SMCB-E-08102005-0551.R2 7 cost,the mean running times of all functions other than G02 and the four criteria. G12 are shorter than 0.5 second. The 4 engineering design problems are solved by OEA with CHc only.Table V summarizes the experimental results.OEA V.PARAMETER ANALYSES OF OEA provides 100%feasible solutions for all functions.As described in Table V.the standard deviations of the 4 problems are There are 4 parameters in OEA,namely,No,AS,CS,and Maxos.To study the sensitivity of OEA to these parameters,a relatively small,so the performance of OEA is very stable in series of empirical experiments are carried out on OEA with solving these problems.Moreover,the mean running times of different values of No,AS,CS,and Maxos. all the 4 functions are shorter than 50 milliseconds,which illustrates that the computational cost of OEA is very low. A.Effects of N on the Performance of OEA In the whole,OEA with CHp outperforms OEA with CHc In general,the fitness landscape of COPs is more complicated But the penalty coefficient in OEA with CHp needs to be tuned than that of UCOPs.such that algorithms are of a high for different problems,and this is inconvenient in real probability of trapping into the local optima.Therefore,many applications.On the contrary,although the performance of OEA researchers focus on developing new methods to handle the with CHc is not as good as that of OEA with CHp,the solution constraints.But the results of Tables IV-IX indicate that OEA quality of OEA with CHc is still competitive with that of OEA with a simple penalty term can obtain very good results.We with CHp.Moreover,no parameter needs to be tuned in OEA think this benefits mainly from the large population,that is, with CHc. N=1500,and the search mechanism of OEA.A common B.Comparison between OEA and RY opinion among researchers is that a large population can make The comparison on G01-G13 is shown in Table VI.As can be algorithm have a little chance to get trapped into the local seen,for G03,G04,G08,G11,and G12,both OEA and RY optima,but maybe reduce the convergence rate.Therefore,to have consistently found the global optima for all trials.For G02 study the effects of N.on the performance ofOEA with CHp,N G06,and G09,both OEA with CHp and CHc outperform RY in increases from 150 to 1500 in steps of 450.Here,Maxos=20, AS-0.8.and CS=0.6.Fig.4 shows the evolutionary process of terms of all the four criteria.For the other functions,OEA the mean best solutions over 50 trials for the 4 more difficult outperforms RY in terms of some criteria while is also functions,G01,G05,G06,and G10. outperformed by RY in terms of some criteria.For G05,OEA with CHp and RY find the same best solution,which is even As can be seen,for the 4 functions,N=150'displays a faster convergence rate in the beginning,but it is overtaken by better than the global optimum.This is the consequence of using N。-600',W。=1050',andN。-l500'quickly.As a result, inequalities to approximate equalities,although a very small is N=150'is obvious worse than the 3 others on the whole used.In the computational cost,the mean running times of OEA evolutionary process.In order to compare N.=600',N.=1050', are significantly shorter than those of RY on all functions,so RY definitely needs much more computational cost than OEA. and N=1500'further,we magnify them in the top-right part of each graph. The comparison on the 4 engineer design problems is shown in Table VII.As can be seen,except that the Best FV of RY for On the whole,the evolutionary process of a small population is much easier trapped into the local optima than that of a large Welded Beam Design is better than that of OEA,OEA population,and OEA with a large population can also converge outperforms RY on other functions in terms of all the four fast.We think this benefits mainly from the search mechanism criteria.Moreover,the mean running times of OEA are still of OEA.OEA searches around each leader by the annexing much shorter than those of RY. operator,which is equivalent to performing a hill-climbing To summarize,both OEA with CHp and CHc are competitive operator on the individual.As a result,once a promising with RY,which illustrate that OEA has an effective search ability,and OEA incorporated simple constraints handling individual appears in the population,it can quickly climb to the crest with a high probability. techniques can obtain good performances in solving COPs. B.Effects of AS and CS on the Performance of OEA C.Comparison between OEA and SMES on G01-G13 Benchmark functions,F01-F15,can be divided into four The comparison is shown in Table VIll where the results of groups,that is,unconstrained unimodal functions, SMES are obtained from [18].As can be seen,under the same unconstrained multimodal functions,equality constraints amount of the NFE,the performance of OEA is competitive functions and inequality constraints functions.So two functions with that of SMES are chosen from each group to use in this subsection,and A.S and D.Comparison between OEA and SCA on the 4 Engineering CS are analyzed on the criterion of Average Success Ratio, Design Problems Definition 4:If a trial satisfies (21).then the trial is called The comparison is shown in Table IX where the results of success;otherwise failure, SCA are obtained from [20].The NFE and the solution vector lF'-Fs Ke.IFI F≠0 for the Best FV are also compared.As can be seen,OEA (21) F*=0 significantly outperforms SCA for the 4 problems in terms of all IFkn Ke where Fbes denotes the best solution found by the trial,and FSMCB-E-08102005-0551.R2 7 cost, the mean running times of all functions other than G02 