# A Modified jSO Algorithm for Solving Constrained Engineering Problems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. JADE

#### 2.2. SHADE

_{F}and M

_{CR}and their updating mechanisms. SHADE is very representative of the state-of-the-art DE variants, and how it works is described in detail below. The basic steps for SHADE are as follows:

**x**

_{i,G}is the current individual and

**x**

_{pbest,G}is randomly selected from the former NP × p

_{i}(where p

_{i}∈ [0, 1]) optimal individuals of thegeneration population. The vectors

**x**

_{r1,G}are randomly selected individuals from thegeneration population,

**x**

_{r2,G}are randomly selected individuals from the combination of thegeneration population and the external archive A, r

_{1}≠ r

_{2}≠ r

_{3}≠ i, F

_{i}is the scaling factor, NP is the population size, rand() is the uniform random distribution, G is the current iteration, and

**v**

_{i,G}is the generated variation vector. Regarding “current-to- pbest/1”, the greed of the mutation strategy depends on the control parameters of p

_{i}, and the calculation formula for p

_{i}is as shown in (5) and (6). It balances exploration and exploitation (a small P is greedier). The scaling factor F

_{i}is generated by the following formula:

_{i}() is a Cauchy distribution, M

_{F,ri}is randomly selected from the historical memory M

_{F}, and r

_{i}is a uniformly distributed random value of [1, H]. If the generated F

_{i}> 1, then F

_{i}= 1; and if F

_{i}≤ 0, Equation (7) is repeated to generate effective values.

_{j,i,G}of the variation vector is outside the search range $\left[{x}_{min},{x}_{max}\right]$, the following correction is performed, as shown in Formula (8):

_{i}or j = jrand, the dimension of the test vector u inherits the dimension of the variation vector v; otherwise, the dimension of the original vector x is inherited. The purpose of setting j = jrand is to provide a protection mechanism so that at least one dimension of the test vector is inherited from the variation vector.

_{i}is generated using the following formula:

_{i}() is a Gaussian distribution, M

_{CR,ri}is randomly selected from historical memory M

_{CR}, and r

_{i}is a uniformly distributed random value of [1, H]. If the generated CR

_{i}> 1, then let CR

_{i}= 1; and if CR

_{i}< 0, then let CR

_{i}= 0.

**x**

_{i,G}will be stored in the external archive of elimination solution A. If the external archive exceeds its capacity, one of them will be deleted randomly to make room for the subsequent elimination solution.

_{F}and M

_{CR}are initialized according to Formula (3), but the contents inside will change as the algorithm iterates. These memories store the scaling factor F and crossover rate CR of “success”, where “success” means that the trial vector u rather than the original vector x is selected in the selection process to be part of the next generation. In each iteration, these “successful” F and CR values are first stored in the arrays S

_{F}and S

_{CR}, respectively. After each iteration, the historical memories of M

_{F}and M

_{CR}are updated by one unit. The updated cell is represented by K. It is initialized as 1, 1 is added after each iteration, and it is reset to 1 when K exceeds the memory capacity H. The ${K}_{th}$ unit of historical memory is updated using the following formula:

_{F}= S

_{CR}= ∅, the historical memory does not update. The weighted average W

_{A}and the weighted Lehmer average W

_{L}are respectively calculated by the following formulas:

#### 2.3. LSHADE

_{new}is smaller than the current population size NP, the population is sorted according to the value of the objective function, and the worst NP-NP

_{new}individuals are discarded. The size of external archive A also decreases as the population size increases.

#### 2.4. iLSHADE

- iLSHADE uses a larger M
_{F}= 0.8 in the evolution of the initialization phase and a smaller population size $N{P}_{init}$= 12·D. - In the iLSHADE algorithm, the last entry in the H-entry pool records constant control parameter pairs, which are M
_{F}= 0.9 and M_{CR}= 0.9, respectively. These two parameters remain unchanged throughout evolution. - At different stages of the evolution, the F and CR of each individual are set to different fixed values, as shown in Equations (18) and (19).
- The value of the degree of greed control parameter P of the variation strategy in iLSHADE increases linearly as the number of fitness function evaluations increases (see Equation (20)).$${F}_{i}=\left\{\begin{array}{c}min\left({F}_{i},0.7\right)ifFES0.25\ast MAXFES\\ min\left({F}_{i},0.8\right)ifFES0.5\ast MAXFES\\ min\left({F}_{i},0.9\right)ifFES0.75\ast MAXFES\end{array}\right.$$$$C{r}_{i}=\left\{\begin{array}{c}min\left(C{r}_{i},0.5\right)ifFES0.25\ast MAXFES\\ min\left(C{r}_{i},0.25\right)ifFES0.5\ast MAXFES\end{array}\right.$$$$p={p}_{min}+\frac{FES}{MAXFES}\left({p}_{max}-{p}_{min}\right)$$

#### 2.5. jSO

_{max}= 0.25, p

_{min}= p

_{max}/2, the initial population size $N{P}_{init}=25\sqrt{D}\mathrm{log}D$, and the historical memory capacity H = 5. In addition, all parameter values in the historical memory M

_{F}and M

_{CR}are set to 0.3 and 0.8, respectively, and the weighted current mutation strategy current-to-pBest-w/1 was used.

_{w}is calculated by the following formula:

## 3. MjSO

#### 3.1. A Parameter Control Strategy Based on a Symmetric Search Process

#### 3.2. A Novel Parameter Adaptive Mechanism Based on Cosine Similarity

#### 3.3. A Novel Opposition-Based Learning Restart Mechanism

Algorithm 1: A novel OBL restart mechanism |

1: if λ = ξ2: for $i=1:NP$ do3: Generate the opposite vector $O{P}_{i}^{\prime}$ using Equation (27) 4: Calculate the fitness value $O{P}_{i}^{\prime}$; 5: $FES++$ 6: Replace ${P}_{i}$ with a fitter one between ${P}_{i}$ and $O{P}_{i}^{\prime}$ 7: end for8: end if |

Algorithm 2: MjSO |

1:$g\leftarrow 1;$ Archive $A$ ← ∅ $;$ $FES=0;$ 2: Initialize population ${P}_{g}$= (${x}_{i,g}$. . . , ${x}_{NP,g}$) randomly 3: Set all values in ${M}_{F}$ to 0.5 4: Set all values in ${M}_{CR}$ to 0.5 5: $k\leftarrow 1$//index counter 6: while the termination criteria are not meet do7: ${S}_{CR}$← ∅, ${S}_{F}$ ← ∅ 8: for $i=1$ to $NP$ do9: ${r}_{i}$ ← select from $\left[1,H\right]$ randomly 10: if ${r}_{i}$= $H$ then11: ${M}_{F,ri}$← 0.9 12: ${M}_{CR,ri}$← 0.9 13: end if14: if ${M}_{CR,ri}$< 0 then15: $C{R}_{i,g}\leftarrow 0$ 16: else17: $C{R}_{i,g}$← ${N}_{i}$(${M}_{CR,ri}$, 0.1) 18: end if19: if $g$ < 0.25 $\ast MAX\_FES$ then20: $C{R}_{ig}$← max ($C{R}_{i,g},$ 0.7) 21: else if $g$ < 0.5$\ast MAX\_FES$ then22: $C{R}_{i,g}$ ← max ($C{R}_{i,g}$, 0.6) 23: end if24: if $FES<MAX\_FES/2$25: ${F}_{i,g}=0.45+0.1\ast rand$ 26: else27: ${F}_{i,g}$← ${C}_{i}$(${M}_{F,ri}$, 0.1) 28: if $g$ < 0.6$\ast MAX\_FES$ and ${F}_{i,g}$ > 0.7 then29: ${F}_{i,g}$ ← 0.7 30: end if31: end if32: ${u}_{i,j}$ ← current-to-pBest-w/1/bin using Equation (21) 33: end for34: for i = 1 to $NP$ do35: if $f\left({u}_{i,j}\right)$≤ $f\left({x}_{i,g}\right)$then36: ${x}_{i,g+1}$← ${u}_{i,j}$ 37: else38: ${x}_{i,g+1}$← ${x}_{i,g}$ 39: end if40: if $f\left({u}_{i,j}\right)$≤ $f\left({x}_{i,g}\right)$then41: ${x}_{i,g}$→$A$, ${x}_{i,g}$ → ${S}_{CR}$, ${F}_{i,g}$ → ${S}_{F}$ 42: end if43: Shrink $A$, if necessary 44: Update ${M}_{CR}$ and ${M}_{F}$ 45: Apply LPSR strategy//linear population size reduction46: Apply Algorithm 147: Update $p$ using Equation (20) 48: end for49: $g\leftarrow g+1$ 50: end while |

## 4. The Experimental Setup

_{min}, x

_{max}] = [−100, 100], and 51 independent repeated experiments are conducted for each function. In order to statistically compare the quality of the solutions of different algorithms, two nonparametric statistical hypothesis tests were used to analyze the results: (1) the Friedman test was used to sort all the comparison algorithms [54]; (2) all of the comparison algorithms were evaluated using the Wilcoxon’s signed rank test with a significance level α = 0.05.

