An Enhanced Differential Evolution Based Algorithm with Simulated Annealing for Solving Multiobjective Optimization Problems

Table 2

Test problems used in the paper.

Problem	𝑛	Variable bounds	Objective functions	Optimal solutions	Comments
ZDT1	30	[0,1]	𝑓1=𝑥1	𝑥1∈[0,1]	Convex
			𝑓2=𝑔(𝑥)1−𝑥1𝑔(𝑥)	𝑥=0; 𝑖=2,…,𝑛
			𝑔(𝑥)9∑=1+^𝑛𝑖=2𝑥𝑖𝑛−1
ZDT2	30	[0,1]	𝑓1=𝑥1	𝑥1∈[0,1]	Nonconvex
			𝑓2=𝑔(𝑥)1−𝑥1𝑔(𝑥)^2	𝑥=0; 𝑖=2,…,𝑛
			𝑔(𝑥)9∑=1+^𝑛𝑖=2𝑥𝑖𝑛−1
ZDT3	30	[0,1]	𝑓1=𝑥1	𝑥1∈[0,1]	Convex disconnected
			𝑓2=𝑔(𝑥)1−𝑥1𝑔(𝑥)−𝑥1𝑔(𝑥)sin(10𝜋𝑥1)	𝑥=0
			∑𝑔(𝑥)=1+9^𝑛𝑖=2𝑥𝑖𝑛−1	𝑖=2,…,𝑛
ZDT4	10	𝑥1∈[0,1]	𝑓1=𝑥1	𝑥1∈[0,1]	Nonconvex
		𝑥𝑖∈[−5,5]	𝑓2=𝑔(𝑥)1−𝑥1𝑔(𝑥)	𝑥=0
			𝑔(𝑥)=1+10(𝑛−1)+∑^𝑛𝑖=2𝑥^2𝑖−10cos(4𝜋𝑥𝑖)	𝑖=2,…,𝑛
	ZDT6	10	[0,1]	𝑓1=1−exp(−4𝑥𝑖)sin^6(6𝜋𝑥1)		𝑥1∈[0,1]	Nonconvex
𝑓2𝑓=𝑔(𝑥)1−^1𝑔(𝑥)^2				𝑥=0
∑𝑔(𝑥)=1+9^𝑛𝑖=2𝑥𝑖𝑛−1^0.25				𝑖=2,…,𝑛
DTLZ1	𝑚+4	[0,1]	𝑓1=12𝑥1𝑥2…𝑥𝑚−1(1+𝑔(𝑥𝑚))	𝑥𝑖=0.5; 𝑖=1,2,…,𝑛
			𝑓2=12𝑥1𝑥2…𝑥𝑚−1(1−𝑥𝑚−1)(1+𝑔(𝑥𝑚))
			⋮
			𝑓𝑚−1=12𝑥1(1−𝑥2)(1+𝑔(𝑥𝑚))
			𝑓𝑚=12(1−𝑥1)(1+𝑔(𝑥𝑚))
			𝑔𝑥𝑚=100[|𝑥𝑚|+∑𝑥𝑖𝜖𝑥𝑚𝑥𝑖−0.5^2−cos20𝜋𝑥𝑖−0.5]
DTLZ2	𝑚+9	[0,1]	𝑓1=1+𝑔𝑥𝑚𝑥cos^1𝜋2𝑥cos^2𝜋2𝑥⋯cos^𝑚−2𝜋2𝑥cos^𝑚−1𝜋2	𝑥𝑖=0.5; 𝑖=1,2,…,𝑛
			𝑓2=1+𝑔𝑥𝑚𝑥cos^1𝜋2𝑥cos^2𝜋2𝑥⋯cos^𝑚−2𝜋2𝑥sin^𝑚−1𝜋2
			𝑓3=1+𝑔𝑥𝑚𝑥cos^1𝜋2𝑥cos^2𝜋2𝑥⋯sin^𝑚−2𝜋2
			⋮
			𝑓𝑚=1+𝑔𝑥𝑚𝑥sin^1𝜋2
			𝑔𝑥𝑚=∑𝑥𝑖𝜖𝑥𝑚(𝑥𝑖−0.5)^2