Elite Opposition-Based Cognitive Behavior Optimization Algorithm for Global Optimization

4.1.1 Benchmark Test Functions

To verify the effectiveness of the algorithm, 23 standard test functions [6, 15, 17] were tested to reflect the objectivity of the experimental results.

These 23 standard test functions can be divided into three types (high-dimensional unimodal function, high-dimensional multimodal function, and low-dimensional function), where f₀₁ to f₀₇ are high-dimensional unimodal functions, f₀₈ to f₁₂ are high-dimensional multimodal functions, and f₁₃ to f₂₃ are low-dimensional functions. f₀₅ is a classical test function whose global minimum value is at the bottom of a parabola and the fitness value of the position near the bottom of the trough changes little, so it is difficult to find the global minimum of the test function. f₀₇ is a multidimensional multimodal function with a large number of local minima in its domain, which increases the difficulty of searching the objective function. In the 23 standard test functions selected in this chapter, the optimal value of most functions is zero, which can fully verify the optimization ability of the algorithm and also select the standard test function with some nonzero optimal values. The test functions and related configurations are shown in Table 1.

4.1.2 Results of Test Results Analysis

In the test, the number of population N is set to 50, the maximum number of iterations is 1000, independently run 30 times. In the test results table, Best, Mean, Worst, and Std, respectively, in 30 independent experiments, are the optimal solution, mean, worst-case solution, and standard deviation of the 30 run results. In addition, the dimension of the test function is described in the table. The rank of each test function is sorted by the variance value. Sum1 is the number of the first rankings in the statistics, and the last rank is the average of the variance. It is worth noting in the table the boldfaced underlined values for the test function of the optimal value.

Table 2 shows the results of the high-dimensional unimodal functions test. Tables 3 and 4 show the results of high-dimensional multimodal functions and low-dimensional functions, respectively. To verify the superiority of the function test results, the algorithm is compared to ABC [18], Cuckoo search (CS) [27], FPA [28], and Grey wolf optimizer (GWO) [19]. The following are the parameter settings of the comparison algorithms:

1. ABC: limit=5D;

2. CS: β=1.5, ρ₀=1.5;

3. FPA: ρ=0.8;

4. GWO: Linearly α decreased from 2 to 0.

In Table 2, in the high-dimensional single-peak test functions, f₀₁ to f₀₄ found the optimal value, and its mean and Std are also 0; whereas the other comparison algorithms cannot test the five benchmark functions, it still found the optimal solution in a 30-dimensional space. f₀₅ ranked third, while it is best to find that the results are smaller than the rest of the algorithm. In the f₀₆ function, obviously, GWO find the results better than the rest of the algorithm. The f₀₇ test results can be found, although the standard test results are not achieved, but the variance of the ECOA is optimal, and its optimal value is also obvious.

Figures 1–14 show the convergence and ANOVA test plots of the ECOA and other comparison algorithms in the f₀₁ to f₀₇ test functions. The logarithm of the Y-axis is taken in the convergence plot, and it is easy to find that the convergence rate is faster in f₀₁ to f₀₄, f₀₆ than in other algorithms. Although f₀₅ convergence accuracy is good for GWO, the ECOA converges earlier than GWO. In the ANOVA test plots, although f₀₅ is not as good as GWO and the COA, f₀₁ to f₀₄, f₀₆ are very stable compared to other contrasting algorithms.

Figure 1:

D=30, Evolution Curves of Fitness Value for f₀₁.

Figure 2:

D=30, ANOVA Test of Global Minimum for f₀₁.

Figure 3:

D=30, Evolution Curves of Fitness Value for f₀₂.

Figure 4:

D=30, ANOVA Test of Global Minimum for f₀₂.

Figure 5:

D=30, Evolution Curves of Fitness Value for f₀₃.

Figure 6:

D=30, ANOVA Test of Global Minimum for f₀₃.

Figure 7:

D=30, Evolution Curves of Fitness Value for f₀₄.

Figure 8:

D=30, ANOVA Test of Global Minimum for f₀₄.

Figure 9:

D=30, Evolution Curves of Fitness Value for f₀₅.

Figure 10:

D=30, ANOVA Test of Global Minimum for f₀₅.

Figure 11:

D=30, Evolution Curves of Fitness Value for f₀₆.

Figure 12:

D=30, ANOVA Test of Global Minimum for f₀₆.

Figure 13:

D=30, Evolution Curves of Fitness Value for f₀₇.

Figure 14:

D=30, ANOVA Test of Global Minimum for f₀₇.