#### 4.1. Experimental Environment

#### 4.2. Clustering Analysis

- The core point distance is that Eps = 1% of the decision space. For the CEC2017 benchmark set, Eps = 2.
- The minimum number of clusters MinPts = 4 (minimum number of individuals with mutations).
- The distance measurement is equal to Chebyshev distance [59]. If the distance between any corresponding attributes of two individuals is greater than 1% of the decision space, they are not considered to be directly dense-reachable.

#### 4.3. Population Diversity

## 5. Experimental Results and Analysis

## 6. MjSO for Engineering Problems

#### 6.1. Pressure Vessel Design Problem

#### 6.2. Tension/Compression Spring Design Problem

#### 6.3. Welded Beam Design Problem

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data availability Statement

## Conflicts of Interest

## Appendix A

No | jSO | MjSO | ||||
---|---|---|---|---|---|---|

#runs | Mean CO | Mean PD | #runs | Mean CO | Mean PD | |

f1 | 51 | 5.67E+01 | 3.93E+01 | 51 | 4.58E+01 | 3.17E+01 |

f2 | 51 | 8.84E+01 | 3.06E+01 | 51 | 6.99E+01 | 2.92E+01 |

f3 | 51 | 8.64E+01 | 7.00E+00 | 51 | 6.97E+01 | 7.19E+00 |

f4 | 51 | 6.01E+01 | 1.18E+01 | 51 | 4.70E+01 | 1.13E+01 |

f5 | 48 | 1.19E+03 | 3.28E+01 | 50 | 1.11E+03 | 3.11E+01 |

f6 | 51 | 9.35E+01 | 8.47E+00 | 51 | 7.21E+01 | 8.84E+00 |

f7 | 49 | 1.40E+03 | 1.05E+01 | 46 | 1.32E+03 | 1.02E+01 |

f8 | 50 | 1.19E+03 | 3.40E+01 | 51 | 1.06E+03 | 3.32E+01 |

f9 | 51 | 9.00E+01 | 7.45E+00 | 51 | 6.95E+01 | 7.80E+00 |

f10 | 47 | 1.37E+03 | 8.31E+01 | 50 | 1.29E+03 | 7.08E+01 |

f11 | 51 | 3.90E+02 | 2.13E+01 | 51 | 3.35E+02 | 1.98E+01 |

f12 | 51 | 2.76E+02 | 4.31E+01 | 51 | 1.86E+02 | 3.40E+01 |

f13 | 51 | 8.30E+02 | 1.35E+01 | 51 | 6.52E+02 | 1.35E+01 |

f14 | 51 | 8.12E+02 | 2.75E+01 | 47 | 6.06E+02 | 3.25E+01 |

f15 | 51 | 3.26E+02 | 1.81E+01 | 51 | 2.62E+02 | 1.80E+01 |

f16 | 49 | 8.87E+02 | 1.77E+01 | 41 | 5.95E+02 | 2.61E+01 |

f17 | 1 | 1.89E+03 | 1.47E+01 | 2 | 1.72E+03 | 2.55E+01 |

f18 | 51 | 3.00E+02 | 2.57E+01 | 51 | 2.40E+02 | 2.40E+01 |

f19 | 51 | 5.11E+02 | 3.04E+01 | 51 | 3.77E+02 | 2.78E+01 |

f20 | 47 | 7.52E+02 | 3.69E+01 | 42 | 6.50E+02 | 3.44E+01 |

f21 | 51 | 5.70E+02 | 4.52E+01 | 51 | 5.78E+02 | 5.50E+01 |

f22 | 51 | 5.68E+01 | 7.83E+00 | 51 | 4.50E+01 | 1.11E+01 |

f23 | 51 | 8.08E+02 | 2.96E+01 | 51 | 6.38E+02 | 2.44E+01 |

f24 | 51 | 8.42E+02 | 2.89E+01 | 51 | 6.77E+02 | 3.64E+01 |

f25 | 51 | 7.75E+01 | 1.85E+01 | 51 | 6.37E+01 | 2.34E+01 |

f26 | 51 | 6.12E+01 | 6.77E+00 | 51 | 5.09E+01 | 7.10E+00 |

f27 | 51 | 1.05E+02 | 1.74E+01 | 51 | 8.44E+01 | 1.98E+01 |

f28 | 51 | 8.25E+01 | 2.42E+01 | 51 | 6.89E+01 | 2.71E+01 |

f29 | 24 | 1.59E+03 | 4.41E+01 | 24 | 1.48E+03 | 3.85E+01 |

f30 | 51 | 1.60E+02 | 1.32E+01 | 51 | 1.24E+02 | 1.31E+01 |

No | jSO | MjSO | ||||
---|---|---|---|---|---|---|

#runs | Mean CO | Mean PD | #runs | Mean CO | Mean PD | |

f1 | 51 | 1.07E+02 | 2.37E+01 | 51 | 9.13E+01 | 2.82E+01 |

f2 | 51 | 2.81E+02 | 1.77E+01 | 51 | 2.21E+02 | 1.77E+01 |

f3 | 51 | 1.90E+02 | 7.05E+00 | 51 | 1.72E+02 | 6.93E+00 |

f4 | 51 | 1.40E+02 | 8.45E+00 | 51 | 1.11E+02 | 8.20E+00 |

f5 | 47 | 2.32E+03 | 4.46E+01 | 48 | 2.34E+03 | 4.15E+01 |

f6 | 51 | 1.82E+02 | 7.46E+00 | 51 | 1.52E+02 | 7.31E+00 |

f7 | 34 | 2.49E+03 | 1.47E+01 | 36 | 2.54E+03 | 1.29E+01 |

f8 | 48 | 2.26E+03 | 5.00E+01 | 42 | 3.34E+03 | 4.45E+01 |

f9 | 51 | 1.77E+02 | 7.19E+00 | 51 | 1.47E+02 | 7.05E+00 |

f10 | 17 | 2.76E+03 | 1.42E+02 | 22 | 2.59E+03 | 1.48E+02 |

f11 | 51 | 1.17E+03 | 2.70E+01 | 51 | 1.22E+03 | 2.87E+01 |

f12 | 51 | 4.77E+02 | 1.85E+01 | 51 | 5.09E+02 | 2.00E+01 |

f13 | 51 | 8.72E+02 | 9.56E+00 | 51 | 9.66E+02 | 9.88E+00 |

f14 | 29 | 2.01E+03 | 2.38E+01 | 26 | 2.88E+03 | 1.80E+01 |

f15 | 51 | 9.12E+02 | 1.45E+01 | 51 | 1.05E+03 | 1.71E+01 |

f16 | 30 | 2.86E+03 | 1.61E+01 | 29 | 2.80E+03 | 1.65E+01 |

f17 | 12 | 2.82E+03 | 5.36E+01 | 7 | 3.74E+03 | 7.99E+01 |

f18 | 20 | 2.50E+03 | 6.02E+00 | 15 | 3.20E+03 | 8.67E+00 |

f19 | 51 | 1.41E+03 | 1.46E+01 | 51 | 1.42E+03 | 1.30E+01 |

f20 | 14 | 2.93E+03 | 3.33E+01 | 8 | 2.85E+03 | 6.11E+01 |

f21 | 48 | 2.36E+03 | 4.28E+01 | 50 | 2.25E+03 | 4.27E+01 |

f22 | 51 | 1.15E+02 | 6.68E+00 | 51 | 9.66E+01 | 6.54E+00 |

f23 | 50 | 2.01E+03 | 4.25E+01 | 51 | 1.86E+03 | 3.62E+01 |

f24 | 51 | 1.82E+03 | 4.62E+01 | 51 | 1.79E+03 | 3.94E+01 |

f25 | 51 | 1.27E+02 | 7.39E+00 | 51 | 1.04E+02 | 7.14E+00 |

f26 | 51 | 1.68E+03 | 3.07E+01 | 51 | 1.15E+03 | 1.80E+01 |

f27 | 51 | 3.44E+02 | 1.59E+01 | 51 | 2.53E+02 | 2.03E+01 |

f28 | 51 | 1.58E+02 | 1.15E+01 | 51 | 1.31E+02 | 9.79E+00 |

f29 | 29 | 2.77E+03 | 5.36E+01 | 24 | 2.72E+03 | 4.26E+01 |

