The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms

Table 1

Test problems.

Problem	𝑛	Variable bounds	Objective functions	Optimal solutions
ZDT1	30	[0, 1]	𝑓1(𝑥)=𝑥1𝑓2(𝑥)=𝑔(𝑥)[1−𝑥19∑/𝑔(𝑥)]𝑔(𝑥)=1+^𝑛𝑖=2𝑥𝑖/(𝑛−1)	𝑥1𝑥∈[0,1]𝑖=0,𝑖=2,…,𝑛
ZDT2	30	[0, 1]	𝑓1(𝑥)=𝑥1𝑓2(𝑥)=𝑔(𝑥)[1−(𝑥1/𝑔(𝑥))^2]𝑔(𝑥)=1+9∑^𝑛𝑖=2𝑥𝑖/(𝑛−1)	𝑥1𝑥∈[0,1]𝑖=0,𝑖=2,…,𝑛
ZDT3	30	[0, 1]	𝑓1(𝑥)=𝑥1𝑓2(𝑥)=𝑔(𝑥)[1−𝑥1/𝑔(𝑥)−𝑥1/𝑔(𝑥)sin(10𝜋𝑥19∑)]𝑔(𝑥)=1+^𝑛𝑖=2𝑥𝑖/(𝑛−1)	𝑥1𝑥∈[0,1]𝑖=0,𝑖=2,…,𝑛
ZDT4	10	𝑥1𝑥∈[0,1]𝑖∈[−5,5],𝑖=2,…,𝑛	𝑓1(𝑥)=𝑥1𝑓2(𝑥)=𝑔(𝑥)[1−𝑥1/𝑔(𝑥)]𝑔(𝑥)=1+10(𝑛−1)+∑^𝑛𝑖=2[𝑥^2𝑖−10cos(4𝜋𝑥𝑖)]	𝑥1𝑥∈[0,1]𝑖=0,𝑖=2,…,𝑛
ZDT6	10	[0, 1]	𝑓1(𝑥)=1−exp(−4𝑥1)sin^6(6𝜋𝑥1)𝑓2(𝑥)=𝑔(𝑥)[1−(𝑓1(𝑥)/𝑔(𝑥))^2]𝑔(𝑥)=1+9∑^𝑛𝑖=2𝑥𝑖/(𝑛−1)^0.25	𝑥1𝑥∈[0,1]𝑖=0,𝑖=2,…,𝑛