Improved Barebones Particle Swarm Optimization with Neighborhood Search and Its Application on Ship Design

Table 1

The twelve benchmark problems.

Problems	𝐷	Search range
𝑓1(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖	30	[−100, 100]
𝑓2(𝑥)=∑^𝐷−1𝑖=1100𝑥^2𝑖−𝑥𝑖+1^2+𝑥𝑖−1^2	30	[−2.048, 2.048]
𝑓3(𝑥)=−20exp−0.21𝐷^𝐷𝑖∑=1𝑥^2𝑖1−exp𝐷^𝐷𝑖∑=1cos2𝜋𝑥𝑖+20+𝑒	30	[−32.768, 32.768]
𝑓4(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖−4000^𝐷∏𝑖=1𝑥cos^𝑖√𝑖+1	30	[−600, 600]
𝑓5(𝑥)=∑^𝐷∑𝑖=1^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘𝑥𝑖+0.5−𝐷∑^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘⋅0.5𝑎=0.5,𝑏=3,𝑘max=20	30	[−0.5, 0.5]
𝑓6(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10	30	[−5.12, 5.12]
𝑓7(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖−10cos2𝜋𝑦𝑖𝑦+10𝑖=𝑥𝑖,𝑥𝑖<12round2𝑥𝑖2,𝑥𝑖≥12	30	[−5.12, 5.12]
𝑓8(𝑥)=418.9829⋅𝐷−∑^𝐷𝑖=1𝑥𝑖sin𝑥𝑖	30	[−500, 500]
𝑓9(𝑥)=−20exp−0.21𝐷^𝐷𝑖∑=1𝑦^2𝑖1−exp𝐷^𝐷𝑖∑=1cos2𝜋𝑦𝑖+20+𝑒𝑦=𝐌∗𝑥	30	[−32.768, 32.768]
𝑓10(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖−4000^𝐷∏𝑖=1𝑦cos^𝑖√𝑖+1𝑦=𝐌∗𝑥	30	[−600, 600]
𝑓11(𝑥)=∑^𝐷∑𝑖=1^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘𝑦𝑖+0.5−𝐷∑^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘⋅0.5𝑦=𝐌∗𝑥	30	[−0.5, 0.5]
𝑓12(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖−10cos2𝜋𝑦𝑖+10𝑦=𝐌∗𝑥	30	[−5.12,  5.12]