f30 | 51 | 2.85E+02 | 1.26E+01 | 51 | 2.15E+02 | 1.34E+01 |

No | jSO | MjSO | ||||
---|---|---|---|---|---|---|

#runs | Mean CO | Mean PD | #runs | Mean CO | Mean PD | |

f1 | 51 | 1.20E+02 | 9.52E+00 | 51 | 1.32E+02 | 1.74E+01 |

f2 | 51 | 3.95E+02 | 1.28E+01 | 51 | 3.68E+02 | 1.32E+01 |

f3 | 51 | 2.43E+02 | 7.88E+00 | 51 | 2.52E+02 | 7.55E+00 |

f4 | 51 | 1.84E+02 | 9.18E+00 | 51 | 1.70E+02 | 8.41E+00 |

f5 | 44 | 2.97E+03 | 5.62E+01 | 44 | 2.83E+03 | 5.57E+01 |

f6 | 51 | 1.95E+02 | 7.93E+00 | 51 | 1.98E+02 | 7.67E+00 |

f7 | 35 | 2.99E+03 | 1.86E+01 | 34 | 3.03E+03 | 1.77E+01 |

f8 | 45 | 3.00E+03 | 5.46E+01 | 44 | 2.90E+03 | 5.47E+01 |

f9 | 51 | 1.90E+02 | 7.75E+00 | 51 | 1.89E+02 | 7.50E+00 |

f10 | 14 | 3.49E+03 | 1.29E+02 | 17 | 3.43E+03 | 1.54E+02 |

f11 | 51 | 1.84E+03 | 3.05E+01 | 51 | 1.85E+03 | 2.97E+01 |

f12 | 51 | 3.67E+02 | 1.08E+01 | 51 | 4.43E+02 | 1.25E+01 |

f13 | 51 | 7.42E+02 | 1.12E+01 | 51 | 9.27E+02 | 1.15E+01 |

f14 | 51 | 1.67E+03 | 1.33E+01 | 50 | 1.84E+03 | 1.43E+01 |

f15 | 51 | 8.34E+02 | 1.20E+01 | 51 | 1.13E+03 | 1.65E+01 |

f16 | 41 | 3.47E+03 | 8.42E+00 | 37 | 3.41E+03 | 1.12E+01 |

f17 | 15 | 3.58E+03 | 7.67E+00 | 10 | 3.52E+03 | 2.62E+01 |

f18 | 51 | 8.67E+02 | 9.25E+00 | 51 | 1.14E+03 | 9.77E+00 |

f19 | 51 | 1.46E+03 | 1.23E+01 | 51 | 1.63E+03 | 1.37E+01 |

f20 | 24 | 3.43E+03 | 2.04E+01 | 20 | 4.44E+03 | 2.19E+01 |

f21 | 47 | 2.94E+03 | 5.73E+01 | 43 | 2.93E+03 | 5.60E+01 |

f22 | 37 | 1.64E+03 | 3.82E+01 | 36 | 9.58E+02 | 3.10E+01 |

f23 | 48 | 2.69E+03 | 5.35E+01 | 47 | 2.64E+03 | 4.69E+01 |

f24 | 51 | 2.59E+03 | 4.76E+01 | 50 | 2.12E+03 | 3.38E+01 |

f25 | 51 | 1.94E+02 | 8.44E+00 | 51 | 1.69E+02 | 7.77E+00 |

f26 | 51 | 2.36E+03 | 3.69E+01 | 51 | 1.40E+03 | 1.43E+01 |

f27 | 51 | 3.00E+02 | 1.05E+01 | 51 | 2.15E+02 | 8.77E+00 |

f28 | 51 | 1.71E+02 | 9.04E+00 | 51 | 2.54E+02 | 1.03E+01 |

f29 | 32 | 3.34E+03 | 5.63E+01 | 23 | 3.34E+03 | 5.09E+01 |

f30 | 51 | 3.28E+02 | 1.11E+01 | 51 | 4.22E+02 | 3.08E+01 |

No | jSO | MjSO | ||||
---|---|---|---|---|---|---|

#runs | Mean CO | Mean PD | #runs | Mean CO | Mean PD | |

f1 | 51 | 1.37E+02 | 9.05E+00 | 51 | 1.94E+02 | 1.15E+01 |

f2 | 51 | 5.10E+02 | 1.09E+01 | 51 | 5.80E+02 | 1.04E+01 |

f3 | 51 | 3.75E+02 | 9.86E+00 | 51 | 4.08E+02 | 9.16E+00 |

f4 | 51 | 2.01E+02 | 9.29E+00 | 51 | 2.32E+02 | 9.04E+00 |

f5 | 50 | 3.93E+03 | 7.09E+01 | 49 | 4.03E+03 | 5.59E+01 |

f6 | 51 | 2.06E+02 | 9.48E+00 | 51 | 2.79E+02 | 8.85E+00 |

f7 | 46 | 4.12E+03 | 2.19E+01 | 43 | 4.02E+03 | 2.07E+01 |

f8 | 45 | 4.04E+03 | 6.76E+01 | 51 | 3.90E+03 | 6.33E+01 |

f9 | 51 | 2.00E+02 | 9.04E+00 | 51 | 2.62E+02 | 8.70E+00 |

f10 | 13 | 4.46E+03 | 3.06E+02 | 13 | 4.52E+03 | 2.13E+02 |

f11 | 51 | 5.27E+02 | 9.79E+00 | 51 | 1.17E+03 | 1.14E+01 |

f12 | 51 | 3.30E+02 | 1.02E+01 | 51 | 4.53E+02 | 9.08E+00 |

f13 | 51 | 7.08E+02 | 1.34E+01 | 51 | 9.23E+02 | 1.39E+01 |

f14 | 51 | 1.01E+03 | 1.02E+01 | 51 | 1.28E+03 | 1.30E+01 |

f15 | 51 | 4.84E+02 | 1.03E+01 | 51 | 9.30E+02 | 9.53E+00 |

f16 | 49 | 4.43E+03 | 1.16E+01 | 47 | 4.60E+03 | 9.92E+00 |

f17 | 33 | 4.66E+03 | 2.45E+01 | 49 | 4.04E+03 | 7.65E+01 |

f18 | 51 | 5.82E+02 | 9.80E+00 | 51 | 5.88E+02 | 1.40E+01 |

f19 | 51 | 6.38E+02 | 1.07E+01 | 51 | 1.48E+03 | 1.20E+01 |

f20 | 26 | 4.64E+03 | 1.16E+01 | 26 | 4.54E+03 | 7.67E+01 |

f21 | 49 | 3.87E+03 | 6.44E+01 | 51 | 2.41E+02 | 9.42E+00 |

f22 | 14 | 4.39E+03 | 2.67E+01 | 49 | 3.95E+03 | 6.22E+01 |

f23 | 50 | 1.72E+03 | 2.38E+01 | 51 | 7.70E+02 | 9.25E+00 |

f24 | 51 | 1.01E+03 | 1.18E+01 | 51 | 2.12E+02 | 8.49E+00 |

f25 | 51 | 2.11E+02 | 9.56E+00 | 51 | 2.53E+02 | 9.72E+00 |

f26 | 51 | 7.98E+02 | 9.50E+00 | 51 | 2.68E+02 | 8.62E+00 |

f27 | 51 | 3.31E+02 | 9.28E+00 | 51 | 2.93E+02 | 8.93E+00 |

f28 | 51 | 2.23E+02 | 9.42E+00 | 51 | 2.31E+02 | 8.99E+00 |

f29 | 51 | 4.38E+03 | 4.21E+01 | 51 | 4.13E+03 | 4.22E+01 |

f30 | 51 | 5.09E+02 | 1.08E+01 | 51 | 4.55E+02 | 1.01E+01 |

Rank | Name | F-Rank |
---|---|---|

0 | MjSO | 2.62 |

1 | jSO | 3.43 |

2 | SALSHADE-cnEPSin | 3.55 |

3 | EBLSHADE | 3.6 |

4 | ELSHADE-SPACMA | 3.82 |

5 | LSHADE | 3.98 |

Rank | Name | F-Rank |
---|---|---|

0 | MjSO | 2.1 |

1 | jSO | 3.4 |

2 | EBLSHADE | 3.43 |

3 | SALSHADE-cnEPSin | 3.95 |

4 | LSHADE | 4.05 |

5 | ELSHADE-SPACMA | 4.07 |

Rank | Name | F-Rank |
---|---|---|

0 | MjSO | 1.9 |

1 | ELSHADE-SPACMA | 3.27 |

2 | jSO | 3.4 |

3 | SALSHADE-cnEPSin | 4.05 |

4 | LSHADE | 4.15 |

5 | EBLSHADE | 4.23 |

Rank | Name | F-Rank |
---|---|---|

0 | MjSO | 1.87 |

1 | ELSHADE-SPACMA | 3.35 |

2 | SALSHADE-cnEPSin | 3.4 |

3 | jSO | 3.78 |

4 | EBLSHADE | 3.93 |

5 | LSHADE | 4.67 |

