# On the Efficacy of Ensemble of Constraint Handling Techniques in Self-Adaptive Differential Evolution

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## Abstract

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## 1. Introduction

## 2. Constrained Optimization Problem and ECHT

#### 2.1. Constrained Optimization Problem (COP)

#### 2.2. Superiority of Feasible Solutions (SF)

- Parent, ${\mathbf{x}}^{i}$ is feasible and offspring, ${\mathbf{x}}^{j}$ is infeasible.
- Both parent and offspring, ${\mathbf{x}}^{i}$ and ${\mathbf{x}}^{j}$ are feasible, but parent, ${\mathbf{x}}^{i}$ has minimum fitness value than the offspring, ${\mathbf{x}}^{j}$.
- Both ${\mathbf{x}}^{i}$ and ${\mathbf{x}}^{j}$ are infeasible, and overall constraints’ violation $v\left({\mathbf{x}}^{i}\right)$ of parent, ${\mathbf{x}}^{i}$ is less than overall constraints’ violation $v\left({\mathbf{x}}^{j}\right)$ of offspring, ${\mathbf{x}}^{j}$, where $v\left({\mathbf{x}}^{i}\right)$ and $v\left({\mathbf{x}}^{j}\right)$ are calculated by using Equation (2).

#### 2.3. Self-Adaptive Penalty (SP)

#### 2.4. The $\epsilon $-Constraint (EC) Handling Technique

#### 2.5. Stochastic Ranking (SR)

## 3. JADE-ECHT and SaDE-ECHT

#### 3.1. JADE-ECHT

**Step****1:**- generate initial population P, set the generation number $t=1$, initial crossover probability ${\mu}_{CR}=0.5$, initial mutation factor ${\mu}_{F}=0.5$, the set of archive inferior solutions ${A}_{i}=\varnothing $, the sets of successful mutation factors and crossovers, ${S}_{F}^{i}=\varnothing $, ${S}_{CR}^{i}=\varnothing $, respectively, where $i=1,\dots ,4$.
**Step****2:**- divide population P into four subpopulations, ${P}_{i},i=1,\dots ,4$ each of size $PS$ (population size to be tackled by each CHT). Set parameters ${PAR}_{i}$, $i=1,\dots ,4$ of $PS$ individuals each with dimension D according to the rules of JADE and corresponding CHT. Also, calculate ${F}_{l}^{i}$ and $C{R}_{l}^{i},\forall \phantom{\rule{0.166667em}{0ex}}l\in \{1,\dots ,PS\}$, where $C{R}_{l}^{i}=rand{n}_{l}^{i}\phantom{\rule{0.166667em}{0ex}}({\mu}_{CR},0.1)$, ${F}_{l}^{i}=rand{n}_{l}^{i}\phantom{\rule{0.166667em}{0ex}}({\mu}_{CR},0.1)$.
**Step****3:****Step****4:**- each parent subpopulation $({P}_{i},i=1,\dots ,4)$ generates offspring subpopulation $(OF{F}_{i},i=1,\dots ,4)$ as a result of applying mutation and crossover operators, respectively as follows [15]:$${\mathbf{v}}_{l,t}^{i}={\mathbf{x}}_{l,t}^{i}+{F}_{l}^{i}\cdot ({\mathbf{x}}_{pbest,t}^{i}-{\mathbf{x}}_{l,t}^{i})+{F}_{l}^{i}\cdot ({\mathbf{x}}_{{r}_{1}^{l},t}^{i}-{\tilde{\mathbf{x}}}_{{r}_{2}^{l},t}^{i}),$$$$\begin{array}{c}\hfill {u}_{l,t}^{i}=\left(\right)open="\{"\; close>\begin{array}{cc}{v}_{l,t}^{i},\hfill & \mathrm{if}(ran{d}_{j}[0,1]{\mu}_{CR}or(j={j}_{rand})\hfill \\ {x}_{l,t}^{i},\hfill & \mathrm{otherwise}.\hfill \end{array}\end{array}$$In Equation (4), ${v}_{l,t}^{i}$ and ${x}_{l,t}^{i}$ are the $l\mathrm{th}$ components of the $i$ mutant and trial vectors in generation t.
**Step****5:****Step****6:**- each parent subpopulation is grouped together with its own offspring and the offspring produced by the remaining three subpopulations corresponding to different CHTs. This way four different groups of populations are generated.
**Step****7:**- Parents population P for the next generation is selected from the four groups according to the rule of each CHT. Unsuccessful parents are added to the archive ${A}_{i}$. All successful crossover probabilities from $C{R}_{PS}^{i}$ and mutation factors from ${F}_{PS}^{i}$ are added to ${S}_{CR}^{i}$ and ${S}_{F}^{i}$.
**Step****8:**- Remove solutions randomly from ${A}_{i}$ so that $|{A}_{i}|\le PS$. Update ${\mu}_{CR}$ and ${\mu}_{F}$ adopting the formulations of [28].
**Step****9:**- If the stopping criteria are not met, go to Step 2; otherwise, stop.

