*3.2. SaDE-ECHT*

SaDE [10] is also an unconstrained optimization algorithm. Like JADE and other EAs, it also needs some additional mechanisms to solve COPs. In this work, the four CHTs of ECHT are used in the selection scheme of SaDE for solving COPs. The whole procedure of proposed SaDE-ECHT, shown in Figure 2, is as follow:

**Figure 2.** Flowchart of JADE-ECHT.


$$p\_i = \frac{ns\_i}{ns\_i + nf\_i}, i = 1,2,3,4\tag{5}$$


**Table 1.**

## **4. Experimental Results**

The performances of JADE-ECHT and SaDE-ECHT were evaluated on the suit of CEC'06, which contains twenty four benchmark functions. The PC configuration and parameters' settings are given in Tables 1 and 2.


Configuration

 of the PC.


**Table 2.** Parameters' settings.

## *4.1. Result Achieved*

In Tables 3–6, a comparison of both algorithms after 5 × 10<sup>5</sup> *FEs* is shown. All the obtained results are gathered according to CEC'06 [30] algorithms' evaluation criteria for problems g01 to g24. The criteria include collecting statistics of the best (minimum), worst (maximum), median, mean and standard deviation of the function error values *f*(**x**) − *f*(**x**<sup>∗</sup>), where *f*(**x**) is the best objective function value obtained by the algorithm after 5 × 10<sup>5</sup> *FEs* and *f*(**x**<sup>∗</sup>) is the know objective function value at the optimal solution. The numbers in parenthesis after the objective function value show the number of violated constraints, whereas *c* determines the number of violated constraints at the median solution with violation greater than 0.1, 0.001, 0.0001. *v* shows mean violation at median solution, *FR* is the feasibility rate which is defined as the number of feasible runs over total runs, and *SR* is success rate given by the number of successful runs over total runs. A run is called a feasible run, if the algorithm attains in *max*\_*FEs* at least one feasible solution. Likewise, a run is successful, if the algorithm gets a feasible solution for which the function error value is smaller than 0.0001 in *max*\_*FEs*.

Table 3 compares the experimental results achieved by JADE-ECHT and SaDE-ECHT for problems g01–g06. This table shows that SaDE-ECHT achieved better statistics in terms of best, median, mean and standard deviation values than JADE-ECHT on problems g01 and g03, whereas JADE-ECHT surpasses SaDE-ECHT on problems g02 and g05 except the best value of g02. It can also be observed from the same table that both algorithms show comparable performance on problems g04 and g06. The table also shows that both algorithms have achieved 100% FR on all six problems, as can be confirmed from the 0*s* in parenthesis after the objective function values, and columns for *c* and *v*. The SR of SaDE-ECHT on problems g01–g03 is higher than JADE-ECHT. JADE-ECHT's SR is better than SaDE-ECHT on problem g05, while both algorithms obtained the same SR of 100% on problems g04 and g06.

**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.


Table 4 presents the experimental statistics achieved by JADE-ECHT and SaDE-ECHT for problems g07–g12. The results of this table show that both algorithms obtained comparable statistics for problems g08, g11 and g12. This table also shows superior performance of SaDE-ECHT in terms median, mean and standard deviation values than JADE-ECHT on the problems g07, g09 and g10 except the best values on problems g07 and g10, where JADE-ECHT go<sup>t</sup> better best values. The table also confirms that both algorithms have achieved 100% FR on all six problems, as can be seen from the 0*s* in parenthesis after the objective function values, and columns for *c* and *v*. The SR of JADE-ECHT on problems g07 and g10 is higher than SaDE-ECHT. SaDE-ECHT's SR is better than JADE-ECHT on problem g09, while both algorithms obtained the same SR of 100% on problems g08, g11 and g12.