D | Chi-sq’ | Prob > Chi-sq’(p) | Critical Value |
---|---|---|---|

10 | 13.73819163 | 1.74E-02 | 11.07 |

30 | 31.07891492 | 9.04E-06 | 11.07 |

50 | 38.65745856 | 2.78E-07 | 11.07 |

100 | 38.16356513 | 3.50E-07 | 11.07 |

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ID | Functions | Optima |
---|---|---|

F1 | Shifted and Rotated Bent Cigar Function | 100 |

F2 | Shifted and Rotated Sum of Differential Power Function | 200 |

F3 | Shifted and Rotated Zakharov Function | 300 |

F4 | Shifted and Rotated Rosenbrock’s Function | 400 |

F5 | Shifted and Rotated Rastrigin’s Function | 500 |

F6 | Shifted and Rotated Expanded Scaffer’s F6 Function | 600 |

F7 | Shifted and Rotated Lunacek Bi_Rastrigin Function | 700 |

F8 | Shifted and Rotated Non-Continuous Rastrigin’s Function | 800 |

F9 | Shifted and Rotated Levy Function | 900 |

F10 | Shifted and Rotated Schwefel’s Function | 1000 |

F11 | Hybrid Function 1 (N = 3) | 1100 |

F12 | Hybrid Function 2 (N = 3) | 1200 |

F13 | Hybrid Function 3 (N = 3) | 1300 |

F14 | Hybrid Function 4 (N = 4) | 1400 |

F15 | Hybrid Function 5 (N = 4) | 1500 |

F16 | Hybrid Function 6 (N = 4) | 1600 |

F17 | Hybrid Function 6 (N = 5) | 1700 |

F18 | Hybrid Function 6 (N = 5) | 1800 |

F19 | Hybrid Function 6 (N = 5) | 1900 |

F20 | Hybrid Function 6 (N = 6) | 2000 |

F21 | Composition Function 1 (N = 3) | 2100 |

F22 | Composition Function 2 (N = 3) | 2200 |

F23 | Composition Function 3 (N = 4) | 2300 |

F24 | Composition Function 4 (N = 4) | 2400 |

F25 | Composition Function 5 (N = 5) | 2500 |

F26 | Composition Function 6 (N = 5) | 2600 |

F27 | Composition Function 7 (N = 6) | 2700 |

F28 | Composition Function 8 (N = 6) | 2800 |

F29 | Composition Function 9 (N = 3) | 2900 |

F30 | Composition Function 10 (N = 3) | 3000 |

Parameter Setting |
---|

MjSO $N{P}_{init}$ = $25\mathrm{log}D$, $N{P}_{fin}$ = 4, $H$ = 5, ${M}_{F}$ = 0.5, ${M}_{CR}$ = 0.5, $\left|A\right|$ = $NP$, $\xi ={10}^{-8},{P}_{max}$ = 0.25 ${P}_{min}$ = ${P}_{max}\u22152$ jSO $N{P}_{init}$ = $25\mathrm{log}D$, $N{P}_{fin}$ = 4, $H$ = 5, ${M}_{F}$ = 0.3, ${M}_{CR}$ = 0.8, $\left|A\right|$ = $NP$, $,{P}_{max}$ = 0.25 ${P}_{min}$ = ${P}_{max}\u22152$ LSHADE $N{P}_{init}$ = $18D$, $N{P}_{fin}$ = 4, $H$ = 6, ${M}_{F}$ = 0.5, ${M}_{CR}$ = 0.5, $\left|A\right|$ = $NP$, P = 0.11 EBLSAHDE $N{P}_{init}$ = $18D$, $N{P}_{fin}$ = 4, $H$ = 5, ${M}_{F}$ = 0.5, ${M}_{CR}$ = 0.5, $\left|A\right|$ = $NP$, P = 0.11 ELSHADE-SPACMA $N{P}_{init}$ = $18D$, $N{P}_{fin}$ = 4, $H$ = 5, ${F}_{cp}$ =0.5, $c$ = 0.8, ${P}_{init}$ = 0.3, ${p}_{min}$ = 0.15 SALSHADE-cnEPSin $N{P}_{init}$ = $18D$, $N{P}_{fin}$ = 4, $H$ = 5, ${M}_{F}$ = 0.5, ${M}_{CR}$ = 0.5, $freq=0.5$, $ps=0.5$, $pc=0.4$ |

**Table 3.**Algorithm comparison between five powerful differential evolution (DE) variants and our MjSO algorithm on $\mathrm{D}=10$ optimization under f1–f30 of our test suite.

NO | EBLSHADE | SALSHADE-cnEPSin | jSO | LSHADE | ELSHADE-SPACMA | MjSO | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |

f1 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f2 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f3 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f4 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f5 | >2.52E+00 | 8.98E-01 | >1.99E+00 | 6.62E-01 | >1.76E+00 | 7.60E–01 | >2.57E+00 | 8.37E-01 | >3.87E+00 | 2.02E+00 | 1.35E+00 | 9.10E-01 |

f6 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f7 | >1.22E+01 | 7.09E-01 | >1.19E+01 | 5.67E-01 | >1.18E+01 | 6.07E–01 | >1.22E+01 | 7.05E-01 | >1.33E+01 | 1.75E+00 | 1.15E+01 | 5.99E-01 |

f8 | >2.23E+00 | 9.02E-01 | >1.99E+00 | 7.63E-01 | >1.95E+00 | 7.44E–01 | >2.52E+00 | 6.99E-01 | >4.10E+00 | 2.51E+00 | 1.37E+00 | 5.87E-01 |

f9 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f10 | >2.50E+01 | 3.99E+01 | <7.36E+00 | 4.49E+01 | >3.59E+01 | 5.55E+01 | >3.86E+01 | 5.50E+01 | >2.27E+01 | 4.84E+01 | 1.53E+01 | 3.27E+01 |

f11 | >2.74E-01 | 5.48E-01 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | >3.65E-01 | 6.94E-01 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f12 | <1.22E+01 | 3.59E+01 | <1.19E+02 | 7.49E+01 | <2.66E+00 | 1.68E+01 | >3.89E+01 | 5.68E+01 | >2.85E+01 | 5.14E+01 | 2.79E+01 | 5.02E+01 |

f13 | >3.64E+00 | 2.23E+00 | >4.83E+00 | 2.30E+00 | >2.96E+00 | 2.35E+00 | >3.88E+00 | 2.37E+00 | >3.57E+00 | 2.21E+00 | 2.49E+00 | 2.50E+00 |

f14 | >5.38E-01 | 8.03E-01 | ≈0.00E+00 | 2.36E-01 | >5.85E–02 | 2.36E–01 | >7.73E-01 | 9.05E-01 | >7.80E-02 | 2.70E-01 | 0.00E+00 | 0.00E+00 |

f15 | ≈1.44E-01 | 2.03E-01 | ≈2.70E-01 | 2.03E+00 | ≈2.21E–01 | 2.00E–01 | ≈1.99E-01 | 2.12E-01 | ≈2.51E-01 | 2.17E-01 | 1.92E-01 | 2.20E-01 |

