Migration NSGA: method to improve a non-elitist searching of Pareto front, with application in magnetics

Figures & data

Figure 1. Three different Pareto fronts and relevant starting individual sets for the same optimization problem. f₁ is the magnetic field uniformity and f₂ is the solution sensitivity. Both quantities are dimensionless.

Figure 2. The flow chart of the first variant of migration NSGA algorithm: MNSGA-1; the migration occurs after the genetic operator.

Figure 3. The flow chart of the second variant of migration NSGA algorithm: M-NSGA-2; the migration occurs before the genetic operator.

Figure 4. Test problem #1a: shape of the f₁ and f₂ functions in the domain.

Figure 5. Test problem #1: (a) and (c) analytical Pareto front and (b) and (d) analytical inverse image of Pareto front for problem #1a and #1b, respectively.

Figure 6. (a) Geometry of the device with eight design variables and three uncertain parameters (parameters underlined). (b) Mesh detail.

Figure 7. (a) Pareto front and (b) design variable space representation obtained using NSGA-II algorithm considering a population of N_p = 20 individuals, N_g = 20 generations and three different initial sets of design variables. (c) Pareto front and (d) design variable space representation obtained using NSGA-II algorithm considering N_p = 50 individuals, N_g = 20 generations and three different initial sets of design variables.

Figure 8. Comparison of (a) the Pareto front obtained using classical NSGA-II, MNSGA-1 and MNSGA-2 algorithm considering N_g = 20, N_p = 20, N_m = 10, and T_m = 5. (b) Comparison of the design variable space obtained using classical NSGA-II, MNSGA-1 and MNSGA-2 algorithms. Crowding distance on (c) objective space and (d) on design variable space.

Figure 9. Comparison of (a) the Pareto front obtained using classical NSGA-II, MNSGA-1 and MNSGA-2 algorithm considering N_g = 20, N_p = 20, N_m = 20, and T_m = 5. (b) Comparison of the design variable space obtained using classical NSGA-II, MNSGA-1 and MNSGA-2 algorithms. Crowding distance on (c) objective space and (d) on design variable space.

Figure 10. Comparison of (a) the Pareto front obtained using classical NSGA-II, MNSGA-1 and MNSGA-2 algorithm considering N_g = 50, N_p = 20, N_m = 10, and T_m = 2. (b) Comparison of the design variable space obtained using classical NSGA-II, MNSGA-1 and MNSGA-2 algorithms. Crowding distance on (c) objective space and (d) on design variable space.

Figure 11. Comparison of the Pareto front and the design variable space obtained using classical NSGA-II, MNSGA-1 and MNSGA-2 algorithm considering N_g = 20, N_p = 20, N_m = 10 (a) and (e) T_m = 2 and (b) and (f) T_m = 5. Crowding distance on (c) and (g) objective space and (d) and (h) on design variable space for T_m = 2 and 5, respectively.

Table 1. Results of the tests performed: average of the rms error evaluated on inverse image using (19) and objective space using (20) of 30 optimization runs.

	f_objective		Np	Ng	Pm	Tm	errrms(x)	errrms(f)
Case #1	(1) and (2)	NSGA	20	20	–	–	0.21	0.08
		MNSGA_1	20	20	10	5	0.17	0.05
		MNSGA_2	20	20	10	5	0.15	0.04
Case #2	(1) and (2)	NSGA	20	20	–	–	0.34	0.14
		MNSGA_1	20	20	10	2	0.41	0.20
		MNSGA_2	20	20	10	2	0.33	0.14
Case #3	(1) and (2)	NSGA	20	50	–	–	0.12	0.04
		MNSGA_1	20	50	10	5	0.08	0.02
		MNSGA_2	20	50	10	5	0.09	0.02
Case #4	(5) and (6)	NSGA	20	20	–	–	0.59	0.48
		MNSGA_1	20	20	10	5	0.48	0.39
		MNSGA_2	20	20	10	5	0.48	0.39
Case #5	(5) and (6)	NSGA	20	20	–	–	0.47	0.28
		MNSGA_1	20	20	10	2	0.38	0.25
		MNSGA_2	20	20	10	2	0.39	0.27
Case #6	(5) and (6)	NSGA	20	50	–	–	0.43	0.29
		MNSGA_1	20	50	10	5	0.31	0.15
		MNSGA_2	20	50	10	5	0.35	0.24

Table 2. Results of the T-test (p-value) in order to compare rms error data resumed in Table 1 of MNSGA-1 and NSGA-II, MNSGA-2 and NSGA-II, and MNSGA-1 and MNSGA-2.

	f_objective		Np	Ng	Pm	Tm	T-test	T-test
								errrms (f)
							errrms (x)
Case #1	(1) and (2)	MNSGA_1-NSGA	20	20	–	–	0.016	0.005
	Problem #1a	MNSGA_2-NSGA	20	20	10	5	0.0001	0.0002
		MNSGA_1-MNSGA_2	20	20	10	5	0.04	0.11
Case #2	(1) and (2)	MNSGA_1-NSGA	20	20	–	–	0.00004	0.0010
	Problem #1a	MNSGA_2-NSGA	20	20	10	2	0.00007	0.0005
		MNSGA_1-MNSGA_2	20	20	10	2	0.79	0.82
Case #3	(1) and (2)	MNSGA_1-NSGA	20	50	–	–	0.0002	0.0007
	Problem #1a	MNSGA_2-NSGA	20	50	10	5	0.0029	0.0049
		MNSGA_1-MNSGA_2	20	50	10	5	0.18	0.34
Case #4	(5) and (6)	MNSGA_1-NSGA	20	20	–	–	0.064	0.42
	Problem #1b	MNSGA_2-NSGA	20	20	10	5	0.059	0.39
		MNSGA_1-MNSGA_2	20	20	10	5	0.96	0.96
Case #5	(5) and (6)	MNSGA_1-NSGA	20	20	–	–	0.081	0.60
	Problem #1b	MNSGA_2-NSGA	20	20	10	2	0.083	0.85
		MNSGA_1-MNSGA_2	20	20	10	2	0.84	0.74
Case #6	(5) and (6)	MNSGA_1-NSGA	20	50	–	–	0.006	0.031
	Problem #1b	MNSGA_2-NSGA	20	50	10	5	0.092	0.50
		MNSGA_1-MNSGA_2	20	50	10	5	0.19	0.08

Figure 12. (a) The approximated Pareto front using classical NSGA-II algorithm with N_p = 50 and N_g = 50, (b) the B–H curves estimated using design variables of solutions in the Pareto set, (c) the approximated Pareto front using classical NSGA-II algorithm with N_p = 20 and N_g = 50 and (d) the B–H curves estimated using design variables of solutions in the Pareto set.

Figure 13. The approximated Pareto front using (a) MNSGA-1 and (c) MNSGA-2. Algorithms setting N_p = 20, N_g = 50, N_m = 10 and T_m = 5. (b) and (d) are the B–H curves estimated using design variables of solutions in the Pareto set.

Figure 14. (a) The approximated Pareto front using (a) MNSGA-1 and (c) MNSGA-2. Algorithms setting N_p = 20, N_g = 20, N_m = 10 and T_m = 2. (b) and (d) are the B–H curves estimated using design variables of solutions in the Pareto set.

Figure 15. Approximated Pareto front using NSGA-II (circles) and MNSGA-1 (squares) algorithm starting from the same initial population (triangles) for the problem #3.