#### 3.2. SaDE-ECHT

**Step****1:**- generate initial population P, initiate the generation counter $t=1$, initial crossover probability ${\mu}_{CR}=0.5$, initial mutation factor ${\mu}_{F}=0.5$.
**Step****2:**- divide population P into four subpopulations, ${P}_{i},i=1,\dots ,4$ each of size $PS$. Set parameters ${PAR}_{i}$, where $i=1,\dots ,4$ of $PS$ individuals each with dimension D and generate F in [0,2] and $CR$ in (0,1) by using normal distribution according to the rules of SaDE and corresponding CHT.
**Step****3:****Step****4:**- each parent in each subpopulation produces offspring by using one of the four mutation strategies, DE/rand/1, DE/current-to-best/2, DE/rand/2, and DE/current-to-rand/1 (for details of these strategies, please see [10]) and crossover given in Equation (4). For first 20 generations, probabilities are fixed and set to ${p}_{1}={p}_{2}={p}_{3}={p}_{4}=0.25$. Afterwards, the Roulette Wheel selection is adopted to update the respective probability ${p}_{i}$ as follows [10]:$${p}_{i}=\frac{n{s}_{i}}{n{s}_{i}+n{f}_{i}},i=1,2,3,4$$
**Step****5:****Step****6:**- each parent subpopulation is grouped together with its own offspring and the offspring produced by the remaining three subpopulations corresponding to different CHTs. This way four different groups of populations are generated.
**Step****7:**- parents population P for the next generation are selected from the four groups according to the rule of each CHT.
**Step****8:**- recalculate crossover probability after every five generations according to the mean of recorded $CR$ values.
**Step****9:**- if the stopping criteria are not met, go to Step 2; otherwise, stop.

## 4. Experimental Results

#### Result Achieved

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Flowchart of self-adaptive differential evolution with optional external archive (JADE)-ensemble of constraint handling techniques (ECHT).

**Figure 4.**Constraint violation comparison of JADE-ECHT and self-adaptive differential evolution (SaDE)-ECHT for g01–g06.

System | Windows 8 |
---|---|

CPU | 3.00 GHz |

Ram | 2 GB |

Language | MATLAB 2012, 8.0.0.783 |

Parameters’ Description | Parameters’ Settings |
---|---|

Population size for each CHT | $PS=25$ |

Whole population size | $NP=4\ast PS=100$ |

Maximum number of generations | $t=2500$ |

Total number of runs | $runs=25$ |

Initial value of mutation factor | ${\mu}_{F}=0.5$ |

Initial value of crossover probability | ${\mu}_{CR}=0.5$ |

Termination criterion based on maximum function evaluations | $max\_FEs=\mathrm{500,000}$. |

**Table 3.**Comparison of self-adaptive differential evolution with optional external archive (JADE)-ensemble of constraint handling techniques (ECHT) and self-adaptive differential evolution (SaDE)-ECHT after FES = 500,000 for g01–g06. The bold numbers indicate the better results.

Prob | Algorithm | Best | Median | Worst | c | $\overline{\mathit{v}}$ | Mean | Std | FR | SR |
---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{g}01$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $2.0000\left(0\right)$ | $0,0,0$ | 0 | $0.0800$ | $0.4000$ | $100\%$ | $96\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $100\%$ | 100% | |

$\mathrm{g}02$ | JADE-ECHT | $0.0001\left(0\right)$ | $\mathbf{0.0004}\left(\mathbf{0}\right)$ | $\mathbf{0.0276}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0.0064}$ | $\mathbf{0.0091}$ | $100\%$ | $16\%$ |

SaDE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $0.0110\left(0\right)$ | $0.1263\left(0\right)$ | $0,0,0$ | 0 | $0.0191$ | $0.0254$ | $100\%$ | 24% | |

$\mathrm{g}03$ | JADE-ECHT | $0.0250\left(0\right)$ | $0.1015\left(0\right)$ | $0.4245\left(0\right)$ | $0,0,0$ | 0 | $0.1385$ | $0.1036$ | $100\%$ | $0\%$ |

SaDE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0.0122}\left(\mathbf{0}\right)$ | $\mathbf{0.1524}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0.0243}$ | $\mathbf{0.0343}$ | $100\%$ | 12% | |

$\mathrm{g}04$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ | |

$\mathrm{g}05$ | JADE-ECHT | $0\left(0\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $100\%$ | 100% |

SaDE-ECHT | $0\left(0\right)$ | $91.4773\left(0\right)$ | $515.4900\left(0\right)$ | $0,0,0$ | 0 | $110.1546$ | $101.2496$ | $100\%$ | $4\%$ | |

$\mathrm{g}06$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

**Table 4.**Comparison of JADE-ECHT and SaDE-ECHT after FES = 500,000 for g07–g12. The bold numbers indicate the better results.

