SMCB-E-08102005-0551.R2 6 Because FEP [39]and OGA/Q [40]obtained good -93.4804189,which is even much worse than the worst FV performances in solving UCOPs,comparisons are made among (-99.4393027)of OEA.On the other hand.since the NFE of FEP,OGA/Q,and OEA.Since the termination criteria used in OEA and OGA/Q are similar,and the mean running times of [39]and [40]are different significantly,to establish a fair OEA are shorter than those of OGA/Q on all the 6 functions. comparison basis,we implemented FEP and OGA/Q,and ran OGA/Q needs more computational cost than OEA.OGA/Q the three algorithms in the same environment.The parameters of applies orthogonal design to select offspring when each time FEP and OGA/Q are set according to [39]and [40],respectively, performing the orthogonal crossover,which increases the and those of OEA are set as follows:N=150,Maxos-20, computational cost. A.S-0.8,and CS=0.6.Since the number of function evaluations (NFE)of OEA and OGA/Q is different in each generation,FEP, OGA/Q,and OEA are terminated when the NFE is larger than IV.EXPERIMENTAL STUDIES ON CONSTRAINED OPTIMIZATION 300 000.The experimental results of FEP,OGA/Q,and OEA PROBLEMS are obtained over 50 independent trials.All the three algorithms In this section,13 benchmark functions (G01-G13)and 4 are realized by Delphi 5 on a 2.4GHz Pentium PC with 1G well-studied engineering design problems (Welded Beam RAM,and the operating system is Windows XP. Design,Spring Design,Speed Reducer Design,and Three-Bar A.Experimental Results of OEA Truss Design)are used to validate the performance of OEA in Table I summarizes the experimental results of OEA,which solving COPs.These functions are described in [19]and [20]. include the best,the mean,the standard deviation,and the worst The equality constraints have been converted into inequality function value (FV)found.Both the mean running time and the constraints,(x)0,using the degree of violation =104,the mean NFE are given.Table I also shows the global optimum for same with that of [19]. each function.What should be noted is the global optimum for Since RY [19]was a new kind of constraints handling F14 given in [40]is-99.2784,but the Best FV found by OEA is technique and obtained a good performance,we first 99.5643815 implemented RY and made a thorough comparison between RY As can be seen,the best,the mean,and the worst FVs of all and OEA on both G01-G13 and the 4 engineering design functions other than F05 are equal or very close to the global problems.Then,comparisons were made between SMES [18] optima,and the standard deviations are relatively small.The and OEA on G01-G13 and between SCA [20]and OEA on the 4 mean running times of the functions other than F12 and F14 are engineering design problems.The parameters of RY are set between 0.5 and 2 seconds.Especially,those of F01,F02,and according to [19],and those of OEA are set as follows:N=1500, F04-F07 are shorter than one second.Although that of F14 is Maxos,AS,and CS are the same with those of Section IIl.Both 6.570s,it is mainly due to the higher computational cost RY and OEA are terminated when the NFE is larger then required by the function itself. 240000 for G01-G13 and 24000 for the 4 engineer design problems.The experimental results of RY and OEA are B.Comparison between OEA and FEP obtained over 50 independent trials.The running environment is The comparison is shown in Table II.As can be seen,the the same with that of Section III. mean FVs of OEA are better than those of FEP on all functions A.Experimental Results of OEA other than F06.For F06,both OEA and FEP find the exact global optimum.On the other hand,since the NFE of OEA and G01-G13 are solved by OEA with two kinds of constraints FEP are similar,and the mean running times of OEA are handling techniques.The penalty coefficient A in(4)adopted by significantly shorter than those of FEP on all functions,FEP each function is shown in(20),and Table IV summarizes the definitely needs more computational cost than OEA.Although experimental results ofOEA,which include the best,the median, FEP is simple and has no complicated computation,FEP needs the mean,the standard deviation,and the worst FV found.Both to generate normally and Cauchy distributed random numbers, the mean running time and the mean NFE are given.Both OEA which spends lots of computational cost. with CHp and CHe provide 100%feasible solutions for all functions. C.Comparison between OEA and OGA/O Ae01-0.5 402=100 Aeo3=105 A0u=104 The comparison is shown in Table III.Because OGA/Q used 4G05=10 Aeo6-=5000 Am=1000 Aos=1000 the orthogonal design to generate the initial population,when (20) the global optima are at the middle point of the search space, Ae09-500 A10-5×10°Am=10 AG2-100 such as F01-F04,F06,F07,and F09-F11,the global optima 43-0.05 always exist in the initial population of OGA/Q.Therefore,the As described in Table IV,both OEA with CHp and CHc have comparison between OEA and OGA/Q is only made on F05, consistently found the global optima for all trials for G03,G04, F08.andF12-F15. G06,G08,G11,and G12.For GO1,G05,G10,and G13,OEA As can be seen,the mean FVs of OEA are better than those of with CHp outperforms OEA with CHc in terms of all the four OGA/Q on all the 6 functions.Especially for F14,the mean FV criteria.For G02,G09,and G07,OEA with CHc outperforms of OEA is -99.5024042 while that of OGA/Q is only OEA with CHp in terms of some criteria.In the computationalSMCB-E-08102005-0551.R2 6 Because FEP [39] and OGA/Q [40] obtained good performances in solving