( 𝐱 2 𝐱 3 ) ( 𝐱 3 𝐱 1 ) + ( 𝐱 3 + 𝐱 4 + 𝐱 3 𝐱 4 ) + ( 𝐱 2 + 𝐱 3 𝐱 3 ) 41.7688e-4
6 𝐱 1 𝐱 1 + ( ( 𝐱 3 + 𝐱 4 ) + 𝐱 4 𝐱 3 ) + ( 𝐱 4 𝐱 3 ( 𝐱 4 𝐱 4 ) ) ( 𝐱 2 𝐱 3 ) 41.7720e-4
7 ( ( 𝐱 1 𝐱 1 ) 𝐱 1 + 𝐱 1 ) + ( 𝐱 4 𝐱 3 𝐱 4 𝐱 2 ) + ( 𝐱 3 + 𝐱 4 𝐱 4 ) 31.8062e-4
8 ( ( 𝐱 1 𝐱 1 ) + 𝐱 4 𝐱 3 ) + ( ( 𝐱 1 + 𝐱 3 ) + 𝐱 3 𝐱 3 ) + 𝐱 3 𝐱 3 𝐱 4 𝐱 4 + 𝐱 4 31.7902e-4
9 ( 𝐱 1 𝐱 1 𝐱 1 ) + 𝐱 1 𝐱 2 + 𝐱 3 + 𝐱 2 + 𝐱 4 𝐱 3 + 𝐱 3 𝐱 3 + 𝐱 1 41.8638e-4
10 ( 𝐱 3 + 𝐱 4 𝐱 3 ) + 𝐱 1 𝐱 1 𝐱 1 𝐱 1 + ( 𝐱 4 + 𝐱 4 𝐱 3 ) ( ( 𝐱 2 + 𝐱 2 ) 𝐱 3 𝐱 2 ) 41.7611e-4