Prob | Algorithm | Best | Median | Worst | c | $\overline{\mathit{v}}$ | Mean | Std | FR | SR |
---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{g}07$ | JADE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $0.0879\left(0\right)$ | $\mathbf{0.2651}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $0.0976$ | $\mathbf{0.0726}$ | $100\%$ | 4% |

SaDE-ECHT | $0.0001\left(0\right)$ | $\mathbf{0.0114}\left(\mathbf{0}\right)$ | $0.3230\left(0\right)$ | $0,0,0$ | 0 | $\mathbf{0.0518}$ | $0.0850$ | $100\%$ | $0\%$ | |

$\mathrm{g}08$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ | |

$\mathrm{g}09$ | JADE-ECHT | $0\left(0\right)$ | $0.0039\left(0\right)$ | $0.0714\left(0\right)$ | $0,0,0$ | 0 | $0.0132$ | $0.0192$ | $100\%$ | $20\%$ |

SaDE-ECHT | $0\left(0\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0.0006}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0.0001}$ | $\mathbf{0.0002}$ | $100\%$ | 76% | |

$\mathrm{g}10$ | JADE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $133.9677\left(0\right)$ | $343.5425\left(0\right)$ | $0,0,0$ | 0 | $143.0809$ | $105.4501$ | $100\%$ | 4% |

SaDE-ECHT | $0.0012\left(0\right)$ | $\mathbf{0.1709}\left(\mathbf{0}\right)$ | $\mathbf{11.9004}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{1.1748}$ | $\mathbf{3.0352}$ | $100\%$ | $0\%$ | |

$\mathrm{g}11$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ | |

$\mathrm{g}12$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

**Table 5.**Comparison of JADE-ECHT and SaDE-ECHT after FES = 500,000 for g13–g18. The bold numbers indicate the better results.

Prob | Algorithm | Best | Median | Worst | c | $\overline{\mathit{v}}$ | Mean | Std | FR | SR |
---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{g}13$ | JADE-ECHT | $0.3849\left(0\right)$ | $0.9118\left(0\right)$ | $0.9459\left(0\right)$ | $0,0,0$ | 0 | $0.8275$ | $\mathbf{0.1750}$ | $100\%$ | $0\%$ |

SaDE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0.3870}\left(\mathbf{0}\right)$ | $\mathbf{0.8491}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0.3608}$ | $0.2828$ | $100\%$ | 4% | |

$\mathrm{g}14$ | JADE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0.0174}\left(\mathbf{0}\right)$ | $5.5402\left(0\right)$ | $0,0,0$ | 0 | $1.9415$ | $2.2940$ | $100\%$ | 40% |

SaDE-ECHT | $0.4527\left(0\right)$ | $1.6397\left(0\right)$ | $\mathbf{3.3912}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{1.7600}$ | $\mathbf{0.6956}$ | $100\%$ | $0\%$ | |

$\mathrm{g}15$ | JADE-ECHT | $0\left(0\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $100\%$ | 100% |

SaDE-ECHT | $0\left(0\right)$ | $0.0009\left(0\right)$ | $2.5449\left(0\right)$ | $0,0,0$ | 0 | $0.3333$ | $0.6971$ | $100\%$ | $44\%$ | |

$\mathrm{g}16$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ | |

$\mathrm{g}17$ | JADE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{74.0580}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{8.8870}$ | $\mathbf{24.5623}$ | $100\%$ | 88% |

SaDE-ECHT | $7.9251\left(0\right)$ | $91.2351\left(0\right)$ | $297.1687\left(0\right)$ | $0,0,0$ | 0 | $92.9967$ | $50.4589$ | $100\%$ | $0\%$ | |

$\mathrm{g}18$ | JADE-ECHT | $0\left(0\right)$ | $0.0001\left(0\right)$ | $0.0206\left(0\right)$ | $0,0,0$ | 0 | $0.0011$ | $0.0041$ | $100\%$ | $52\%$ |

SaDE-ECHT | $0\left(0\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{0}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $100\%$ | 100% |

**Table 6.**Comparison of JADE-ECHT and SaDE-ECHT after FES = 500,000 for g19–g24. The bold numbers indicate the better results.