f16 | ≈4.34E-01 | 2.23E-01 | ≈6.25E-01 | 2.59E-01 | ≈5.69E–01 | 2.64E–01 | ≈3.83E-01 | 1.64E-01 | ≈5.62E-01 | 2.55E-01 | 5.64E-01 | 2.76E-01 |

f17 | ≈1.23E-01 | 1.51E-01 | ≈1.77E-01 | 2.41E-01 | ≈5.02E–01 | 3.48E–01 | ≈1.06E-01 | 1.31E-01 | ≈1.39E-01 | 1.44E-01 | 3.54E-01 | 3.11E-01 |

f18 | ≈1.79E-01 | 1.95E-01 | ≈4.49E-01 | 5.43E+00 | ≈3.08E–01 | 1.95E–01 | ≈2.15E-01 | 1.97E-01 | ≈7.10E-01 | 2.80E+00 | 3.06E-01 | 1.99E-01 |

f19 | ≈9.06E-03 | 1.09E-02 | ≈1.97E-02 | 3.01E-02 | ≈1.07E–02 | 1.25E–02 | ≈1.05E-02 | 1.10E-02 | ≈1.55E-02 | 1.14E-02 | 1.60E-02 | 2.30E-02 |

f20 | <1.22E-02 | 6.12E-02 | ≈3.12E-01 | 4.01E-01 | ≈3.43E–01 | 1.29E–01 | <1.22E-02 | 6.06E-02 | ≈1.41E-01 | 1.57E-01 | 3.12E-01 | 0.00E+00 |

f21 | >1.56E+02 | 5.12E+01 | ≈1.00E+02 | 5.13E+01 | >1.32E+02 | 4.84E+01 | >1.50E+02 | 5.14E+01 | ≈1.02E+02 | 1.48E+01 | 1.10E+02 | 3.11E+01 |

f22 | ≈1.00E+02 | 1.01E-01 | ≈1.00E+02 | 6.26E-02 | ≈1.00E+02 | 0.00E+00 | ≈1.00E+02 | 4.01E-02 | ≈1.00E+02 | 1.21E-01 | 1.00E+02 | 8.17E-14 |

f23 | >3.03E+02 | 1.71E+00 | >3.01E+02 | 1.43E+00 | >3.01E+02 | 1.59E+00 | >3.03E+02 | 1.65E+00 | >3.04E+02 | 2.30E+00 | 3.00E+02 | 9.63E-01 |

f24 | >3.16E+02 | 5.46E+01 | >3.29E+02 | 7.97E+01 | >2.97E+02 | 7.93E+01 | >3.21E+02 | 4.52E+01 | >2.91E+02 | 9.54E+01 | 2.42E+02 | 1.14E+02 |

f25 | >4.15E+02 | 2.24E+01 | >4.43E+02 | 2.22E+01 | >4.06E+02 | 1.75E+01 | >4.09E+02 | 1.95E+01 | >4.13E+02 | 2.18E+01 | 3.95E+02 | 1.37E+01 |

f26 | ≈3.00E+02 | 0.00E+00 | ≈3.00E+02 | 0.00E+00 | ≈3.00E+02 | 0.00E+00 | ≈3.00E+02 | 0.00E+00 | ≈3.00E+02 | 0.00E+00 | 3.00E+02 | 0.00E+00 |

f27 | >3.89E+02 | 1.39E-01 | >3.88E+02 | 1.66E+00 | >3.89E+02 | 2.26E–01 | >3.89E+02 | 1.78E-01 | >3.89E+02 | 1.67E-01 | 3.87E+02 | 1.63E+00 |

f28 | >3.47E+02 | 1.10E+02 | ≈3.00E+02 | 1.23E+02 | >3.39E+02 | 9.65E+01 | >3.58E+02 | 1.18E+02 | >3.25E+02 | 1.04E+02 | 3.00E+02 | 0.00E+00 |

f29 | >2.33E+02 | 2.65E+00 | ≈2.28E+02 | 1.56E+00 | >2.34E+02 | 2.96E+00 | >2.34E+02 | 2.78E+00 | ≈2.30E+02 | 2.26E+00 | 2.30E+02 | 2.10E+00 |

f30 | <3.24E+04 | 1.60E+05 | ≈3.94E+02 | 9.42E+04 | ≈3.95E+02 | 4.50E–02 | >4.05E+02 | 2.08E+01 | >4.02E+02 | 1.77E+01 | 3.96E+02 | 9.44E+00 |

**Table 4.**Algorithm comparison between five powerful DE variants and our MjSO algorithm on $\mathrm{D}=30$ optimization under f1–f30 of our test suite.

NO | EBLSHADE | SALSHADE-cnEPSin | jSO | LSHADE | ELSHADE-SPACMA | MjSO | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |

f1 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f2 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f3 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f4 | ≈5.86E+01 | 3.11E-14 | <4.90E+01 | 3.32E+00 | ≈5.87E+01 | 7.78E–01 | ≈5.86E+01 | 3.22E-14 | ≈5.86E+01 | 0.00E+00 | 5.86E+01 | 3.66E-14 |

f5 | ≈6.26E+00 | 1.29E+00 | >1.24E+01 | 2.39E+00 | >8.56E+00 | 2.10E+00 | ≈6.41E+00 | 1.52E+00 | >1.86E+01 | 8.04E+00 | 7.45E+00 | 2.20E+00 |

f6 | >6.04E-09 | 2.71E-08 | ≈0.00E+00 | 8.66E-08 | >6.04E–09 | 2.71E–08 | >2.68E-08 | 1.52E-07 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f7 | ≈3.73E+01 | 1.44E+00 | >4.32E+01 | 2.18E+00 | >3.89E+01 | 1.46E+00 | ≈3.71E+01 | 1.55E+00 | >3.89E+01 | 3.43E+00 | 3.75E+01 | 2.11E+00 |

f8 | ≈6.66E+00 | 1.54E+00 | >1.36E+01 | 2.21E+00 | >9.09E+00 | 1.84E+00 | ≈7.15E+00 | 1.58E+00 | >1.61E+01 | 7.46E+00 | 7.98E+00 | 1.62E+00 |

f9 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f10 | ≈1.42E+03 | 2.01E+02 | >1.47E+03 | 2.35E+02 | >1.53E+03 | 2.77E+02 | >1.50E+03 | 1.73E+02 | >1.70E+03 | 4.06E+02 | 1.42E+03 | 2.73E+02 |

f11 | >2.63E+01 | 2.82E+01 | >3.93E+00 | 1.76E+01 | >3.04E+00 | 2.65E+00 | >3.22E+01 | 2.85E+01 | >7.80E+00 | 1.42E+01 | 1.56E+00 | 1.36E+00 |

f12 | >9.45E+02 | 3.73E+02 | >3.43E+02 | 2.19E+02 | >1.70E+02 | 1.02E+02 | >1.00E+03 | 3.59E+02 | >2.47E+02 | 1.28E+02 | 1.32E+02 | 8.57E+01 |

f13 | >1.55E+01 | 4.88E+00 | >1.70E+01 | 5.24E+00 | >1.48E+01 | 4.83E+00 | >1.61E+01 | 4.97E+00 | >1.57E+01 | 5.03E+00 | 1.25E+01 | 8.99E+00 |

f14 | ≈2.11E+01 | 4.24E+00 | >2.20E+01 | 3.85E+00 | >2.18E+01 | 1.25E+00 | ≈2.14E+01 | 2.98E+00 | >2.37E+01 | 5.25E+00 | 2.16E+01 | 4.72E+00 |

f15 | >2.67E+00 | 1.43E+00 | >3.65E+00 | 1.75E+00 | ≈1.09E+00 | 6.91E–01 | >3.22E+00 | 1.32E+00 | ≈1.86E+00 | 1.29E+00 | 1.80E+00 | 1.27E+00 |

f16 | >3.84E+01 | 2.64E+01 | >1.88E+01 | 3.98E+01 | >7.89E+01 | 8.48E+01 | >6.52E+01 | 7.46E+01 | >6.68E+01 | 8.35E+01 | 1.55E+01 | 5.56E+00 |

