SMCB-E-08102005-0551.R2 the global optimum.Suppose Ms out of M trials succeed,and VI.CONCLUSION then Average Success Ratio is given as R =M./M A new numerical optimization algorithm,OEA,has been The smaller gis,the better the solution is.Here,gis set to 103 proposed in this paper.The experimental results in Tables I-III for G13 and 10>for the 7 other functions.N is set to 150 for indicate that OEA outperforms the two famous algorithms,FEP F02,F05,F08,and F10,and set to 1500 for G04,G08,G11,and and OGA/Q,in solving UCOPs.OEA can find better solutions G13.Maxos is set to 20 for all the 8 functions.For G04,G08, using a lower computational cost.Furthermore,the standard G11,and G13,OEA with CHp is used.The experiments are deviation ofOEA is small.which indicates that OEA has a more carried out as follows:A.S and CS increase from 0 to 1 in steps of stable performance.The experimental results in Tables IV-IX 0.05,with 21x21=441 groups of parameters obtained.50 trials indicate that both OEA with CHp and CHc obtain good are carried out for each selected function at each group of performances in a wide range of benchmark problems functions parameters,and the results are shown in Fig.5 in solving COPs,and the performance is competitive with the As can be seen,for different functions,Ra value change with recently published approaches,RY,SMES,and SCA. AS and CS in different manners.When AS is smaller than 0.1. Systematic analyses have been made on the 4 parameters of Rs values of the 8 functions are small.When AS is larger than OEA.To summarize,a large population can reduce the 0.9.R values of F05.F08 and F10 are also small.On the other probability of OEA trapped into the local optima,at the same hand,even in the case of g=105,Ras values of F02,F08,F10. time,the search mechanism of OEA can make a large G04,and G08 still achieve to 1 and R values of F05,G11,and population also have a fast convergence rate.The performance G13 also achieve to a high value at many groups of parameters. ofOEA is less sensitive to the parameters,AS,CS,and Maxos. On the whole,Fig.5 shows that AS has a larger effect on the Although the best values of AS,CS,and Maxos are different for performance of OEA than CS.The performance of OEA is less different functions,OEA can perform stably in a wide range of sensitive to AS and CS,and OEA can perform well in a wide values for AS,CS,and Maxos,which prove OEA is quite robust range of values for AS and CS.Since AS and CS control the and easy to use. probability of using AnStr1,AnStr2 in the annexing operator On the whole,OEA obtains a good performance for both and CoStrl,CoStr2 in the cooperating operator,the results also UCOPs and COPs.This benefits mainly from the following two show that combining the two strategies in the two operators can aspects.One is the structured population,and the other is the make OEA perform better.Therefore,it is better to select AS three evolutionary operators. and CS from 0.2-0.9. In the structured population in OEA,the individuals that can generate offspring are not selected from the whole population, C.Effects of Maxos on the Performance ofOEA but from an organization.This guarantees the diversity as well The functions,the criterion,and the parameter settings of e as the qualities of the selected individuals. and N.used in this subsection are the same with those of the The annexing operator is equivalent to performing a local above subsection.AS is set to 0.8,and CS is set to 0.6.For G04. search.The cooperating operator is equivalent to performing a G08,G11,and G13.OEA with CHp is also used.The global search.The splitting operator controls the size of each experiments are carried out as follows:Maxos increases from 5 organization so that the computational cost is reasonably to 150 in steps of 5 for the unconstrained functions and from 50 distributed between the local searching and the global searching. to 1500 in steps of 50 for the constrained functions.50 trials are In such a manner,once a promising individual appears in the carried out for each function at each group of parameters. population,it has a high probability to generate better The results show that Rs values of F08,F10,G04,and G08 individuals. achieve to 1 at each sampled value of Maxos.For the 4 other What should be also noted is OEA obtains good functions,Maxos has a larger effect on the performance of OEA. performances in solving COPs by only incorporating two simple and the results are shown in Fig.6.As can be seen,for F02 and existing constraints handling techniques.This also illustrates G11,R values are larger than 0.8 at each sampled value of that OEA has an effective searching mechanism.However,from Maxos,and Maxos does not affect them apparently.For F05,Rs another viewpoint,the performance of OEA on COPs is still value is larger than 0.6 only when Maxos is in 10-35.For G13, worse than that of some state-of-the-arts algorithms,such as the Rs value is larger than 0.6 only when Maxos is smaller than 100. algorithm based on evolution strategies and differential In order to investigate the effects of Maxos on G11 and G13 variation [41].Therefore,improving the performance of OEA further,Maxos increases from 5 to 100 in steps of 5,and the by developing novel constraints handling techniques is our results are shown in the top-right corner of Fig.6(b).For G13, further work. when Maxos is smaller than 40,Ras value is larger. On the whole,all results show that the performance of OEA is ACKNOWLEDGMENT less sensitive to Maxos,and OEA can perform well when Maxos The authors are grateful to the reviewers for their helpful is in 5-40.Maxos restricts the size of an organization which will comments and valuable suggestions. be annexed.Thus,selecting Maxos from this range can prevent OEA from doing too many searches around the same leader.SMCB-E-08102005-0551.R2 8 the global optimum. Suppose MS out of M trials succeed, and then Average Success Ratio is given as Ras s = M M . The smaller ε is, the better the solution is. Here, ε is set to 10-3 for G13 and 10-5 for the 7 other functions. No is set to 150 for F02, F05, F08, and F10, and set to 1500 for G04, G08, G11, and G13. MaxOS is set to 20 for all the 8 functions. For G04, G08, G11, and G13, OEA with CHp is used. The experiments are carried out as follows: AS and CS increase from 0 to 1 in steps of 0.05, with 21×21=441 groups of parameters obtained. 