Table 5 demonstrates the experimental results achieved by JADE-ECHT and SaDE-ECHT for problems g13–g18. The results of this table show that both algorithms performed similar on problem g16. This table also shows superior performance of SaDE-ECHT in terms best, median, mean and standard deviation values than JADE-ECHT on problems g13 and g18 except the standard deviation of g13, while JADE-ECHT performed better than SaDE-ECHT on problems g14, g15 and g17 except the mean and standard deviation values of problem g14, where SaDE-ECHT go<sup>t</sup> better values for the two quantities. The table also confirms that both algorithms have achieved 100% FR on all six problems, as can be seen from the 0*s* in parenthesis after the objective function values, and columns for *c* and *v*. The SR of JADE-ECHT on problems g14, g15, and g17 is higher than SaDE-ECHT. SaDE-ECHT's SR is better than JADE-ECHT on problems g13 and g18, while both algorithms obtained the same SR of 100% on problem g16.


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

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


Table 6 presents the experimental results achieved by JADE-ECHT and SaDE-ECHT for problems g19–g24. The results of this table show that both algorithms performed similar on problem g24. This table also shows superior performance of JADE-ECHT in terms best, median, mean and standard deviation values than SaDE-ECHT on problems g19, g20 g21 and g23, except the best value of problem g20 and standard deviation value of problem g23, while SaDE-ECHT performed better than JADE-ECHT on problem g22. The table also confirms that both algorithms have achieved 100% FR on problems g19 and g24, as can be seen from the 0*s* in parenthesis after the objective function values, and columns for *c* and *v*. Both algorithms are unsuccessful in solving problems g20 and g20. The FR of JADE-ECHT on problem g21 is lower than SaDE-ECHT, while the situation is vice versa in case of SR. The FR and SR of JADE-ECHT on problem g23 is higher than SaDE-ECHT.


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

Figure 3 compares the convergence graphs of JADE-ECHT and SaDE-ECHT for problems g01–g06. This figure shows that JADE-ECHT converges faster than SaDE-ECHT on problems g01, g05 and g06, as less number of *FEs* have been used by it. In case of problem g04, the convergence of SaDE-ECHT is speedy than JADE-ECHT, while in case of problems g02, g03 both algorithms converge at the same rate.

**Figure 3.** Convergence comparison of JADE-ECHT and SaDE-ECHT for g01–g06.

Figure 4 compares the constraints' violations vs *FES* graphs of JADE-ECHT and SaDE-ECHT for problems g01–g06. This figure shows that both algorithms converge quickly to the feasible region and the optimal solution (s) thus has zero constraints' violations.

Figure 5 compares the convergence graphs of JADE-ECHT and SaDE-ECHT for problems g07–g12. This figure shows that both JADE-ECHT and SaDE-ECHT converge at the same rate for all six problems except g11, where JADE-ECHT converges faster than SaDE-ECHT.

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

**Figure 5.** Convergence comparison of JADE-ECHT and SaDE-ECHT for g07–g12.

Figure 6 compares the constraints' violations vs *FES* graphs of JADE-ECHT and SaDE-ECHT for problems g07–g12. This figure too shows that both algorithms converge quickly to the feasible region and optimal solution(s) thus has zero constraints' violations.

Figure 7 compares the convergence graphs of JADE-ECHT and SaDE-ECHT for problems g13–g18. This figure shows that both JADE-ECHT and SaDE-ECHT converge at the same rate for all six problems except g15, where JADE-ECHT converges faster than SaDE-ECHT.

**Figure 6.** Constraint violation comparison of JADE-ECHT and SaDE-ECHT for g07–g12.

**Figure 7.** Convergence comparison of JADE-ECHT and SaDE-ECHT for g13–g18.

Figure 8 compares the constraints' violations vs *FES* graphs of JADE-ECHT and SaDE-ECHT for problems g13–g18. This figure shows that both algorithms explore the infeasible region for about 1000 iterations and then converge to the feasible region. As a result, optimal solution(s) thus obtained has zero constraints' violations.

**Figure 8.** Constraint violation comparison of JADE-ECHT and SaDE-ECHT for g13–g18.

Figure 9 compares the convergence graphs of JADE-ECHT and SaDE-ECHT for problems g19–g24. This figure shows that both JADE-ECHT and SaDE-ECHT converge almost at the same rate for all six problems and utilize the maximum function evaluations.