f17 | >3.32E+01 | 5.24E+00 | >2.83E+01 | 5.88E+00 | >3.29E+01 | 8.08E+00 | >3.28E+01 | 6.36E+00 | >2.97E+01 | 6.76E+00 | 2.70E+01 | 6.12E+00 |

f18 | ≈2.07E+01 | 3.93E+00 | ≈2.06E+01 | 9.07E-01 | ≈2.04E+01 | 2.87E+00 | >2.21E+01 | 9.87E-01 | ≈2.09E+01 | 3.03E+00 | 2.08E+01 | 3.21E-01 |

f19 | >5.32E+00 | 1.65E+00 | >5.91E+00 | 1.89E+00 | >4.50E+00 | 1.73E+00 | >5.21E+00 | 1.58E+00 | >4.61E+00 | 1.35E+00 | 4.22E+00 | 1.32E+00 |

f20 | >3.08E+01 | 5.80E+00 | >3.08E+01 | 5.96E+00 | >2.94E+01 | 5.85E+00 | >3.10E+01 | 6.54E+00 | >2.73E+01 | 4.56E+00 | 2.63E+01 | 6.29E+00 |

f21 | >2.11E+02 | 1.67E+00 | >2.13E+02 | 2.07E+00 | >2.09E+02 | 1.96E+00 | ≈2.07E+02 | 1.47E+00 | >2.22E+02 | 6.64E+00 | 2.07E+02 | 1.72E+00 |

f22 | ≈1.00E+02 | 1.00E-13 | ≈1.00E+02 | 1.00E-13 | ≈1.00E+02 | 0.00E+00 | ≈1.00E+02 | 1.00E-13 | ≈1.00E+02 | 0.00E+00 | 1.00E+02 | 0.00E+00 |

f23 | >3.48E+02 | 2.81E+00 | >3.54E+02 | 4.11E+00 | >3.51E+02 | 3.30E+00 | >3.50E+02 | 3.10E+00 | >3.69E+02 | 1.05E+01 | 3.45E+02 | 3.66E+00 |

f24 | >4.25E+02 | 1.89E+00 | >4.29E+02 | 2.71E+00 | >4.26E+02 | 2.47E+00 | >4.26E+02 | 1.44E+00 | >4.41E+02 | 7.84E+00 | 4.22E+02 | 2.90E+00 |

f25 | ≈3.87E+02 | 2.71E-02 | ≈3.87E+02 | 6.82E-03 | ≈3.87E+02 | 7.68E–03 | ≈3.87E+02 | 2.47E-02 | ≈3.87E+02 | 9.60E-03 | 3.87E+02 | 5.67E-03 |

f26 | >8.97E+02 | 3.13E+01 | >9.51E+02 | 4.74E+01 | >9.20E+02 | 4.30E+01 | >9.51E+02 | 3.79E+01 | >1.08E+03 | 8.68E+01 | 8.91E+02 | 3.48E+01 |

f27 | >5.01E+02 | 5.44E+00 | >5.03E+02 | 4.01E+00 | >4.98E+02 | 7.00E+00 | >5.05E+02 | 4.81E+00 | >4.99E+02 | 6.15E+00 | 4.96E+02 | 5.69E+00 |

f28 | >3.26E+02 | 4.66E+01 | ≈3.05E+02 | 4.21E+01 | >3.09E+02 | 3.03E+01 | >3.33E+02 | 5.24E+01 | ≈3.02E+02 | 1.60E+01 | 3.02E+02 | 2.49E+01 |

f29 | >4.38E+02 | 6.17E+00 | >4.38E+02 | 1.05E+01 | >4.34E+02 | 1.36E+01 | >4.34E+02 | 8.45E+00 | >4.33E+02 | 1.56E+01 | 4.28E+02 | 1.06E+01 |

f30 | ≈1.98E+03 | 3.07E+01 | ≈1.97E+03 | 4.42E+01 | ≈1.97E+03 | 1.90E+01 | ≈1.99E+03 | 5.24E+01 | ≈1.98E+03 | 3.34E+01 | 1.97E+03 | 1.24E+01 |

**Table 5.**Algorithm comparison between five powerful DE variants and our MjSO algorithm on $\mathrm{D}=50$ optimization under f1–f30 of our test suite.

NO | EBLSHADE | SALSHADE-cnEPSin | jSO | LSHADE | ELSHADE-SPACMA | MjSO | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |

f1 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f2 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f3 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f4 | >7.00E+01 | 4.58E+01 | >3.69E+01 | 4.17E+01 | >5.62E+01 | 4.88E+01 | >7.07E+01 | 4.97E+01 | >4.36E+01 | 3.62E+01 | 3.00E+01 | 2.70E+01 |

f5 | ≈1.42E+01 | 1.78E+00 | >2.81E+01 | 5.30E+00 | >1.64E+01 | 3.46E+00 | ≈1.38E+01 | 2.95E+00 | ≈1.39E+01 | 5.55E+00 | 1.48E+01 | 3.28E+00 |

f6 | >6.94E-05 | 3.32E-04 | >9.52E-07 | 1.40E-06 | >1.09E–06 | 2.62E–06 | >6.12E-05 | 3.09E-04 | <0.00E+00 | 0.00E+00 | 1.83E-08 | 4.41E-08 |

f7 | ≈6.29E+01 | 1.98E+00 | >7.73E+01 | 5.54E+00 | ≈6.65E+01 | 3.47E+00 | ≈6.30E+01 | 1.85E+00 | ≈6.15E+01 | 3.86E+00 | 6.58E+01 | 3.26E+00 |

f8 | ≈1.22E+01 | 2.00E+00 | >2.64E+01 | 5.85E+00 | >1.70E+01 | 3.14E+00 | ≈1.20E+01 | 2.11E+00 | >1.79E+01 | 7.47E+00 | 1.29E+01 | 2.17E+00 |

f9 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f10 | >3.17E+03 | 3.06E+02 | >3.36E+03 | 3.02E+02 | >3.14E+03 | 3.67E+02 | >3.15E+03 | 2.59E+02 | >3.69E+03 | 6.07E+02 | 2.99E+03 | 3.14E+02 |

f11 | >4.37E+01 | 7.27E+00 | >2.81E+01 | 1.89E+00 | >2.79E+01 | 3.33E+00 | >4.80E+01 | 8.24E+00 | >2.62E+01 | 3.76E+00 | 2.44E+01 | 2.96E+00 |

f12 | >2.02E+03 | 5.00E+02 | >1.28E+03 | 3.64E+02 | >1.68E+03 | 5.23E+02 | >2.24E+03 | 5.22E+02 | >1.36E+03 | 3.42E+02 | 8.51E+02 | 3.87E+02 |

f13 | >6.40E+01 | 3.47E+01 | >8.68E+01 | 2.97E+01 | >3.06E+01 | 2.12E+01 | >6.41E+01 | 2.65E+01 | >3.68E+01 | 1.72E+01 | 2.56E+01 | 1.96E+01 |

f14 | >2.78E+01 | 2.27E+00 | >2.65E+01 | 2.35E+00 | ≈2.50E+01 | 1.87E+00 | >2.98E+01 | 3.01E+00 | >3.07E+01 | 3.95E+00 | 2.52E+01 | 2.53E+00 |

f15 | >3.41E+01 | 9.07E+00 | >2.63E+01 | 3.57E+00 | >2.39E+01 | 2.49E+00 | >4.01E+01 | 1.06E+01 | >2.28E+01 | 2.20E+00 | 2.11E+01 | 1.67E+00 |

f16 | >3.54E+02 | 1.09E+02 | >3.29E+02 | 1.11E+02 | >4.51E+02 | 1.38E+02 | >3.77E+02 | 1.24E+02 | >4.15E+02 | 1.77E+02 | 2.85E+02 | 1.22E+02 |

f17 | >2.64E+02 | 6.33E+01 | >2.76E+02 | 5.33E+01 | >2.83E+02 | 8.61E+01 | ≈2.51E+02 | 5.71E+01 | ≈2.30E+02 | 9.68E+01 | 2.48E+02 | 8.68E+01 |

f18 | >3.31E+01 | 7.86E+00 | >2.50E+01 | 2.09E+00 | >2.43E+01 | 2.02E+00 | >3.98E+01 | 8.64E+00 | >2.51E+01 | 2.56E+00 | 2.24E+01 | 1.45E+00 |