Prob | Algorithm | Best | Median | Worst | c | $\overline{\mathit{v}}$ | Mean | Std | FR | SR |
---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{g}19$ | JADE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{1.4028}\left(\mathbf{0}\right)$ | $\mathbf{3.6498}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $\mathbf{1.5502}$ | $\mathbf{1.0136}$ | $100\%$ | 12% |

SaDE-ECHT | $0.3671\left(0\right)$ | $1.7022\left(0\right)$ | $6.6604\left(0\right)$ | $0,0,0$ | 0 | $2.3120$ | $1.9699$ | $100\%$ | $0\%$ | |

$\mathrm{g}20$ | JADE-ECHT | $3.2029\left(9\right)$ | $\mathbf{6.2057}\left(\mathbf{8}\right)$ | $\mathbf{15.4062}\left(\mathbf{12}\right)$ | $\mathbf{1},\mathbf{1},\mathbf{2}$ | $\mathbf{1.1209}$ | $\mathbf{7.2582}$ | $\mathbf{3.5087}$ | $0\%$ | $0\%$ |

SaDE-ECHT | $\mathbf{2.4461}\left(\mathbf{11}\right)$ | $14.8045\left(9\right)$ | $18.3511\left(11\right)$ | $2,4,4$ | $3.1946$ | $13.1617$ | $4.8304$ | $0\%$ | $0\%$ | |

$\mathrm{g}21$ | JADE-ECHT | $0\left(0\right)$ | $\mathbf{0.0633}\left(\mathbf{0}\right)$ | $263.7866\left(1\right)$ | $0,0,0$ | 0 | $\mathbf{39.1073}$ | $63.9006$ | $96\%$ | 44% |

SaDE-ECHT | $0\left(0\right)$ | $77.3185\left(0\right)$ | $\mathbf{110.2441}\left(0\right)$ | $0,0,0$ | 0 | $71.8631$ | $\mathbf{25.3368}$ | 100% | $4\%$ | |

$\mathrm{g}22$ | JADE-ECHT | $390.4334\left(4\right)$ | $\mathrm{10,565.5111}\left(3\right)$ | $\mathrm{19,715.2233}\left(4\right)$ | $3,3,3$ | $\mathrm{175,401.6096}$ | $\mathrm{10,557.6213}$ | $6162.3243$ | $0\%$ | $0\%$ |

SaDE-ECHT | $\mathbf{292.6511}\left(\mathbf{3}\right)$ | $\mathbf{8834.7836}\left(\mathbf{3}\right)$ | $\mathbf{19,258.8965}\left(\mathbf{3}\right)$ | $3,3,3$ | $\mathbf{90,196.1317}$ | $\mathbf{9289.3437}$ | $\mathbf{4998.2886}$ | $0\%$ | $0\%$ | |

$\mathrm{g}23$ | JADE-ECHT | $\mathbf{0}\left(\mathbf{0}\right)$ | $\mathbf{8.5726}\left(\mathbf{0}\right)$ | $601.1293\left(0\right)$ | $0,0,0$ | 0 | $\mathbf{117.5730}$ | $198.2664$ | 36% | 36% |

SaDE-ECHT | $182.7482\left(0\right)$ | $357.7081\left(0\right)$ | $\mathbf{518.9083}\left(\mathbf{0}\right)$ | $0,0,0$ | 0 | $344.3397$ | $\mathbf{87.4764}$ | $0\%$ | $0\%$ | |

$\mathrm{g}24$ | JADE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

SaDE-ECHT | $0\left(0\right)$ | $0\left(0\right)$ | $0\left(0\right)$ | $0,0,0$ | 0 | 0 | 0 | $100\%$ | $100\%$ |

**Table 7.**Comparison of JADE-ECHT and SaDE-ECHT in terms of feasibility rate (FR) and success rate (SR) with algorithms of CEC 2006.

Algorithms | FR | SR |
---|---|---|

DE | 95.65% | 78.09% |

DMS-PSO | 100% | 90.61% |

$\u03f5$ DE | 100% | 95.65% |

GDE | 92.00% | 77.39% |

jDE-2 | 95.65% | 80.00% |

MDE | 95.65% | 87.65% |

MPDE | 94.96% | 87.65% |

PCX | 95.65% | 94.09% |

PESO+ | 95.48% | 67.83% |

SaDE | 100% | 87.13% |

JADE-ECHT | 95.30% | 57.04% |

SaDE-ECHT | 95.65% | 46.43% |

**Table 8.**Comparison of median values of JADE-ECHT, SaDE-ECHT and CEC’2006 algorithms achieved after 500,000 FEs. The bold numbers indicate the better results.

