Mathematical Problems in Engineering

Research Article

A Hybrid Global Optimization Algorithm Based on Wind Driven Optimization and Differential Evolution

Table 5

Comparison of optimal solution of algorithms in low dimension.


Test functions	Numbers of	Algorithm						Theoretical optimal
Test functions	iterations	PSO	BA	DE	WDO	GAPSO	WDO-DE	solution

	50	(3.129031847760726, 3.134768665032161)	(3.141553103558010, −14.166037126201127)	(3.171893641024008, 3.131824693854047)	(0.0312746919874603, 0.0313915996157868)	(3.10191140474814, 3.10191140474814)	(3.14140267832352, 3.14201661395468)
	100	(3.140173624854166, 3.142584336878753)	(3.141553103558010, −14.166037126201127)	(3.141594887583755, 3.141597340960960)	(0.031336085130091, 0.0314633376198137)	(3.1630840677713, 3.1630840677713)	(3.14159264694396, 3.14159265459054)
()	200	(3.141597860814148, 3.141592422127699)	(3.141553103558010, −14.166037126201127)	(3.141592746546202, 3.141592336946040)	(0.031336085130091, 0.0314633376198137)	(3.14537991099944, 3.14537991099944)	(3.14159264694396, 3.14159265459054)	()
	500	(3.141592649491149, 3.141592658550973)	(3.141553103558010, −14.166037126201127)	(3.141592746546202, 3.141592336946040)	(0.031336085130091, 0.0314633376198137)	(3.14380966948824, 3.14380966948824)	(3.14159264694396, 3.14159265459054)
	1000	(3.141592649491149, 3.141592658550973)	(3.141643909560600, −14.166039618443593)	(3.141592746546202, 3.141592336946040)	(0.031336085130091, 0.0314633376198137)	(3.14241959184766, 3.14241959184766)	(3.14159264694396, 3.14159265459054)

	50	(−0.000274778503056, −1.000060523567713)	(−0.000239117085666, −1.000035528802250)	(−0.000019254003131, −1.000054781208720)	(0.0282917779333518, −0.448857319887312)	(, −0.999978563799561)	(, −1.00000055778692)
	100	(−0.000274778503056, −1.000060523567713)	(−0.000170307371325, −1.000123380488044)	(−0.000000003959147, −0.999999996915653)	(0.0282917779333518, −0.448857319887312)	(, −0.999999998297379)	( − , −1.00000000299309)
()	200	(0.000000218751157, −0.999999566676768)	(−0.000109683118045, −1.000005656513546)	(0.000000000326953, −1.000000001047802)	(0.0282917779333518, −0.448857319887312)	(, −1.00000000037874)	(, −1.00000000221316)	(0, −1)
	500	(−0.000000003246821, −1.000000000641675)	(0.000010008823714, −1.000028736158479)	(−0.000000000328438, −1.000000001077549)	(0.0282917779333518, −0.448857319887312)	(, −1.00000000037874)	(, −1.00000000221316)
	1000	(−0.000000001007721, −0.999999999978878)	(0.000010008823714, −1.000028736158479)	(−0.000000005887967, −1.000000003780139)	(0.0282917779333518, −0.448857319887312)	(, −0.999999995696904)	(, −1.00000000221316)

	50	(−0.090537017481104, 0.713861505393922)	(0.089934924086771, −0.712819423269000)	(−0.089842014825500, 0.712656400621485)	(0.0150932106229768, −0.139673825192312)	(−0.0898421518832255, 0.712656330384267)	(−0.0898489505500046, 0.712661110362659)	(0.0898, −0.7126) and (−0.0898, 0.7126)
	100	(−0.089579213574223, 0.712890938477391)	(0.089669828873280, −0.712754046171649)	(−0.089842014825500, 0.712656400621485)	(0.0150932106229768, −0.139673825192312)	(−0.0898420125705887, 0.712656401610307)	(−0.0898420134082198, 0.71265640418782)
()	200	(−0.089839286217455, 0.712655891026959)	(0.089744843168425, −0.712675910202981)	(−0.089842014825500, 0.712656400621485)	(0.0150932106229768, −0.139673825192312)	(−0.0898420139216395, 0.712656402819083)	(−0.0898420134082198, 0.71265640418782)
	500	(−0.089842013891696, 0.712656404507059)	(0.089744843168425, −0.712675910202981)	(−0.089842014825500, 0.712656400621485)	(0.0150932106229768, −0.139673825192312)	(−0.0898420139216395, 0.712656402819083)	(−0.0898420134082198, 0.71265640418782)
	1000	(−0.089842013891696, 0.712656404507059)	(0.089806395772405, −0.712611171310040)	(−0.089842014825500, 0.712656400621485)	(0.0150932106229768, −0.139673825192312)	(−0.0898420139216395, 0.712656402819083)	(−0.0898420134082198, 0.71265640418782)

	50	(0, 0.555683378893354, 0.853290531318392)	(0.113849817052731, 0.555603849176208, 0.852558071570959)	(0.111559931280739, 0.555576225589055, 0.852504656091373)	(0.344200172275741, 0.191217218742259, 0.719829205828109)	(0.114453784913746, 0.555632415452041, 0.852564399900014)	(0.114490931509466, 0.555654876198981, 0.852559993186743)
	100	(0, 0.555636156053527, 0.853073594865928)	(0.113849817052731, 0.555603849176208, 0.852558071570959)	(0.114589394834657, 0.555648863896550, 0.852546999524024)	(0.344200172275741, 0.191217218742259, 0.719829205828109)	(0.114588883123683, 0.555648895909646, 0.852546984663178)	(0.114588896201262, 0.555648895689609, 0.852546983169442)
()	200	(0, 0.555678232858826, 0.853059878827053)	(0.115127379365939, 0.555614295721604, 0.852539734887561)	(0.114588865815882, 0.555648895017175, 0.852546985276585)	(0.344200172275741, 0.191217218742259, 0.719829205828109)	(0.114588890069778, 0.555648895154575, 0.852546983865954)	(0.114588880154869, 0.555648895361092, 0.852546984294326)	(0.114614, 0.555649, 0.852547)
	500	(0, 0.555678096423990, 0.853059974727304)	(0.115127379365939, 0.555614295721604, 0.852539734887561)	(0.114588865815882, 0.555648895017175, 0.852546985276585)	(0.344200172275741, 0.191217218742259, 0.719829205828109)	(0.114588890069778, 0.555648895154575, 0.852546983865954)	(0.114588880154869, 0.555648895361092, 0.852546984294326)
	1000	(0, 0.555678096423990, 0.853059974727304)	(0.115127379365939, 0.555614295721604, 0.852539734887561)	(0.114588865815882, 0.555648895017175, 0.852546985276585)	(0.344200172275741, 0.191217218742259, 0.719829205828109)	(0.114588890069778, 0.555648895154575, 0.852546983865954)	(0.114588880154869, 0.555648895361092, 0.852546984294326)