Page 201 - 《软件学报》2026年第3期

1164 软件学报 2026 年第 37 卷第 3 期

表 5 Two-Arch/IS 和其他算法在低维问题上获得的 IGD 结果 (续)

测试问题 目标数目 Two-Arch/IS KnEA IBEA GrEA
2 4.0221E+0 (2.74E+0) 1.5605E–1 (3.57E–1)+ 3.7296E–1 (1.50E–1) + 1.4562E–1 (3.18E–1) +
DTLZ3
3 4.8346E+0 (3.12E+0) 1.6175E–1 (1.95E–1) + 4.8071E–1 (7.62E–3) + 2.6765E–1 (3.38E–1) +
2 1.0370E–2 (2.40E–3) 1.8314E–1 (3.14E–1) – 3.3046E–1 (3.66E–1) – 8.3635E–2 (2.23E–1) =
DTLZ4
3 9.0088E–2 (8.53E–2) 9.4265E–2 (1.61E–1) – 8.0132E–2 (2.92E–3) + 1.4634E–1 (1.81E–1) –
2 1.0414E–2 (2.44E–3) 4.5989E–2 (1.35E–2) – 1.6464E–2 (1.36E–3) – 1.0558E–2 (1.59E–4) =
DTLZ5
3 1.1542E–2 (2.10E–3) 9.3796E–3 (1.62E–3) + 1.6437E–2 (1.38E–3) – 2.1783E–2 (1.36E–3) –
2 9.3458E–3 (1.02E–3) 7.4819E–2 (1.54E–2) – 2.9108E–2 (2.96E–3) – 1.0710E–2 (4.12E–5) –
DTLZ6
3 9.4169E–3 (1.09E–3) 1.2975E–2 (8.02E–3) = 2.6852E–2 (3.20E–3) – 2.2272E–2 (1.79E–4) –
2 6.4720E–3 (7.98E–4) 3.2909E–2 (7.83E–2) – 4.9855E–2 (1.34E–1) – 4.1663E–2 (7.71E–2) –
DTLZ7
3 6.0493E–2 (3.98E–3) 9.4944E–2 (8.81E–2) – 1.4384E–1 (1.30E–1) – 1.1471E–1 (9.16E–2) –
+/=/– － 5/1/13 4/1/14 4/3/12
为进一步展示种群中个体的分布情况, 选择和本文方法较为类似的 Two_Arch2 算法作为对比, 图 8 给出了
Two-Arch/IS 和 Two_Arch2 算法在 2 目标和 3 目标 DTLZ1 测试问题上的种群个体分布图. 从图 8(a) 和图 8(b) 可
以观察到, 在 2 目标的 DTLZ1 问题上, Two-Arch/IS 算法展现出了更好的多样性, 个体均匀覆盖在帕累托前沿面
上, 而 Two_Arch2 算法则存在局部区域未收敛. 从图 8(c) 和图 8(d) 可以观察到, 在 3 目标的 DTLZ1 问题上, Two-
Arch/IS 依旧展现出了较好的多样性, 而 Two_Arch2 算法多样性较差.

0.50 0.5
0.45
0.40 0.4
0.35
0.30 0.3
f 2 0.25 f 2
0.20 0.2
0.15
0.10 0.1
0.05
0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 0.1 0.2 0.3 0.4 0.5
f 1 f 1
(a) Two-Arch/IS on 2-DTLZ1 (b) Two_Arch2 on 2-DTLZ1

0.5 0.5
0.4 0.4
0.3 0.3
f 3 f 3
0.2 0.2
0.1 0.1
0 0
0 0 0 0
0.1 0.1
0.2 0.2 0.2 0.2
0.3 0.3 f 1
0.4 0.4 f 2 0.4 0.4 f 1
f 2
(c) Two-Arch/IS on 3-DTLZ1 (d) Two-Arch2 on 3-DTLZ1
图 8 Two-Arch/IS 和 Two_Arch2 在 2 目标和 3 目标 DTLZ1 问题上的种群个体分布图