Prob | DE | DMS-PSO | $\mathit{\u03f5}$ DE | GDE | jDE-2 | MDE | MPDE | PCX | PESO+ | SaDE | JADE-ECHT | SaDE-ECHT |
---|---|---|---|---|---|---|---|---|---|---|---|---|

g01 | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) |

g02 | 5.1700 $\times {10}^{-8}$ (0) | 0(0) | 3.0933 $\times {10}^{-8}$(0) | 2.3251 $\times {10}^{-7}$(0) | 3.3051 $\times {10}^{-9}$(0) | 0.017460(0) | 3.5608 $\times {10}^{-6}$ | 0(0) | 1.4314 $\times {10}^{-6}$(0) | 3.0800 $\times {10}^{-9}$(0) | 0.0004(0) | 0.0110(0) |

g03 | 6.7110 $\times {10}^{-1}$(0) | 0(0) | −4.4409 $\times {10}^{-16}$(0) | 9.3634 $\times {10}^{-1}$(0) | 0.3481(0) | 0(0) | −2.8866 $\times {10}^{-15}$ | 0(0) | 1.5890 $\times {10}^{-7}$ (0) | 1.7770 $\times {10}^{-8}$(0) | 0.1015(0) | 0.0122(0) |

g04 | 7.6398 $\times {10}^{-11}$(0) | 0(0) | 0(0) | 8.0036 $\times {10}^{-11}$(0) | 0(0) | 0(0) | 3.6380 $\times {10}^{-12}$ | 0(0) | 1.0000 $\times {10}^{-10}$(0) | 2.1667 $\times {10}^{-7}$(0) | 0(0) | 0(0) |

g05 | −9.0949 $\times {10}^{-13}$(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 91.4773(0) |

g06 | 4.5475 $\times {10}^{-11}$ | 0(0) | 1.1823 $\times {10}^{-11}$(0) | 6.1846 $\times {10}^{-11}$(0) | 1.1823 $\times {10}^{-11}$(0) | 0(0) | 1.0914 $\times {10}^{-11}$ | 0(0) | 1.0000 $\times {10}^{-10}$(0) | 4.5475 $\times {10}^{-11}$(0) | 0(0) | 0(0) |

g07 | 7.9783 $\times {10}^{-11}$(0) | 0(0) | −1.8474 $\times {10}^{-13}$(0) | 3.6402 $\times {10}^{-10}$(0) | −1.8829 $\times {10}^{-13}$(0) | 0(0) | −1.8474 $\times {10}^{-13}$ | 0(0) | 9.4367 $\times {10}^{-6}$(0) | 1.4608 $\times {10}^{-7}$(0) | 0.0879(0) | 0.0114(0) |

g08 | 8.1964 $\times {10}^{-11}$(0) | 0(0) | 4.1633 $\times {10}^{-17}$(0) | 8.1964 $\times {10}^{-11}$(0) | 4.1633 $\times {10}^{-17}$(0) | 0(0) | 4.1633 $\times {10}^{-17}$ | 0(0) | 1.0000 $\times {10}^{-10}$(0) | 8.1964 $\times {10}^{-11}$(0) | 0(0) | 0(0) |

g09 | −9.8112 $\times {10}^{-11}$(0) | 0(0) | 0(0) | −9.7884 $\times {10}^{-11}$(0) | 2.2737 $\times {10}^{-13}$(0) | 0(0) | 1.1369 $\times {10}^{-13}$ | 0(0) | 1.0000 $\times {10}^{-10}$(0) | 3.7440 $\times {10}^{-7}$(0) | 0.0039(0) | 0(0) |

g10 | 6.2755 $\times {10}^{-11}$(0) | 1.0124 $\times {10}^{-8}$(0) | −9.0949 $\times {10}^{-13}$(0) | 6.9122 $\times {10}^{-11}$(0) | −9.0949 $\times {10}^{-13}$(0) | 0(0) | −9.0949 $\times {10}^{-13}$ | 0(0) | 1.3432 $\times {10}^{-3}$(0) | 1.8120 $\times {10}^{-6}$(0) | 133.9677(0) | 0.1709(0) |

g11 | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) |

g12 | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) |

g13 | 3.8486 $\times {10}^{-1}$(0) | 0(0) | 9.7145 $\times {10}^{-17}$(0) | 3.8486 $\times {10}^{-1}$(0) | 0.6800(0) | 0(0) | 3.8486 $\times {10}^{-1}$ | 0(0) | 1.1200 $\times {10}^{-8}$(0) | 4.1898 $\times {10}^{-11}$(0) | 0.9118(0) | 0.3870(0) |