**Figure 9.** Convergence comparison of JADE-ECHT and SaDE-ECHT for g19–g24.

Figure 10 compares the constraints' violations vs *FES* graphs of JADE-ECHT and SaDE-ECHT for problems g19–g24. This figure clearly shows that both algorithms failed to obtain any feasible solution in case of problems g20 and g22, although maximum function evaluations have been used.

**Figure 10.** Constraint violation comparison of JADE-ECHT and SaDE-ECHT for g19–g24.

Figures 3,5,7 and 9 show the comparison of the convergence graphs vs *FES* of both algorithms for all problems g01-g24, whereas Figures 4,6,8 and 10 demonstrate their comparison graphs of the constraints' violations vs *FEs*.

Overall, it can be concluded from the tabulated results and figures that both algorithms have achieved feasible solution (s) and near optimal solution (s) on 22 problems out of 24 except problems g20 and g22. The tables show that the FR of JADE-ECHT on 20 problems out of 24 is 100% and that of SaDE-ECHT on 22 problems out of 24 is 100%. The SR of JADE-ECHT on most of the problems is better than SaDE-ECHT. On two problems g20 and g22, the FR and SR of both algorithms are 0%. The dimension of these two problems is higher than other 22 problems. Also, these two problems had a large number of equality constraints. It can be noted from our experiments and some other literature review that equality constraints were hard to handle.

Table 7 compares the FR and SR of JADE-ECHT and SaDE-ECHT with other competing algorithms of CEC'2006. It can be seen from the said table that both JADE-ECHT and SaDE-ECHT achieved better FR, and can be placed at positions second and fourth, respectively. However, they failed to achieve better SR than the competing algorithms. A reason of failure could be the use of four different CHTs, where the resources (*FEs*) are distributed based on the success of each individual CHT, while the competing algorithms used just one CHT. The same can also be observed from Tables 8 and 9, where the median and standard deviation values obtained after 5 × 10<sup>5</sup> *FEs* of JADE-ECHT and SaDE-ECHT are compared with other competing algorithms (the values of the two quantities for the competing algorithms are taken from each source paper). Another reason of low SR could be observed from the figures showing constraints' violations vs *FES* graphs. It can be noticed from these graphs that both algorithms converge quickly to the feasible region. As a result, they less explore the infeasible region and consequently suffer from stagnation and premature convergence.


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


**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.

## *Mathematics* **2019**, *7*, 635

**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.


## **5. Conclusions and Future Work**

This paper employed ECHT in the frameworks of two self-adaptive variants of DE, JADE and SaDE. Thus, constrained versions of the two algorithms, denoted by JADE-ECHT and SaDE-ECHT were developed. The proposed algorithms JADE-ECHT and SaDE-ECHT were tested and compared on CEC'06 benchmark test suit. The experimental results show that the SR of JADE-ECHT on most of the tested problems is better than SaDE-ECHT, while SaDE-ECHT surpasses JADE-ECHT in terms of FR. Both algorithms, like other algorithms in the literature, failed to solve problems g20 and g22 due to the hard nature of these problems. In the future, we intend to design ECHT of some other CHTs, embed it then in DE and swarm based algorithms to develop constrained evolutionary algorithms and finally test these newly developed algorithms on some real-world and engineering optimization problems. In addition to that we are going to use [34] for multipath routing protocols and for video streaming systems [35] in order to ge<sup>t</sup> the advantages of these plus the benefits of the proposed work would be very beneficent and demanded.

**Author Contributions:** Conceptualization, H.J. and M.A.J.; methodology, H.K., M.A.J. and W. K.M.; software, R.A.K., N.T. and H.S.; validation, H.U.K. and M.S.; formal analysis, H.J., M.A.J., R.A.K. and H.S.; investigation, H.J., M.A.J. and M.S.; resources, N.T. and H.S.; writing—original draft preparation, H.J. and M.A.J.; writing—review and editing, H.U.K. and M.S.; supervision, M.A.J. and W.K.M.; project administration, N.T. and H.S.; funding acquisition, N.T. and H.S.

**Funding:** The authors would like to thank King Khalid University of Saudi Arabia for supporting this research under the gran<sup>t</sup> number R.G.P.2/7/38.

**Conflicts of Interest:** The authors declare no conflict of interest.