and G12 are shorter than 0.5 second. The 4 engineering design problems are solved by OEA with CHc only. Table V summarizes the experimental results. OEA provides 100% feasible solutions for all functions. As described in Table V, the standard deviations of the 4 problems are relatively small, so the performance of OEA is very stable in solving these problems. Moreover, the mean running times of all the 4 functions are shorter than 50 milliseconds, which illustrates that the computational cost of OEA is very low. In the whole, OEA with CHp outperforms OEA with CHc. But the penalty coefficient in OEA with CHp needs to be tuned for different problems, and this is inconvenient in real applications. On the contrary, although the performance of OEA with CHc is not as good as that of OEA with CHp, the solution quality of OEA with CHc is still competitive with that of OEA with CHp. Moreover, no parameter needs to be tuned in OEA with CHc. B. Comparison between OEA and RY The comparison on G01-G13 is shown in Table VI. As can be seen, for G03, G04, G08, G11, and G12, both OEA and RY have consistently found the global optima for all trials. For G02, G06, and G09, both OEA with CHp and CHc outperform RY in terms of all the four criteria. For the other functions, OEA outperforms RY in terms of some criteria while is also outperformed by RY in terms of some criteria. For G05, OEA with CHp and RY find the same best solution, which is even better than the global optimum. This is the consequence of using inequalities to approximate equalities, although a very small δ is used. In the computational cost, the mean running times of OEA are significantly shorter than those of RY on all functions, so RY definitely needs much more computational cost than OEA. The comparison on the 4 engineer design problems is shown in Table VII. As can be seen, except that the Best FV of RY for Welded Beam Design is better than that of OEA, OEA outperforms RY on other functions in terms of all the four criteria. Moreover, the mean running times of OEA are still much shorter than those of RY. To summarize, both OEA with CHp and CHc are competitive with RY, which illustrate that OEA has an effective search ability, and OEA incorporated simple constraints handling techniques can obtain good performances in solving COPs. C. Comparison between OEA and SMES on G01-G13 The comparison is shown in Table VIII where the results of SMES are obtained from [18]. As can be seen, under the same amount of the NFE, the performance of OEA is competitive with that of SMES. D. Comparison between OEA and SCA on the 4 Engineering Design Problems The comparison is shown in Table IX where the results of SCA are obtained from [20]. The NFE and the solution vector for the Best FV are also compared. As can be seen, OEA significantly outperforms SCA for the 4 problems in terms of all the four criteria. V. PARAMETER ANALYSES OF OEA There are 4 parameters in OEA, namely, No, AS, CS, and MaxOS. To study the sensitivity of OEA to these parameters, a series of empirical experiments are carried out on OEA with different values of No, AS, CS, and MaxOS. A. Effects of No on the Performance of OEA In general, the fitness landscape of COPs is more complicated than that of UCOPs, such that algorithms are of a high probability of trapping into the local optima. Therefore, many researchers focus on developing new methods to handle the constraints. But the results of Tables IV-IX indicate that OEA with a simple penalty term can obtain very good results. We think this benefits mainly from the large population, that is, No=1500, and the search mechanism of OEA. A common opinion among researchers is that a large population can make algorithm have a little chance to get trapped into the local optima, but maybe reduce the convergence rate. Therefore, to study the effects of No on the performance of OEA with CHp, No increases from 150 to 1500 in steps of 450. Here, MaxOS=20, AS=0.8, and CS=0.6. Fig.4 shows the evolutionary process of the mean best solutions over 50 trials for the 4 more difficult functions, G01, G05, G06, and G10. As can be seen, for the 4 functions, ‘No=150’ displays a faster convergence rate in the beginning, but it is overtaken by ‘No=600’, ‘No=1050’, and ‘No=1500’ quickly. As a result, ‘No=150’ is obvious worse than the 3 others on the whole evolutionary process. In order to compare ‘No=600’, ‘No=1050’, and ‘No=1500’ further, we magnify them in the top-right part of each graph. On the whole, the evolutionary process of a small population is much easier trapped into the local optima than that of a large population, and OEA with a large population can also converge fast. We think this benefits mainly from the search mechanism of OEA. OEA searches around each leader by the annexing operator, which is equivalent to performing a hill-climbing operator on the individual. As a result, once a promising individual appears in the population, it can quickly climb to the crest with a high probability. B. Effects of AS and CS on the Performance of OEA Benchmark functions, F01-F15, can be divided into four groups, that is, unconstrained unimodal functions, unconstrained multimodal functions, equality constraints functions and inequality constraints functions. So two functions are chosen from each group to use in this subsection, and AS and CS are analyzed on the criterion of Average Success Ratio, Definition 4: If a trial satisfies (21), then the trial is called success; otherwise failure, | ||| 0 | | 0 best best FF F F F F ε ε ∗ ∗∗ ∗  − <⋅ ≠   < = (21) where Fbest denotes the best solution found by the trial, and F*