In Table 3, f₀₈ to f₁₁ found the theoretical optimal solution, and the variance is 0 in f₀₉ to f₁₁, which shows its high stability. Although f₀₈ found the optimal value, its relatively large base caused by the corresponding variance is relatively large. Although the results of f₁₂ did not meet the standard, the variance is still ranked in the best of six, and the effect is more obvious. It is shown that the ECOA has better stability and robustness in the multimodal function optimization problem.

Figures 15–24 show the convergence and variance plots of the ECOA and other comparison algorithms in the f₀₈ to f₁₂ test functions. In f₀₉ to f₁₂ convergent graphs, the logarithm of the Y-axis is taken. In the f₀₈ function, the logarithm is not taken because the theoretical optimal value is negative. It is easy to find that f₀₈ to f₁₂ converge faster than other algorithms, and in the variance graph, f₀₈ to f₁₂ are stable compared to other algorithms.

Figure 15:

D=30, Evolution Curves of Fitness Value for f₀₈.

Figure 16:

D=30, ANOVA Test of Global Minimum for f₀₈.

Figure 17:

D=30, Evolution Curves of Fitness Value for f₀₉.

Figure 18:

D=30, ANOVA Test of Global Minimum for f₀₉.

Figure 19:

D=30, Evolution Curves of Fitness Value for f₁₀.

Figure 20:

D=30, ANOVA Test of Global Minimum for f₁₀.

Figure 21:

D=30, Evolution Curves of Fitness Value for f₁₁.

Figure 22:

D=30, ANOVA Test of Global Minimum for f₁₁.

Figure 23:

D=30, Evolution Curves of Fitness Value for f₁₂.

Figure 24:

D=30, ANOVA Test of Global Minimum for f₁₂.

In Table 4, the ECOA finds the standard results from the comparison of f₁₃ to f₂₃, the 11 low-dimensional test functions, and the overall ranking is also relatively high, where f₁₃, f₁₄, f₁₆, f₁₈, and f₂₀ are in the variance rank. However, it still found the theoretical optimal value, and the variance is basically in the same series. Although FPA ranked first in f₁₉, it did not find the theoretical optimal value, but both the COA and the ECOA found the optimal value, and the variance and FPA also did not differ much. f₁₅, f₁₇, f₂₁, f₂₂, and f₂₃ are ranked first, showing a stronger search capability, higher accuracy, and high robustness.

Figures 25–46 show the convergence and variance graphs of the ECOA and other comparison algorithms in f₁₃ to f₂₃ test functions, where f₁₃ is logarithmic to the Y-axis, and it is easy to see that it converges faster than other algorithms. In f₁₄ to f₂₃ functions, the theoretical optimal value is negative, so there is no logarithm. The convergence speed of f₁₄ to f₂₀, f₂₃ is relatively fast compared to other contrast algorithms. In the ANOVA, f₁₄ to f₂₃ compared to other contrast algorithms are very stable.

Figure 25:

D=4, Evolution Curves of Fitness Value for f₁₃.

Figure 26:

D=4, ANOVA Test of Global Minimum for f₁₃.

Figure 27:

D=2, Evolution Curves of Fitness Value for f₁₄.

Figure 28:

D=2, ANOVA Test of Global Minimum for f₁₄.

Figure 29:

D=2, Evolution Curves of Fitness Value for f₁₅.

Figure 30:

D=2, ANOVA Test of Global Minimum for f₁₅.

Figure 31:

D=2, Evolution Curves of Fitness Value for f₁₆.

Figure 32:

D=2, ANOVA Test of Global Minimum for f₁₆.

Figure 33:

D=3, Evolution Curves of Fitness Value for f₁₇.

Figure 34:

D=3, ANOVA Test of Global Minimum for f₁₇.

Figure 35:

D=6, Evolution Curves of Fitness Value for f₁₈.

Figure 36:

D=6, ANOVA Test of Global Minimum for f₁₈.

Figure 37:

D=4, Evolution Curves of Fitness Value for f₁₉.

Figure 38:

D=4, ANOVA Test of Global Minimum for f₁₉.

Figure 39:

D=4, Evolution Curves of Fitness Value for f₂₀.

Figure 40:

D=4, ANOVA Test of Global Minimum for f₂₀.

Figure 41:

D=4, Evolution Curves of Fitness Value for f₂₁.

Figure 42:

D=4, ANOVA Test of Global Minimum for f₂₁.

Figure 43:

D=2, Evolution Curves of Fitness Value for f₂₂.

