An Adaptive Reference Vector Adjustment Strategy and Improved Angle-Penalized Value Method for RVEA

Table 1

HV results of the 6 algorithms on benchmarks MaF–MaF9.

	M	A-NSGA-III [32]	MOEA/D [5]	[16]	MaOEA-IT [36]	MOEA/D-PaS [37]	AP-RVEA
MaF1	3	2.0353e − 1 (1.24e − 3)	1.9823e − 1 (1.76e − 6)	2.1216e − 1 (1.81e − 3)	8.0588e − 2 (4.47e − 2)	1.8407e − 1 (1.07e − 3)	2.1699e − 1 (5.89e − 4)
	5	8.3089e − 3 (1.82e − 4)	1.1127e − 2 (3.73e − 4)	4.0294e − 3 (4.33e − 4)	2.6586e − 3 (2.46e − 3)	5.4484e − 3 (1.01e − 4)	6.1155e − 3 (5.10e − 4)
	8	1.1271e − 5 (3.71e − 6)	3.1161e − 6 (9.84e − 7)	7.0858e − 7 (4.00e − 7)	7.5799e − 7 (1.56e − 6)	2.6431e − 5 (1.73e − 6)	2.6546e − 6 (1.19e − 6)
	10	7.9566e − 9 (4.94e − 9)	1.3977e − 8 (1.82e − 9)	5.7247e − 9 (3.03e − 9)	2.6723e − 9 (5.03e − 9)	2.3472e − 8 (3.55e − 8)	6.4159e − 8 (1.93e − 8)
MaF2	3	2.2882e − 1 (5.62e − 4)	2.3886e − 1 (7.14e − 4)	2.4321e − 1 (9.30e − 4)	1.3382e − 1 (1.83e − 2)	2.3105e − 1 (1.28e − 3)	2.2968e − 1 (1.52e − 3)
	5	1.8268e − 1 (1.23e − 3)	1.8704e − 1 (4.46e − 4)	1.7520e − 1 (1.92e − 3)	1.3101e − 1 (3.22e − 2)	9.2288e − 2 (6.42e − 3)	1.9060e − 1 (1.74e − 3)
	8	2.0533e − 1 (3.14e − 3)	2.0719e − 1 (6.02e − 4)	1.7835e − 1 (5.28e − 3)	1.4836e − 1 (3.23e − 2)	7.4638e − 2 (1.84e − 2)	1.7799e − 1 (4.91e − 3)
	10	1.0651e − 1 (9.50e − 3)	1.7083e − 1 (7.94e − 4)	1.0762e − 1 (9.34e − 3)	9.2650e − 2 (1.41e − 2)	4.3372e − 2 (3.33e − 4)	1.7975e − 1 (7.45e − 3)
MaF3	3	9.5716e − 1 (6.89e − 4)	9.5491e − 1 (1.13e − 3)	9.6108e − 1 (8.13e − 4)	8.4815e − 2 (2.60e − 1)	7.1020e − 1 (3.76e − 1)	9.5871e − 1 (1.76e − 3)
	5	9.4447e − 1 (2.12e − 1)	9.8725e − 1 (1.44e − 3)	8.2530e − 1 (3.35e − 1)	0.0000e + 0 (0.00e + 0)	2.1056e − 1 (3.80e − 1)	9.9929e − 1 (3.38e − 4)
	8	7.8110e − 1 (3.94e − 1)	9.6767e − 1 (2.78e − 3)	9.9682e − 1 (3.04e − 4)	0.0000e + 0 (0.00e + 0)	9.5248e − 3 (2.85e − 2)	9.9939e − 1 (2.13e − 2)
	10	9.99454e − 1 (1.28e − 2)	9.4948e − 1 (2.78e − 3)	9.9851e − 1 (4.61e − 4)	0.0000e + 0 (0.00e + 0)	2.8415e − 3 (1.56e − 2)	9.1280e − 1 (2.72e − 1)
MaF4	3	5.2149e − 1 (6.16e − 3)	5.0055e − 1 (3.40e − 3)	5.2054e − 1 (2.85e − 3)	4.6992e − 2 (1.37e − 1)	4.2732e − 1 (1.80e − 1)	4.5632e − 1 (8.95e − 2)
	5	5.8544e − 2 (5.49e − 3)	7.4840e − 3 (2.94e − 3)	3.4932e − 2 (1.10e − 2)	0.0000e + 0 (0.00e + 0)	4.2368e − 2 (2.20e − 2)	6.0373e − 2 (1.06e − 2)
	8	3.8552e − 5 (1.55e − 5)	2.5638e − 6 (3.90e − 7)	3.9599e − 5 (3.54e − 5)	0.0000e + 0 (0.00e + 0)	2.8057e − 3 (1.47e − 4)	1.7934e − 5 (1.69e − 5)
	10	3.0500e − 8 (2.50e − 8)	1.4597e − 8 (1.86e − 9)	2.8142e − 7 (2.42e − 7)	1.1961e − 7 (6.55e − 7)	2.0046e − 4 (4.62e − 5)	1.8383e − 7 (1.28e − 7)
MaF5	3	4.9687e − 1 (9.09e − 2)	4.7843e − 1 (1.35e − 1)	4.5833e − 1 (1.72e − 1)	1.0907e − 1 (3.90e − 2)	4.5198e − 1 (1.56e − 1)	5.5958e − 1 (3.09e − 5)
