An improved multiobjective cultural algorithm with a multistrategy knowledge base

Abstract

Based on the dual-inheritance framework of cultural evolution, an improved multiobjective cultural algorithm (IMOCA) with a multistrategy knowledge base is presented in this paper. Inspired by the original versions of the cultural algorithm (CA), four basic types of knowledge sources, i.e., normative, situational, topographical and historical knowledge, are effectively utilized in the proposed IMOCA. Several modifications with the knowledge base of the IMOCA are made to tackle the characteristics of the multiobjective problem. Situational knowledge is used as an external repository for storing elite individuals, and the redesigned topographical knowledge functions as a search engine to broaden the expansion of the obtained solution set. The historical knowledge used in the IMOCA aims to select a productive knowledge source to generate new individuals. Furthermore, a simple mutation scheme is introduced into the knowledge base as an influence function for the purpose of fine tuning in the late stage of search. After configuring the parameters used in IMOCA, two classic benchmark suites, i.e., WFG and MaF, are used to assess the performance of the IMOCA in approaching the Pareto fronts (PFs) with accuracy and diversity. Nondominated solution sets obtained by the IMOCA are compared with 8 state-of-the-art multiobjective algorithms available in the literature. A statistical analysis is conducted, which reveals that, by modifying the basic knowledge structure of the CA, the proposed multiobjective cultural algorithm is competent enough to handle multiobjective problems with competitive performance.

Appendix: Performance of different knowledge source as IMOCA solving MaF test suite

Here in A, Figs. 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23 show the minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when the IMOCA optimizing the MaF test suite in a random run.

Fig. 9

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF1 using the IMOCA (iteration t ∈ [1,300])

Fig. 10

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF2 using the IMOCA (iteration t ∈ [1,300])

Fig. 11

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF3 using the IMOCA (iteration t ∈ [1,300])

Fig. 12

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF4 using the IMOCA (iteration t ∈ [1,300])

Fig. 13

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF5 using the IMOCA (iteration t ∈ [1,300])

Fig. 14

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF6 using the IMOCA (iteration t ∈ [1,300])

Fig. 15

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF7 using the IMOCA (iteration t ∈ [1,300])

Fig. 16

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ as when optimizing MaF8 using the IMOCA (iteration t ∈ [1,300])

Fig. 17

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF9 using the IMOCA (iteration t ∈ [1,300])

Fig. 18

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF10 using the IMOCA (iteration t ∈ [1,300])

Fig. 19

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF11 using the IMOCA (iteration t ∈ [1,300])

Fig. 20

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF12 using the IMOCA (iteration t ∈ [1,300])

Fig. 21

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF13 using the IMOCA (iteration t ∈ [1,300])

Fig. 22

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF14 using the IMOCA (iteration t ∈ [1,300])

Fig. 23

The minimum values for ${\sum }_{m=1}^{M} f_{m}(\mathbf {x})$ when optimizing MaF15 using the IMOCA (iteration t ∈ [1,300])