Multiobjective Memetic Estimation of Distribution Algorithm Based on an Incremental Tournament Local Searcher

Table 1

Test instances.

Instance	Variable	Objectives	Number of evaluations
ZDT1	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)(𝐱𝑔(𝐱)∑/𝑔)=1+9^𝑛𝑖=2𝑥𝑖/(𝑛−1)	50 000
ZDT2	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1; 𝑓2(𝐱)=𝑔(𝐱)1−(𝑓1(𝐱)/𝑔(𝐱))^2; 𝑔(𝐱)=1+9∑^𝑛𝑖=2𝑥𝑖/(𝑛−1)	50 000
ZDT3	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1; 𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)/𝑔(𝐱)−𝑓1(𝐱)/𝑔(𝐱)sin(10𝜋𝑥1); 𝑔(𝐱)=1+9∑^𝑛𝑖=2𝑥𝑖/(𝑛−1)	50 000
ZDT4	𝑥1∈[0,1]; 𝑥𝑖∈[−5,5]; 𝑖=2,…,10	𝑓1(𝐱)=𝑥1; 𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)/𝑔(𝐱); 𝑔(𝐱)=1+10(𝑛−1)+∑^𝑛𝑖=2𝑥𝑖^2−10cos4𝜋𝑥𝑖	50 000
ZDT6	[0,1]^𝑛; 𝑛=10	𝑓1(𝐱)=1−exp−4𝑥1sin^66𝜋𝑥1; 𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)/𝑔(𝐱)^2; 𝑔(𝐱)=1+9[∑^𝑛𝑖=2𝑥𝑖/(9)]^0.25	50 000
DTLZ1	[0,1]^𝑛; 𝑛=7; 𝐱𝑀=[0,1]^5	𝑓1(𝐱)=0.5𝑥1𝑥2(𝐱)𝑓1+𝑔2(𝐱)=0.5𝑥11−𝑥2(𝐱)𝑓1+𝑔3(𝐱)=0.51−𝑥1(𝐱)𝑔(𝐱)𝐱1+𝑔=100𝑀+∑𝑥𝑖∈𝑋𝑀𝑥𝑖−0.5^2−cos20𝜋𝑥𝑖−0.5	60 000
DTLZ2	[0,1]^𝑛; 𝑛=12	𝑓1(𝐱)=cos0.5𝜋𝑥1cos(0.5𝜋𝑥2𝑓)(1+𝑔)2(𝐱)=cos0.5𝜋𝑥1sin(0.5𝜋𝑥2𝑓)(1+𝑔)3(𝐱)=sin(0.5𝜋𝑥1𝑔(𝐱))(1+𝑔)=∑^𝑛𝑖=3(𝑥𝑖−0.5)^2	50 000
DTLZ3	[0,1]^𝑛; 𝑛=12	𝑓1(𝐱)=cos0.5𝜋𝑥1cos(0.5𝜋𝑥2𝑓)(1+𝑔)2(𝐱)=cos0.5𝜋𝑥1sin(0.5𝜋𝑥2𝑓)(1+𝑔)3(𝐱)=sin(0.5𝜋𝑥1𝑔(𝐱)𝐱)(1+𝑔)=100𝑀+∑^𝑛𝑖=3𝑥𝑖−0.5^2−cos20𝜋𝑥𝑖−0.5	50 000
DTLZ4	[0,1]^𝑛; 𝑛=12	𝑓1(𝐱)=cos0.5𝜋𝑥1^𝛼cos(0.5𝜋𝑥2^𝛼𝑓)(1+𝑔)2(𝐱)=cos0.5𝜋𝑥1^𝛼sin(0.5𝜋𝑥2^𝛼𝑓)(1+𝑔)3(𝐱)=sin(0.5𝜋𝑥1^𝛼𝑔(𝐱))(1+𝑔)=∑^𝑛𝑖=3(𝑥𝑖−0.5)^2𝛼=100	50 000
DTLZ6	[0,1]^𝑛; 𝑛=22; 𝐱𝑀=[0,1]^20	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑥2𝑓3(𝐱)=1+𝑔𝐱𝑀ℎ𝑔𝐱𝑀=1+0.45∑𝑥𝑖𝑥𝑖∈𝐱𝑀ℎ=3−𝑓1/1+𝑔1+sin3𝜋𝑓1−𝑓2/1+𝑔1+sin3𝜋𝑓2	50 000