UCOPs, comparisons are made among FEP, OGA/Q, and OEA. Since the termination criteria used in [39] and [40] are different significantly, to establish a fair comparison basis, we implemented FEP and OGA/Q, and ran the three algorithms in the same environment. The parameters of FEP and OGA/Q are set according to [39] and [40], respectively, and those of OEA are set as follows: No=150, MaxOS=20, AS=0.8, and CS=0.6. Since the number of function evaluations (NFE) of OEA and OGA/Q is different in each generation, FEP, OGA/Q, and OEA are terminated when the NFE is larger than 300 000. The experimental results of FEP, OGA/Q, and OEA are obtained over 50 independent trials. All the three algorithms are realized by Delphi 5 on a 2.4GHz Pentium PC with 1G RAM, and the operating system is Windows XP. A. Experimental Results of OEA Table I summarizes the experimental results of OEA, which include the best, the mean, the standard deviation, and the worst function value (FV) found. Both the mean running time and the mean NFE are given. Table I also shows the global optimum for each function. What should be noted is the global optimum for F14 given in [40] is –99.2784, but the Best FV found by OEA is -99.5643815. As can be seen, the best, the mean, and the worst FVs of all functions other than F05 are equal or very close to the global optima, and the standard deviations are relatively small. The mean running times of the functions other than F12 and F14 are between 0.5 and 2 seconds. Especially, those of F01, F02, and F04-F07 are shorter than one second. Although that of F14 is 6.570s, it is mainly due to the higher computational cost required by the function itself. B. Comparison between OEA and FEP The comparison is shown in Table II. As can be seen, the mean FVs of OEA are better than those of FEP on all functions other than F06. For F06, both OEA and FEP find the exact global optimum. On the other hand, since the NFE of OEA and FEP are similar, and the mean running times of OEA are significantly shorter than those of FEP on all functions, FEP definitely needs more computational cost than OEA. Although FEP is simple and has no complicated computation, FEP needs to generate normally and Cauchy distributed random numbers, which spends lots of computational cost. C. Comparison between OEA and OGA/Q The comparison is shown in Table III. Because OGA/Q used the orthogonal design to generate the initial population, when the global optima are at the middle point of the search space, such as F01-F04, F06, F07, and F09-F11, the global optima always exist in the initial population of OGA/Q. Therefore, the comparison between OEA and OGA/Q is only made on F05, F08, and F12-F15. As can be seen, the mean FVs of OEA are better than those of OGA/Q on all the 6 functions. Especially for F14, the mean FV of OEA is -99.5024042 while that of OGA/Q is only -93.4804189, which is even much worse than the worst FV (-99.4393027) of OEA. On the other hand, since the NFE of OEA and OGA/Q are similar, and the mean running times of OEA are shorter than those of OGA/Q on all the 6 functions, OGA/Q needs more computational cost than OEA. OGA/Q applies orthogonal design to select offspring when each time performing the orthogonal crossover, which increases the computational cost. IV. EXPERIMENTAL STUDIES ON CONSTRAINED OPTIMIZATION PROBLEMS In this section, 13 benchmark functions (G01-G13) and 4 well-studied engineering design problems (Welded Beam Design, Spring Design, Speed Reducer Design, and Three-Bar Truss Design) are used to validate the performance of OEA in solving COPs. These functions are described in [19] and [20]. The equality constraints have been converted into inequality constraints, |h(x)|-δ≤0, using the degree of violation δ=10-4, the same with that of [19]. Since RY [19] was a new kind of constraints handling technique and obtained a good performance, we first implemented RY and made a thorough comparison between RY and OEA on both G01-G13 and the 4 engineering design problems. Then, comparisons were made between SMES [18] and OEA on G01-G13 and between SCA [20] and OEA on the 4 engineering design problems. The parameters of RY are set according to [19], and those of OEA are set as follows: No=1500, MaxOS, AS, and CS are the same with those of Section III. Both RY and OEA are terminated when the NFE is larger then 240000 for G01-G13 and 24000 for the 4 engineer design problems. The experimental results of RY and OEA are obtained over 50 independent trials. The running environment is the same with that of Section III. A. Experimental Results of OEA G01-G13 are solved by OEA with two kinds of constraints handling techniques. The penalty coefficient A in (4) adopted by each function is shown in (20), and Table IV summarizes the experimental results of OEA, which include the best, the median, the mean, the standard deviation, and the worst FV found. Both the mean running time and the mean NFE are given. Both OEA with CHp and CHc provide 100% feasible solutions for all functions. 5 4 G01 G02 G03 G04 G05 G06 G07 G08 6 G09 G10 G11 G12 G13 =0.5 =100 =10 =10 =10 =5000 =1000 =1000 =500 =5 10 =10 =100 =0.05 AAA A AA A A AA A A A     ×   (20) As described in Table IV, both OEA with CHp and CHc have consistently found the global optima for all trials for G03, G04, G06, G08, G11, and G12. For G01, G05, G10, and G13, OEA with CHp outperforms OEA with CHc in terms of all the four criteria. For G02, G09, and G07, OEA with CHc outperforms OEA with CHp in terms of some criteria. In the computational