Figure 44:

D=2, ANOVA Test of Global Minimum for f₂₂.

Figure 45:

D=2, Evolution Curves of Fitness Value for f₂₃.

Figure 46:

D=2, ANOVA Test of Global Minimum for f₂₃.

4.3.1 Pressure Vessel Design Problem

The pressure vessel design problem [10] is a classical hybrid constrained optimization problem. The working pressure is 2000 psi and the maximum capacity is 750 ft³. As shown in Figure 47, both ends of the cylindrical container are covered with a hemispherical head. Using a rolled steel plate, the shell is made into two halves, which are joined by two longitudinal welds to form a cylinder. The goal is to minimize the total cost, including the cost of materials, molding, and welding.

Figure 47:

Pressure Vessel Design Problem.

Minimize f(x)=0.6224x1x3x4+1.7781x2x32+3.1661x12x4+19.8x12x3

Subject to g₁(x)=−x₁+0.0193x₃≤0

g₂(x)=−x₃+0.00954x₃≤0

g3(x)=−πx32x4−43πx33+1296000≤0

g₄(x)=x₄−240≤0

In this paper, the ECOA is used to solve this problem, in which the comparison algorithms run 20 times independently, with GA [8], HS [14], ABC [18], CS [10], GSA [22], CoBiDE [25], DSA [4], and AMO [15] are presented in Table 6.

The ECOA is less costly than other comparison algorithms. Thus, the ECOA used to solve the pressure vessel design problem is feasible.

4.3.2 Cantilever Beam Design Problem

The cantilever beam design problem [11] structure is shown in Figure 48, which minimizes the weight of the cantilever beam with a hollow square block. There are five squares, where the first block is fixed and the fifth block is borne by the vertical load. The five parameters define the shape of the cross-section of the cube. The form of the problem is as follows:

Figure 48:

Cantilever Beam Design Problem.

Minimize f(x)=0.0624(x₁+x₂+x₃+x₄+x₅);

Subject to g(x)=61x13+37x23+19x33+7x43+1x53≤1;

In this paper, the ECOA is used to solve this problem. The algorithms include the method of moving asymptotes (MMA) [5], generalized convex approximation (GCA_I) [5], GCA_II [5], CS [10], symbiotic organisms search (SOS) [2], and MVO [20] run independently 20 times, and the results are presented in Table 7.

It can be found by comparison that the minimum cost of the ECOA in solving the cantilever problem is in the first place, which can verify the superiority of the algorithm.

4.3.3 Welded Beam Design Problem

The welded beam design problem is a problem that is academically widely used in practical engineering design (Figure 49). The purpose of the design is to minimize the manufacturing costs. The design model is affected by the shear stress (τ), the beam bending stress (θ), the rod buckling load (Pc), the beam end deflection (δ), and the normal stress (σ) related to the seven conditional constraints. Specific optimization problems can be described as a formula. The four decision variables x₁ to x₄ defined the domain as the formula:

Figure 49:

Welded Beam Design Problem.

Minimize f(x)=1.10471x12x2+0.04811x3x4(14+x2);

Subject to g₁(x)=τ(x)−τ_max≤0,

g₂(x)=σ(x)−σ_max≤0,

g₃(x)=x₃−x₄≤0,

g₄(x)=0.125−x₁≤0,

g₅=δ(x)−0.25≤0,

g₆=P−P_c(x)≤0,

g7=0.10471x12+0.04811x3x4(14+x2)−5≤0;

Variable range 0.1≤x₁≤2; 0.1≤x₂≤10;0.1≤x₃≤10;0.1≤x₄≤2;

where τ(x)=τ12+2τ1τ2(x22R)+τ22;

τ1=P2x1x2;

τ2=MRJ;

M=P(L+x22);

J(x)=2{2x1x2[x224+(x1+x22)2]};

R=x224+(x1+x32)2;

σ(x)=6PLx4x32;

δ(x)=6PL3Ex33x4;

Pc=4.013Ex32x4636L2(1−x32LE4G).

In this paper, the ECOA is used to solve this problem, which includes GWO [19], GSA [19], CPSO [19], GA (Coello) [10], GA (Deb) [9], GA (Deb) [7], HS [5], Random [21], Simplex [21], David [21], APPROX [21], and BA [11] run 20 times independently and presented in Table 8.

By comparison, we can find that the ECOA has the lowest cost and the most obvious effect in the welded beam design problem, which shows that the ECOA has obvious advantages in the welded beam design problem.