Improving Performance of Evolutionary Algorithms with Application to Fuzzy Control of Truck Backer-Upper System

Table 1

The 12 benchmark functions used in our experimental study; the second column introduces name of functions.

𝑓1(𝑥)	Sphere model	^𝑛∑𝑖=1𝑥^2𝑖
𝑓2(𝑥)	Schwefel's problems 2.22	^𝑛∑𝑖=1𝑥𝑖+∏^𝑛𝑖=1𝑥𝑖
𝑓3(𝑥)	Schwefel's problems 1.2	∑∑^𝑛𝑖=1^𝑖𝑗=1𝑥𝑗^2
𝑓4(𝑥)	Schwefel's problems 2.21	max𝑖𝑥𝑖,1≤𝑖≤𝑛
𝑓5(𝑥)	Generalized Schwefel's problem 2.26	^𝑛∑𝑖=1−𝑥𝑖sin𝑥𝑖
𝑓6(𝑥)	Generalized Rastrigin's function	^𝑛∑𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10
𝑓7(𝑥)	Ackley's function	−20exp−0.21𝑛^𝑛𝑖∑=1𝑥^2𝑖1−exp𝑛^𝑛𝑖∑=1cos2𝜋𝑥𝑖+20+exp(1)
𝑓8(𝑥)	Generalized Griewank function	14000^𝑛𝑖∑=1𝑥^2𝑖−∏^𝑛𝑖=1𝑥cos^𝑖√𝑖+1
𝑓9(𝑥)	Shekel's Foxholes function	1+500^25𝑗∑=11𝑗+∑^2𝑖=1𝑥𝑖−𝑎𝑖𝑗^6
𝑓10(𝑥)	Six-hump camel-back function	4𝑥^21−2.1𝑥^41+13𝑥^61+𝑥1𝑥2−4𝑥^22+4𝑥^42
𝑓11(𝑥)	Hartman's family 2	−∑^4𝑖=1𝑐𝑖exp−∑^4𝑖=1𝑎𝑖𝑗𝑥𝑗−𝑝𝑖𝑗^2
𝑓12(𝑥)	Shekel's family 3	−∑^10𝑖=1𝑥−𝑎𝑖𝑥−𝑎𝑖^𝑇+𝑐𝑖^−1