50 trials are carried out for each selected function at each group of parameters, and the results are shown in Fig.5. As can be seen, for different functions, Ras value change with AS and CS in different manners. When AS is smaller than 0.1, Ras values of the 8 functions are small. When AS is larger than 0.9, Ras values of F05, F08 and F10 are also small. On the other hand, even in the case of ε=10-5, Ras values of F02, F08, F10, G04, and G08 still achieve to 1 and Ras values of F05, G11, and G13 also achieve to a high value at many groups of parameters. On the whole, Fig.5 shows that AS has a larger effect on the performance of OEA than CS. The performance of OEA is less sensitive to AS and CS, and OEA can perform well in a wide range of values for AS and CS. Since AS and CS control the probability of using AnStr1, AnStr2 in the annexing operator and CoStr1, CoStr2 in the cooperating operator, the results also show that combining the two strategies in the two operators can make OEA perform better. Therefore, it is better to select AS and CS from 0.2-0.9. C. Effects of MaxOS on the Performance of OEA The functions, the criterion, and the parameter settings of ε and No used in this subsection are the same with those of the above subsection. AS is set to 0.8, and CS is set to 0.6. For G04, G08, G11, and G13, OEA with CHp is also used. The experiments are carried out as follows: MaxOS increases from 5 to 150 in steps of 5 for the unconstrained functions and from 50 to 1500 in steps of 50 for the constrained functions. 50 trials are carried out for each function at each group of parameters. The results show that Ras values of F08, F10, G04, and G08 achieve to 1 at each sampled value of MaxOS. For the 4 other functions, MaxOS has a larger effect on the performance of OEA, and the results are shown in Fig.6. As can be seen, for F02 and G11, Ras values are larger than 0.8 at each sampled value of MaxOS, and MaxOS does not affect them apparently. For F05, Ras value is larger than 0.6 only when MaxOS is in 10-35. For G13, Ras value is larger than 0.6 only when MaxOS is smaller than 100. In order to investigate the effects of MaxOS on G11 and G13 further, MaxOS increases from 5 to 100 in steps of 5, and the results are shown in the top-right corner of Fig.6 (b). For G13, when MaxOS is smaller than 40, Ras value is larger. On the whole, all results show that the performance of OEA is less sensitive to MaxOS, and OEA can perform well when MaxOS is in 5-40. MaxOS restricts the size of an organization which will be annexed. Thus, selecting MaxOS from this range can prevent OEA from doing too many searches around the same leader. VI. CONCLUSION A new numerical optimization algorithm, OEA, has been proposed in this paper. The experimental results in Tables I-III indicate that OEA outperforms the two famous algorithms, FEP and OGA/Q, in solving UCOPs. OEA can find better solutions using a lower computational cost. Furthermore, the standard deviation of OEA is small, which indicates that OEA has a more stable performance. The experimental results in Tables IV-IX indicate that both OEA with CHp and CHc obtain good performances in a wide range of benchmark problems functions in solving COPs, and the performance is competitive with the recently published approaches, RY, SMES, and SCA. Systematic analyses have been made on the 4 parameters of OEA. To summarize, a large population can reduce the probability of OEA trapped into the local optima, at the same time, the search mechanism of OEA can make a large population also have a fast convergence rate. The performance of OEA is less sensitive to the parameters, AS, CS, and MaxOS. Although the best values of AS, CS, and MaxOS are different for different functions, OEA can perform stably in a wide range of values for AS, CS, and MaxOS, which prove OEA is quite robust and easy to use. On the whole, OEA obtains a good performance for both UCOPs and COPs. This benefits mainly from the following two aspects. One is the structured population, and the other is the three evolutionary operators. In the structured population in OEA, the individuals that can generate offspring are not selected from the whole population, but from an organization. This guarantees the diversity as well as the qualities of the selected individuals. The annexing operator is equivalent to performing a local search. The cooperating operator is equivalent to performing a global search. The splitting operator controls the size of each organization so that the computational cost is reasonably distributed between the local searching and the global searching. In such a manner, once a promising individual appears in the population, it has a high probability to generate better individuals. What should be also noted is OEA obtains good performances in solving COPs by only incorporating two simple existing constraints handling techniques. This also illustrates that OEA has an effective searching mechanism. However, from another viewpoint, the performance of OEA on COPs is still worse than that of some state-of-the-arts algorithms, such as the algorithm based on evolution strategies and differential variation [41]. Therefore, improving the performance of OEA by developing novel constraints handling techniques is our further work. ACKNOWLEDGMENT The authors are grateful to the reviewers for their helpful comments and valuable suggestions