f19 | >1.93E+01 | 3.25E+00 | >1.81E+01 | 3.41E+00 | >1.41E+01 | 2.26E+00 | >2.32E+01 | 5.94E+00 | >1.44E+01 | 2.31E+00 | 1.26E+01 | 2.58E+00 |

f20 | >1.72E+02 | 6.94E+01 | >1.31E+02 | 2.64E+01 | >1.40E+02 | 7.74E+01 | >1.73E+02 | 7.15E+01 | >1.08E+02 | 7.31E+01 | 1.00E+02 | 3.35E+01 |

f21 | >2.21E+02 | 2.55E+00 | >2.26E+02 | 6.04E+00 | >2.19E+02 | 3.77E+00 | ≈2.12E+02 | 2.25E+00 | >2.42E+02 | 9.52E+00 | 2.14E+02 | 3.99E+00 |

f22 | >2.67E+03 | 1.59E+03 | >1.00E+03 | 1.70E+03 | >1.49E+03 | 1.75E+03 | >2.68E+03 | 1.62E+03 | ≈7.86E+02 | 1.64E+03 | 7.67E+02 | 1.42E+03 |

f23 | >4.67E+02 | 4.38E+00 | >4.41E+02 | 7.07E+00 | >4.30E+02 | 6.24E+00 | >4.30E+02 | 4.91E+00 | >4.62E+02 | 1.39E+01 | 4.27E+02 | 5.86E+00 |

f24 | >5.05E+02 | 3.51E+00 | >5.14E+02 | 6.05E+00 | >5.07E+02 | 4.13E+00 | >5.06E+02 | 2.55E+00 | >5.34E+02 | 9.14E+00 | 4.98E+02 | 3.50E+00 |

f25 | >4.88E+02 | 2.01E+01 | >4.88E+02 | 1.54E+00 | ≈4.81E+02 | 2.80E+00 | >4.84E+02 | 1.29E+01 | ≈4.81E+02 | 2.80E+00 | 4.80E+02 | 1.81E-02 |

f26 | >1.13E+03 | 4.45E+01 | >1.25E+03 | 9.13E+01 | >1.13E+03 | 5.62E+01 | >1.14E+03 | 4.93E+01 | >1.34E+03 | 1.38E+02 | 1.05E+03 | 4.64E+01 |

f27 | >5.27E+02 | 1.09E+01 | >5.23E+02 | 8.58E+00 | ≈5.11E+02 | 1.11E+01 | >5.31E+02 | 1.67E+01 | ≈5.10E+02 | 9.52E+00 | 5.18E+02 | 1.43E+01 |

f28 | >4.73E+02 | 2.23E+01 | >4.67E+02 | 6.78E+00 | ≈4.60E+02 | 6.84E+00 | >4.71E+02 | 2.15E+01 | ≈4.60E+02 | 6.84E+00 | 4.59E+02 | 2.91E-13 |

f29 | >3.62E+02 | 1.04E+01 | >3.61E+02 | 1.07E+01 | >3.63E+02 | 1.32E+01 | ≈3.50E+02 | 1.09E+01 | >3.58E+02 | 1.78E+01 | 3.53E+02 | 1.21E+01 |

f30 | >6.54E+05 | 7.78E+04 | >6.48E+05 | 5.85E+04 | ≈6.01E+05 | 2.99E+04 | >6.58E+05 | 8.12E+04 | ≈5.97E+05 | 2.38E+04 | 6.02E+05 | 3.07E+04 |

**Table 6.**Algorithm comparison between five powerful DE variants and our MjSO algorithm on $\mathrm{D}=100$ optimization under f1–f30 of our test suite.

NO | EBLSHADE | SALSHADE-cnEPSin | jSO | LSHADE | ELSHADE-SPACMA | MjSO | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |

f1 | ≈0.00E+00 | 0.00E+00 | >1.36E-08 | 2.95E-08 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f2 | <6.44E+00 | 1.54E+01 | >1.49E+11 | 7.72E+11 | <8.94E+00 | 2.42E+01 | >8.42E+05 | 6.01E+06 | >7.52E+07 | 6.01E+07 | 3.27E+01 | 5.92E+01 |

f3 | >1.56E-06 | 1.46E-06 | <0.00E+00 | 0.00E+00 | >2.39E–06 | 2.73E–06 | >6.90E-06 | 6.77E-06 | ≈1.60E-07 | 4.00E-07 | 8.81E-07 | 7.38E-07 |

f4 | ≈1.84E+02 | 5.87E+01 | >2.01E+02 | 7.91E+00 | ≈1.90E+02 | 2.89E+01 | ≈1.97E+02 | 1.58E+01 | >2.01E+02 | 8.67E+00 | 1.94E+02 | 7.94E+00 |

f5 | >4.14E+01 | 3.80E+00 | >6.19E+01 | 1.05E+01 | >4.39E+01 | 5.61E+00 | >3.83E+01 | 4.90E+00 | >3.78E+01 | 5.85E+00 | 2.82E+01 | 9.53E+00 |

f6 | >1.22E-02 | 6.81E-03 | >5.65E-05 | 3.51E-05 | >2.02E-04 | 6.20E–04 | >5.71E-03 | 3.43E-03 | <0.00E+00 | 1.34E-08 | 1.05E-06 | 9.32E-07 |

f7 | >1.40E+02 | 4.25E+00 | >1.71E+02 | 7.36E+00 | >1.45E+02 | 6.70E+00 | >1.41E+02 | 4.46E+00 | >1.51E+02 | 1.48E+00 | 1.34E+02 | 6.19E+00 |

f8 | >3.73E+01 | 6.67E+00 | >6.20E+01 | 9.99E+00 | >4.22E+01 | 5.52E+00 | >3.86E+01 | 4.47E+00 | >2.98E+01 | 1.32E+01 | 2.80E+01 | 1.00E+01 |

f9 | >6.33E-01 | 4.60E-01 | ≈0.00E+00 | 0.00E+00 | >4.59E-02 | 1.15E–01 | >4.86E-01 | 4.83E-01 | ≈0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

f10 | >1.03E+04 | 4.51E+02 | >1.05E+04 | 5.30E+02 | >9.70E+03 | 6.82E+02 | >1.04E+04 | 5.45E+02 | >1.08E+04 | 9.53E+02 | 9.60E+03 | 7.03E+02 |

f11 | >3.71E+02 | 1.03E+02 | >4.54E+01 | 4.79E+01 | >1.13E+02 | 4.32E+01 | >4.52E+02 | 8.94E+01 | >7.34E+01 | 4.30E+01 | 3.01E+01 | 4.79E+00 |

f12 | >2.28E+04 | 5.65E+03 | >6.72E+03 | 9.17E+02 | >1.84E+04 | 8.35E+03 | >2.49E+04 | 9.97E+03 | >7.79E+03 | 2.92E+03 | 5.38E+03 | 1.32E+03 |

f13 | >2.33E+02 | 6.04E+01 | >1.04E+02 | 3.72E+01 | >1.45E+02 | 3.80E+01 | >5.70E+02 | 4.14E+02 | >1.49E+02 | 3.83E+01 | 5.58E+01 | 2.54E+01 |

f14 | >2.36E+02 | 1.88E+01 | >5.12E+01 | 6.90E+00 | >6.43E+01 | 1.09E+01 | >2.51E+02 | 2.94E+01 | >4.75E+01 | 5.69E+00 | 3.84E+01 | 4.16E+00 |

f15 | >2.65E+02 | 3.97E+01 | >9.37E+01 | 3.19E+01 | >1.62E+02 | 3.81E+01 | >2.57E+02 | 4.02E+01 | >1.08E+02 | 4.34E+01 | 5.55E+01 | 1.54E+01 |

f16 | >1.50E+03 | 3.54E+02 | >1.51E+03 | 1.90E+02 | >1.86E+03 | 3.49E+02 | >1.66E+03 | 2.78E+02 | >1.76E+03 | 4.88E+02 | 1.38E+03 | 3.43E+02 |

f17 | >1.13E+03 | 2.25E+02 | >1.14E+02 | 1.75E+02 | >1.28E+03 | 2.38E+02 | >1.16E+03 | 1.94E+02 | >1.27E+03 | 3.45E+02 | 9.17E+02 | 2.21E+02 |

f18 | >2.65E+02 | 4.96E+01 | >6.90E+01 | 1.55E+01 | >1.67E+02 | 3.65E+01 | >2.41E+02 | 5.66E+01 | >1.05E+02 | 2.56E+01 | 6.00E+01 | 1.36E+01 |