g14 | 8.5123 $\times {10}^{-12}$(0) | 0(0) | 2.1316 $\times {10}^{-14}$(0) | 6.3148 $\times {10}^{-8}$(0) | 2.1316 $\times {10}^{-14}$(0) | 0(0) | −3.9961 $\times {10}^{-4}$ | 0(0) | 3.2912 $\times {10}^{-3}$(0) | 1.4793 $\times {10}^{-5}$(0) | 0.0174(0) | 1.6397(0) |

g15 | 6.0822 $\times {10}^{-11}$(0) | 0(0) | 0(0) | 6.0936 $\times {10}^{-11}$(0) | 0(0) | 0(0) | 0(0) | 0(0) | 1.0000 $\times {10}^{-10}$(0) | 6.0822 $\times {10}^{-11}$(0) | 0(0) | 0.0009(0) |

g16 | 6.5214 $\times {10}^{-11}$(0) | 0(0) | 4.4409 $\times {10}^{-15}$(0) | 6.5216 $\times {10}^{-11}$(0) | 5.1070 $\times {10}^{-15}$(0) | 0(0) | 5.3291 $\times {10}^{-15}$ | 0(0) | 1.0000 $\times {10}^{-10}$(0) | 6.5214 $\times {10}^{-11}$(0) | 0(0) | 0(0) |

g17 | 7.4058 $\times {10}^{1}$(0) | 7.4058 $\times {10}^{1}$(0) | 1.8190 $\times {10}^{-12}$(0) | 7.4052 $\times {10}^{1}$(0) | 10.4896(0) | 0(0) | 7.4058 $\times {10}^{1}$ | 0(0) | 13.9638(0) | 7.4058 $\times {10}^{1}$(0) | 0(0) | 91.2351(0) |

g18 | 1.5561 $\times {10}^{-11}$(0) | 0(0) | 3.3307 $\times {10}^{-16}$(0) | 4.6362 $\times {10}^{-11}$(0) | 4.4408 $\times {10}^{-16}$(0) | 0(0) | 4.4409 $\times {10}^{-16}$ | 0(0) | 4.0000 $\times {10}^{-10}$(0) | 1.5561 $\times {10}^{-11}$(0) | 0.0001(0) | 0(0) |

g19 | 4.6370 $\times {10}^{-11}$(0) | 0(0) | 5.2162 $\times {10}^{-8}$(0) | 5.2669 $\times {10}^{-9}$(0) | 4.2632 $\times {10}^{-14}$(0) | 0.387033(0) | 3.5527 $\times {10}^{-14}$ | 0(0) | 2.6302 $\times {10}^{-2}$(0) | 1.3868 $\times {10}^{-10}$(0) | 1.4028(0) | 1.7022(0) |

g20 | −2.4674 $\times {10}^{-2}$(8) | −7.3330 $\times {10}^{-2}$(17) | −2.4674 $\times {10}^{-2}$(8) | 1.3503 $\times {10}^{1}$(20) | 0.1082(2) | 0.103314(20) | 1.0151 $\times {10}^{1}$ | 0.0675(12) | 3.2600 $\times {10}^{-2}$(8) | 2.3757 $\times {10}^{-1}$(20) | 6.2057(8) | 14.8045(9) |

g21 | −2.8371 $\times {10}^{-10}$(0) | 3.5911 $\times {10}^{-6}$(0) | −2.8422 $\times {10}^{-14}$(0) | 6.5523 $\times {10}^{-8}$(0) | −2.8421 $\times {10}^{-14}$(0) | 0(0) | 1.4211 $\times {10}^{-13}$ | 0(0) | 81.3460(0) | 2.5785 $\times {10}^{-8}$(0) | 0.0633(0) | 77.3185(0) |

g22 | 1.0336 $\times {10}^{4}$(15) | 1.2200 $\times {10}^{2}$ (0) | 1.2332 $\times {10}^{1}$(0) | 9.7885 $\times {10}^{3}$(19) | 8033.6537(8) | 9210.082460(18) | 8.7919 $\times {10}^{3}$ | 9888.6409(15) | 14,198.8059(19) | 4.6907 $\times {10}^{1}$(0) | 10,565.5111(3) | 8834.7836(3) |

g23 | 3.0005 $\times {10}^{2}$(0) | 1.0267 $\times {10}^{-8}$ (0) | 0(0) | 1.0569 $\times {10}^{1}$(0) | 2.2737 $\times {10}^{-13}$(0) | 0(0) | 0(0) | 0(0) | 130.5043(0) | 3.9790 $\times {10}^{-13}$(0) | 8.5726(0) | 357.7081(0) |

g24 | 4.6736 $\times {10}^{-12}$(0) | 0(0) | 5.7732 $\times {10}^{-14}$(0) | 4.7269 $\times {10}^{-12}$(0) | 5.5067 $\times {10}^{-14}$(0) | 0(0) | 7.1054 $\times {10}^{-14}$ | 0(0) | 0(0) | 4.6372 $\times {10}^{-12}$(0) | 0(0) | 0(0) |

**Table 9.**Comparison of standard deviation values of JADE-ECHT, SaDE-ECHT and CEC’2006 algorithms achieved after 500,000 FEs. The bold numbers indicate the better results.

