A Mutual-Evaluation Genetic Algorithm for Numerical and Routing Optimization

Table 1

Twelve benchmark functions.

Benchmark functions	𝑁	𝐷	𝑓min
𝑓1(𝑥)=∑^𝑁𝑖=1𝑥^2𝑖	30	[−100,100]	0
𝑓2(𝑥)=∑^𝑁𝑖=1𝑥𝑖+∏^𝑁𝑖=1𝑥𝑖	30	[−10,10]	0
𝑓3(𝑥)=∑^𝑁𝑖=1∑^𝑖𝑗=1𝑥𝑗^2	30	[−100,100]	0
𝑓4(𝑥)=max𝑥𝑖,𝑖=1,2,…,𝑁	30	[−100,100]	0
𝑓5(𝑥)=∑^𝑁−1𝑖=1100𝑥^2𝑖−𝑥𝑖+1^2+𝑥𝑖−1^2	30	[−5,10]	0
𝑓6(𝑥)=∑^𝑁𝑖=1𝑥^4𝑖+random[0,1)	30	[−1.28,1.28]	0
𝑓7(𝑥)=∑^𝑁𝑖=1−𝑥𝑖sin𝑥𝑖	30	[−500,500]	−12596.5
𝑓8(𝑥)=∑^𝑁𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10	30	[−5.12,5.12]	0
𝑓9(𝑥)=−20exp−0.21/𝑛∑^𝑛𝑖=1𝑥^2𝑖−exp(1/𝑁)∑^𝑁𝑖=1cos2𝜋𝑥𝑖+20+exp(1)	30	[−32,32]	0
𝑓10(𝑥)=(1/4000)∑^𝑁𝑖=1𝑥^2𝑖−∏^𝑁𝑖=1cos𝑥𝑖/√𝑖+1	30	[−600,600]	0
𝑓11(𝑥)=𝜋𝑁10sin^2𝜋𝑦1+𝑦𝑁−1^2+∑^𝑁−1𝑖=1𝑦𝑖−1^2×1+10sin^2𝜋𝑦𝑖+1+∑^𝑁𝑖=1𝑢𝑥𝑖,10,100,4; where𝑦𝑖=1+(1/4)𝑥𝑖+1and𝑢𝑥𝑖,𝑎,𝑘,𝑚=𝑘(𝑥𝑖−𝑎)^𝑚𝑥𝑖>𝑎0−𝑎≤𝑥𝑖≤𝑎𝑘(−𝑥𝑖−𝑎)^𝑚𝑥𝑖<−𝑎	30	[−50,50]	0
𝑓12=(1/10)sin^23𝜋𝑥1+𝑥𝑁−1^21+sin^22𝜋𝑥𝑁+∑^𝑁−1𝑖=1𝑥𝑖−1^21+sin^23𝜋𝑥𝑖+1+∑^𝑁𝑖=1𝑢𝑥𝑖,5,100,4	30	[−50,50]	0

N: problem dimension; D: prescribed search domain; 𝑓_min: global optimum value.