	5	7.3778e − 1 (2.33e − 2)	4.4689e − 1 (1.05e − 1)	8.1174e − 1 (4.40e − 4)	1.2033e − 1 (9.19e − 2)	7.4462e − 1 (3.40e − 2)	8.0819e − 1 (2.53e − 2)
	8	8.2333e − 1 (1.43e − 2)	3.5794e − 1 (8.63e − 2)	8.9065e − 1 (4.49e − 2)	2.5990e − 2 (4.18e − 2)	8.9836e − 1 (6.10e − 3)	8.9915e − 1 (6.73e − 3)
	10	8.7108e − 1 (2.20e − 2)	2.5125e − 1 (7.94e − 2)	8.2530e − 1 (9.06e − 2)	6.7036e − 2 (4.50e − 2)	8.7214e − 1 (1.21e − 1)	8.0365e − 1 (2.30e − 2)
MaF6	3	1.9257e − 1 (1.19e − 3)	1.8185e − 1 (5.24e − 6)	1.8865e − 1 (1.50e − 3)	4.6163e − 2 (1.07e − 2)	1.8937e − 1 (8.73e − 4)	1.7995e − 1 (1.09e − 3)
	5	1.2878e − 1 (4.32e − 4)	1.1532e − 1 (3.49e − 2)	1.0692e − 1 (1.46e − 2)	1.0711e − 1 (4.43e − 2)	1.0778e − 1 (8.34e − 4)	1.3141e − 1 (9.79e − 3)
	8	7.5412e − 2 (4.26e − 2)	9.5638e − 2 (2.45e − 2)	9.4549e − 2 (3.40e − 3)	8.3754e − 2 (3.75e − 2)	6.3906e − 2 (3.70e − 2)	9.5782e − 2 (1.33e − 3)
	10	8.4470e − 2 (5.98e − 3)	7.1992e − 2 (4.28e − 2)	9.4688e − 2 (1.14e − 3)	9.0951e − 3 (2.77e − 2)	8.1006e − 2 (2.76e − 2)	8.5933e − 2 (1.23e − 2)
MaF7	3	2.5256e − 1 (5.32e − 3)	2.5331e − 1 (1.38e − 2)	2.6855e − 1 (1.22e − 2)	1.2596e − 1 (3.78e − 2)	2.1428e − 1 (9.79e − 2)	2.5443e − 1 (5.81e − 3)
	5	2.1329e − 1 (8.04e − 3)	1.4613e − 1 (1.31e − 2)	2.2129e − 1 (3.85e − 3)	4.9265e − 3 (9.10e − 3)	2.1578e − 1 (6.01e − 2)	2.4446e − 1 (2.11e − 3)
	8	9.2868e − 2 (1.71e − 2)	3.9645e − 3 (1.39e − 2)	1.8451e − 1 (2.99e − 3)	0.0000e + 0 (0.00e + 0)	1.8405e − 1 (7.74e − 3)	1.2609e − 1 (1.95e − 2)
	10	1.3202e − 1 (1.55e − 2)	5.3985e − 5 (1.47e − 4)	1.0158e − 1 (4.76e − 2)	0.0000e + 0 (0.00e + 0)	1.5212e − 3 (8.33e − 3)	6.6357e − 2 (1.80e − 2)
MaF8	3	2.5863e − 1 (2.33e − 3)	2.5984e − 1 (6.33e − 3)	2.6481e − 1 (2.65e − 3)	2.4228e − 2 (3.38e − 2)	0.0000e + 0 (0.00e + 0)	2.4098e − 1 (3.60e − 3)
	5	1.0867e − 1 (1.69e − 3)	1.1426e − 1 (2.34e − 3)	8.8484e − 2 (5.55e − 3)	6.8305e − 3 (1.08e − 2)	0.0000e + 0 (0.00e + 0)	1.1673e − 2 (5.91e − 3)
	8	2.4678e − 2 (6.97e − 4)	2.2108e − 2 (4.52e − 4)	1.1092e − 2 (3.72e − 3)	8.5991e − 4 (1.66e − 3)	0.0000e + 0 (0.00e + 0)	1.5505e − 2 (1.89e − 3)
	10	3.8703e − 3 (7.50e − 4)	6.2901e − 3 (2.08e − 4)	2.6006e − 3 (8.94e − 4)	4.9137e − 5 (1.52e − 4)	0.0000e + 0 (0.00e + 0)	7.9081e − 3 (2.71e − 4)
MaF9	3	8.3695e − 1 (1.26e − 3)	8.3183e − 1 (1.52e − 2)	8.4344e − 1 (9.81e − 4)	9.9304e − 3 (5.44e − 2)	7.7608e − 1 (1.48e − 1)	8.5828e − 1 (4.12e − 2)
	5	2.8542e − 1 (3.37e − 3)	3.1093e − 1 (2.91e − 3)	2.3391e − 1 (1.09e − 2)	0.0000e + 0 (0.00e + 0)	2.3636e − 1 (5.28e − 2)	2.5743e − 1 (1.39e − 2)
	8	3.3482e − 2 (5.60e − 3)	4.1322e − 2 (5.03e − 4)	2.2144e − 2 (2.52e − 3)	0.0000e + 0 (0.00e + 0)	4.4560e − 2 (3.99e − 4)	1.9959e − 2 (2.68e − 3)
	10	6.2283e − 3 (1.26e − 3)	1.1816e − 2 (2.24e − 3)	4.6773e − 3 (1.18e − 3)	0.0000e + 0 (0.00e + 0)	1.2047e − 2 (2.47e − 3)	1.2064e − 2 (4.20e − 4)
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