Prob | DE | DMS-PSO | $\mathit{\u03f5}$ DE | GDE | jDE-2 | MDE | MPDE | PCX | PESO+ | SaDE | JADE-ECHT | SaDE-ECHT |
---|---|---|---|---|---|---|---|---|---|---|---|---|

g01 | 6.4146 $\times {10}^{-15}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.4000 | 0 |

g02 | 1.0015 $\times {10}^{-18}$ | 4.6953 $\times {10}^{-3}$ | 1.7523 $\times {10}^{-8}$ | 7.5112 $\times {10}^{-3}$ | 3.0488 $\times {10}^{-3}$ | 0 | 2.7802 $\times {10}^{-3}$ | 0 | 7.1385 $\times {10}^{-3}$ | 4.9786 $\times {10}^{-3}$ | 0.0091 | 0.0254 |

g03 | 5.2098 $\times {10}^{-14}$ | 0 | 2.9582 $\times {10}^{-31}$ | 1.9833 $\times {10}^{-1}$ | 1.0140 $\times {10}^{-1}$ | 0 | 8.3681 $\times {10}^{-2}$ | 0 | 1.5396 $\times {10}^{-6}$ | 3.4743 $\times {10}^{-5}$ | 0.1036 | 0.0343 |

g04 | 3.9644 $\times {10}^{-13}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.8550 $\times {10}^{-12}$ | 0 | 0 |

g05 | 0 | 0 | 0 | 1.6854 $\times {10}^{2}$ | 2.4193 | 0 | 3.6380 $\times {10}^{-13}$ | 0 | 0 | 1.8190 $\times {10}^{-13}$ | 0 | 101.2496 |

g06 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

g07 | 6.4146 $\times {10}^{-15}$ | 0 | 2.1831 $\times {10}^{-15}$ | 3.8948 $\times {10}^{-7}$ | 1.7405 $\times {10}^{-15}$ | 0 | 3.0215 $\times {10}^{-15}$ | 0 | 2.2678 $\times {10}^{-5}$ | 1.4993 $\times {10}^{-5}$ | 0.0726 | 0.0850 |

g08 | 1.0015 $\times {10}^{-18}$ | 0 | 1.2326 $\times {10}^{-32}$ | 0 | 1.2580 $\times {10}^{-32}$ | 0 | 0 | 0 | 0 | 3.8426 $\times {10}^{-18}$ | 0 | 0 |

g09 | 5.2098 $\times {10}^{-14}$ | 0 | 0 | 4.9555 $\times {10}^{-14}$ | 4.2538 $\times {10}^{-14}$ | 0 | 0 | 0 | 0 | 7.9851 $\times {10}^{-14}$ | 0.0192 | 0.0002 |

g10 | 3.9644 $\times {10}^{-13}$ | 8.8141 $\times {10}^{-9}$ | 4.2426 $\times {10}^{-13}$ | 0 | 9.1279 $\times {10}^{-8}$ | 0 | 3.0165 $\times {10}^{-13}$ | 0 | 5.8279 $\times {10}^{-2}$ | 1.5244 $\times {10}^{-6}$ | 105.4501 | 3.0352 |

g11 | 0 | 0 | 0 | 0 | 4.5540 $\times {10}^{-4}$ | 0 | 7.0638 $\times {10}^{-5}$ | 0 | 0 | 0 | 0 | 0 |

g12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

g13 | 0 | 1.8204 $\times {10}^{-6}$ | 0 | 3.8650 $\times {10}^{-1}$ | 2.2376 $\times {10}^{-1}$ | 0 | 2.8719 $\times {10}^{-1}$ | 0 | 6.3801 $\times {10}^{-6}$ | 2.7832 $\times {10}^{-7}$ | 0.1750 | 0.2828 |

g14 | 3.4809 $\times {10}^{-15}$ | 0 | 1.3924 $\times {10}^{-15}$ | 3.8556 $\times {10}^{-3}$ | 3.4809 $\times {10}^{-15}$ | 0 | 7.9441 $\times {10}^{-15}$ | 0 | 2.8553 $\times {10}^{-2}$ | 6.4986 $\times {10}^{-5}$ | 2.2940 | 0.6956 |

g15 | 0 | 0 | 0 | 9.6094 $\times {10}^{-1}$ | 2.2020 $\times {10}^{-2}$ | 0 | 4.3027 $\times {10}^{-6}$ | 0 | 0 | 0 | 0 | 0.6971 |