F1	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)(𝐱𝑔(𝐱)∑/𝑔)=1+9^𝑛𝑖=2𝑥𝑖−𝑥1^2/(𝑛−1)	15 000
F2	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−(𝑓1(𝐱)/𝑔(𝐱))^2𝑔(𝐱)=1+9∑^𝑛𝑖=2𝑥𝑖−𝑥1^2/(𝑛−1)	15 000
F3	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=1−exp−4𝑥1sin^66𝜋𝑥1; 𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)/𝑔(𝐱)^2; 𝑔(𝐱)=1+9[∑^𝑛𝑖=2𝑥𝑖−𝑥1^2/9]^0.25	100 000
F4	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=cos0.5𝜋𝑥1cos(0.5𝜋𝑥2𝑓)(1+𝑔)2(𝐱)=cos0.5𝜋𝑥1sin(0.5𝜋𝑥2𝑓)(1+𝑔)3(𝐱)=sin(0.5𝜋𝑥1𝑔(𝐱))(1+𝑔)=∑^𝑛𝑖=3(𝑥𝑖−𝑥1)^2	35 000
F5	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)(𝐱𝑔(𝐱)∑/𝑔)=1+9^𝑛𝑖=2𝑥𝑖^2−𝑥1^2/(𝑛−1)	15 000
F6	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−(𝑓1(𝐱)/𝑔(𝐱))^2𝑔(𝐱)=1+9∑^𝑛𝑖=2𝑥𝑖^2−𝑥1^2/(𝑛−1)	15 000
F7	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=1−exp−4𝑥1sin^66𝜋𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)/𝑔(𝐱)^2𝑔(𝐱)=1+9[∑^𝑛𝑖=2𝑥𝑖^2−𝑥1^2/9]^0.25	100 000
F8	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=cos0.5𝜋𝑥1cos(0.5𝜋𝑥2𝑓)(1+𝑔)2(𝐱)=cos0.5𝜋𝑥1sin(0.5𝜋𝑥2𝑓)(1+𝑔)3(𝐱)=sin(0.5𝜋𝑥1𝑔(𝐱))(1+𝑔)=∑^𝑛𝑖=3(𝑥𝑖^2−𝑥1)^2	35 000
F9	[0,1]×[0,10]^𝑛−1; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)/𝑔(𝐱)𝑔(𝐱)=0.00025∑^𝑛𝑖=2(𝑥^2𝑖−𝑥1)^2−∏^𝑛𝑖=2cos((𝑥^2𝑖)/√𝑖−1)+2	100 000
F10	[0,1]×[0,10]^𝑛−1; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)(𝐱𝑔(𝐱)/𝑔)=1+10(𝑛−1)+∑^𝑛𝑖=2𝑥𝑖^2−𝑥1^2−10cos2𝜋𝑥𝑖^2−𝑥1	100 000
AF1	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)(𝐱𝑔(𝐱)∑/𝑔)=1+9^𝑛𝑖=2100sin(𝜋𝑥𝑖)−𝑥1^2/(𝑛−1)	30 000
AF2	[0,1]^𝑛; 𝑛=30	𝑓1(𝐱)=1−exp−4𝑥1sin^66𝜋𝑥1𝑓2(𝐱)=𝑔(𝐱)1−𝑓1(𝐱)/𝑔(𝐱)^2𝑔(𝐱)=1+9[∑^𝑛𝑖=26sin𝜋𝑥𝑖−𝑥1^2/9]^0.25	60 000