f19 | >1.62E+02 | 1.78E+01 | >5.71E+01 | 7.301+00 | >1.05E+02 | 2.01E+01 | >1.78E+02 | 2.42E+01 | >6.05E+01 | 7.55E+00 | 4.73E+01 | 5.00E+00 |

f20 | >1.63E+03 | 1.86E+02 | >1.41E+03 | 1.88E+02 | >1.38E+03 | 2.43E+02 | >1.56E+03 | 2.04E+02 | >1.28E+03 | 2.54E+02 | 1.26E+03 | 2.69E+02 |

f21 | >2.59E+02 | 3.33E+00 | >2.88E+02 | 1.45E+01 | >2.64E+02 | 6.43E+00 | >2.59E+02 | 6.03E+00 | >2.96E+02 | 1.65E+01 | 2.51E+02 | 7.79E+00 |

f22 | >1.14E+04 | 5.60E+02 | >1.08E+04 | 5.81E+02 | >1.02E+04 | 2.18E+03 | >1.13E+04 | 5.65E+02 | >9.70E+03 | 1.20E+03 | 3.53E+01 | 6.04E+00 |

f23 | <5.71E+02 | 6.68E+00 | <5.92E+02 | 8.64E+00 | <5.71E+02 | 1.07E+01 | <5.66E+02 | 9.02E+00 | <6.03E+02 | 2.19E+01 | 1.13E+03 | 2.49E+01 |

f24 | >9.01E+02 | 5.70E+00 | >9.19E+02 | 1.21E+01 | >9.02E+02 | 7.89E+00 | >9.20E+02 | 6.78E+00 | >9.32E+02 | 1.90E+01 | 2.95E+02 | 2.18E+01 |

f25 | >7.50E+02 | 2.58E+01 | >7.21E+02 | 4.62E+01 | >7.36E+02 | 3.53E+01 | >7.53E+02 | 2.58E+01 | >7.00E+02 | 3.99E+01 | 6.68E+02 | 1.80E+01 |

f26 | >3.22E+03 | 6.88E+01 | >3.15E+03 | 1.73E+02 | >3.27E+03 | 8.02E+01 | >3.43E+03 | 8.34E+01 | >3.24E+03 | 2.19E+02 | 3.00E+02 | 3.22E-13 |

f27 | <6.19E+02 | 1.68E+01 | <5.88E+02 | 1.75E+01 | <5.85E+02 | 2.17E+01 | <6.43E+02 | 1.70E+01 | <5.62E+02 | 1.75E+01 | 1.67E+03 | 1.35E+02 |

f28 | >5.32E+02 | 2.58E+01 | >5.16E+02 | 1.91E+01 | >5.27E+02 | 2.73E+01 | >5.27E+02 | 2.15E+01 | >5.21E+02 | 2.38E+01 | 5.04E+02 | 1.70E+01 |

f29 | >1.12E+03 | 1.54E+02 | >1.14E+03 | 1.40E+02 | >1.26E+03 | 1.91E+02 | >1.27E+03 | 1.76E+02 | >1.21E+03 | 1.98E+02 | 9.89E+02 | 1.84E+02 |

f30 | <2.39E+03 | 1.39E+02 | <2.33E+03 | 1.66E+02 | <2.33E+03 | 1.19E+02 | <2.41E+03 | 1.52E+02 | <2.25E+03 | 1.11E+02 | 3.32E+03 | 8.57E+01 |

MjSO vs. | $\mathbf{D}=10$ | $\mathbf{D}=30$ | $\mathbf{D}=50$ | $\mathbf{D}=100$ | |
---|---|---|---|---|---|

EBLSHADE | $>$(better) | 14 | 16 | 23 | 24 |

$\approx $(no sig) | 13 | 14 | 7 | 2 | |

$<$(worse) | 3 | 0 | 0 | 4 | |

SALSHADE-cnEPSin | $>$(better) | 8 | 19 | 26 | 25 |

$\approx $(no sig) | 20 | 10 | 4 | 1 | |

$<$(worse) | 2 | 1 | 0 | 4 | |

jSO | $>$(better) | 13 | 20 | 20 | 24 |

$\approx $(no sig) | 16 | 10 | 10 | 2 | |

$<$(worse) | 1 | 0 | 0 | 4 | |

LSHADE | $>$(better) | 16 | 17 | 20 | 25 |

$\approx $(no sig) | 13 | 13 | 10 | 2 | |

$<$(worse) | 1 | 0 | 0 | 3 | |

ELSHADE-SPACMA | $>$(better) | 13 | 18 | 17 | 23 |

$\approx $(no sig) | 17 | 12 | 12 | 3 | |

$<$(worse) | 0 | 0 | 1 | 4 |

Algorithm | Variable | Target Cost | |||
---|---|---|---|---|---|

${\mathit{T}}_{\mathit{s}}$ | ${\mathit{T}}_{\mathit{h}}$ | $\mathit{R}$ | $\mathit{L}$ | ||

DE | 0.8231 | 0.4453 | 42.9230 | 176.7356 | 6301.5664 |

LSHADE | 0.8168 | 0.4472 | 42.1412 | 177.1231 | 6138.8931 |

EBLSHADE | 0.7802 | 0.3856 | 40.4292 | 198.4964 | 5889.3216 |

ELSHADE-SPACMA | 0.8125 | 0.4375 | 42.0913 | 176.7465 | 6061.0777 |

SALSHADE-cnEPSin | 0.7929 | 0.3914 | 41.1773 | 188.3950 | 5912.7115 |

jSO | 0.8036 | 0.3972 | 41.6392 | 182.4120 | 5930.3137 |

MjSO | 0.7782 | 0.3847 | 40.3201 | 199.9975 | 5885.5226 |

Algorithm | Variable | Target Weight | ||
---|---|---|---|---|

d | D | N | ||

DE | 0.0592 | 0.4983 | 8.8980 | 0.0172 |

LSHADE | 0.0524 | 0.3532 | 11.6824 | 0.0133 |

EBLSHADE | 0.0500 | 0.3171 | 14.1417 | 0.0127 |

ELSHADE-SPACMA | 0.0519 | 0.3487 | 11.8145 | 0.0129 |

SALSHADE-cnEPSin | 0.0503 | 0.3159 | 14.250 | 0.0128 |

jSO | 0.0562 | 0.4754 | 6.6670 | 0.0130 |

MjSO | 0.0516 | 0.3597 | 11.2880 | 0.0126 |

Algorithm | Variable | Target Cost | |||
---|---|---|---|---|---|

$\mathit{h}$ | $\mathit{l}$ | $\mathit{t}$ | $\mathit{b}$ | ||

DE | 0.2389 | 3.4067 | 9.6383 | 0.2901 | 2.0701 |

LSHADE | 0.2134 | 3.5601 | 8.4629 | 0.2346 | 1.8561 |

EBLSHADE | 0.2087 | 6.7221 | 9.3673 | 0.4217 | 1.7583 |

ELSHADE-SPACMA | 0.1947 | 3.7831 | 9.1234 | 0.2077 | 1.7796 |

SALSHADE-cnEPSin | 0.2023 | 3.5442 | 9.0366 | 0.2057 | 1.7280 |

jSO | 0.2147 | 3.3841 | 8.8103 | 0.2195 | 1.7890 |

MjSO | 0.2057 | 3.4704 | 9.0366 | 0.2057 | 1.7248 |

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## Share and Cite

**MDPI and ACS Style**

Shen, Y.; Liang, Z.; Kang, H.; Sun, X.; Chen, Q.
A Modified jSO Algorithm for Solving Constrained Engineering Problems. *Symmetry* **2021**, *13*, 63.
https://doi.org/10.3390/sym13010063

**AMA Style**

Shen Y, Liang Z, Kang H, Sun X, Chen Q.
A Modified jSO Algorithm for Solving Constrained Engineering Problems. *Symmetry*. 2021; 13(1):63.
https://doi.org/10.3390/sym13010063

**Chicago/Turabian Style**

Shen, Yong, Ziyuan Liang, Hongwei Kang, Xingping Sun, and Qingyi Chen.
2021. "A Modified jSO Algorithm for Solving Constrained Engineering Problems" *Symmetry* 13, no. 1: 63.
https://doi.org/10.3390/sym13010063