g16 | 2.2092 $\times {10}^{-16}$ | 0 | 1.5777 $\times {10}^{-30}$ | 1.7764 $\times {10}^{-16}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

g17 | 3.0234 $\times {10}^{1}$ | 0 | 1.2117 $\times {10}^{-27}$ | 8.2114 $\times {10}^{1}$ | 3.8319 $\times {10}^{1}$ | 0 | 3.4111 $\times {10}^{1}$ | 0 | 42.286 | 1.6168 $\times {10}^{1}$ | 24.5623 | 50.4589 |

g18 | 4.8817 $\times {10}^{-17}$ | 0 | 2.1756 $\times {10}^{-17}$ | 6.4619 $\times {10}^{-1}$ | 3.6822 $\times {10}^{-17}$ | 0 | 4.1541 $\times {10}^{-17}$ | 0 | 3.8184 $\times {10}^{-2}$ | 5.2898 $\times {10}^{-2}$ | 0.0041 | 0 |

g19 | 1.2568 $\times {10}^{-5}$ | 0 | 1.2568 $\times {10}^{-5}$ | 4.5735 $\times {10}^{-5}$ | 1.0531 $\times {10}^{-13}$ | 0.847517 | 3.5527 $\times {10}^{-14}$ | 0 | 1.6158 $\times {10}^{-1}$ | 7.2976 $\times {10}^{-10}$ | 1.0136 | 1.9699 |

g20 | 4.2362 $\times {10}^{-2}$ | 6.9516 $\times {10}^{-3}$ | 4.2362 $\times {10}^{-2}$ | 1.9580 $\times {10}^{0}$ | 1.1510 $\times {10}^{-2}$ | 0.021688 | 2.8984 $\times {10}^{0}$ | 0.0219 | 4.9951 $\times {10}^{-3}$ | 1.0638 $\times {10}^{-1}$ | 3.5087 | 4.8304 |

g21 | 6.5489 $\times {10}^{1}$ | 2.0784 $\times {10}^{-6}$ | 3.3417 $\times {10}^{-14}$ | 8.6788 $\times {10}^{1}$ | 3.6266 $\times {10}^{1}$ | 0 | 6.2358 $\times {10}^{1}$ | 0 | 67.019 | 1.5058 $\times {10}^{-3}$ | 63.9006 | 25.3368 |

g22 | 5.7875 $\times {10}^{3}$ | 2.7703 $\times {10}^{1}$ | 1.5690 $\times {10}^{1}$ | 5.9865 $\times {10}^{3}$ | 5.1748 $\times {10}^{3}$ | 4808.800969 | 4.8022 $\times {10}^{3}$ | 4421.5326 | 4963.7 | 3.0415 $\times {10}^{1}$ | 6162.3243 | 4998.2886 |

g23 | 8.8097 $\times {10}^{-6}$ | 4.5997 $\times {10}^{-4}$ | 1.1139 $\times {10}^{-14}$ | 1.9119 $\times {10}^{2}$ | 5.9983 $\times {10}^{1}$ | 0 | 5.1159 $\times {10}^{-14}$ | 0 | 87.634 | 3.4116 $\times {10}^{-4}$ | 198.2664 | 87.4764 |

g24 | 8.8525 $\times {10}^{-20}$ | 0 | 2.5244 $\times {10}^{-29}$ | 0 | 1.9323 $\times {10}^{-29}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Javed, H.; Jan, M.A.; Tairan, N.; Mashwani, W.K.; Khanum, R.A.; Sulaiman, M.; Khan, H.U.; Shah, H.
On the Efficacy of Ensemble of Constraint Handling Techniques in Self-Adaptive Differential Evolution. *Mathematics* **2019**, *7*, 635.
https://doi.org/10.3390/math7070635

**AMA Style**

Javed H, Jan MA, Tairan N, Mashwani WK, Khanum RA, Sulaiman M, Khan HU, Shah H.
On the Efficacy of Ensemble of Constraint Handling Techniques in Self-Adaptive Differential Evolution. *Mathematics*. 2019; 7(7):635.
https://doi.org/10.3390/math7070635

**Chicago/Turabian Style**

Javed, Hassan, Muhammad Asif Jan, Nasser Tairan, Wali Khan Mashwani, Rashida Adeeb Khanum, Muhammad Sulaiman, Hidayat Ullah Khan, and Habib Shah.
2019. "On the Efficacy of Ensemble of Constraint Handling Techniques in Self-Adaptive Differential Evolution" *Mathematics* 7, no. 7: 635.
https://doi.org/10.3390/math7070635