A new angle-based preference selection mechanism for solving many-objective optimization problems

Abstract

Many evolutionary multi-objective optimization (EMO) methodologies have been proposed and performed very well on finding a representative set of Pareto-optimal solutions, but this advantage will be weakened with the increasing number of objectives. And in real applications, what decision makers (DMs) want is a unique solution or a set of solutions rather than the overall Pareto-optimal front. It is a difficult task to solve many-objective problems by using preference information provided by decision maker (DM) during optimization process. In this paper, a new angle-based preference selection mechanism is proposed, which replaces the traditional crowding distance with the aid of preference information provided by the DMs. Particularly, we combine it with a multi-objective immune algorithm with non-dominated neighbor-based selection. The proposed method has been extensively compared with other recently proposed preference-based EMO approaches over DTLZ1, DTLZ2, and DTLZ3 test problems with 4–100 objectives. The results of the experiment indicate that the proposed algorithm can achieve competitive and better results.

Appendix

DTLZ1:

$$\begin{aligned}&f_1 (X)=0.5x_1 x_2 \cdots x_{M-1} \left( {1+g(\hbox {X}_M )} \right) \\&f_2 (X)=0.5x_1 x_2 \cdots \left( {1-x_{M-1} } \right) \left( {1+g(\hbox {X}_M )} \right) \\&\cdots \\&f_{M-1} (X)=0.5x_1 \left( {1-x_2 } \right) \left( {1+g(\hbox {X}_M )} \right) \\&f_M (X)=0.5\left( {1-x_1 } \right) \left( {1+g(\hbox {X}_M )} \right) \\ \end{aligned}$$

where $g(X_M )=100\left[ \left| {X_M } \right| +\sum _{x_i \in X_M } \left( \left( {x_i -0.5} \right) ^{2}-\cos \left( {20\pi \left( {x_i -0.5} \right) } \right) \right) \right]$, and $X=(x_1 ,\ldots ,x_n )^{T}\in [0,1]^{n}$. $X_M$ represents the later $k=(n-M+1)$ variables of decision vector, where n and M denote the number of decision variables and the number of objectives, respectively. In our experiments, $k=5$ is set.

DTLZ2:

$$\begin{aligned}&f_1 (X)=\left( {1+g(X_M )} \right) \cos \left( {0.5x_1 \pi } \right) \cdots \cos \left( {0.5x_{M-1} \pi } \right) \\&f_2 (X)=\left( {1+g(X_M )} \right) \cos \left( {0.5x_1 \pi } \right) \cdots \sin \left( {0.5x_{M-1} \pi } \right) \\&\cdots \\&f_{M-1} (X)=\left( {1+g(X_M )} \right) \cos \left( {0.5x_1 \pi } \right) \sin \left( {0.5x_2 \pi } \right) \\&f_M (X)=\left( {1+g(X_M )} \right) \sin \left( {0.5x_1 \pi } \right) \\ \end{aligned}$$

where $g(X_M )=\sum _{x_i \in X_M } {\left( {x_i -0.5} \right) ^{2}}$, and $X=(x_1 ,\ldots ,x_n )^{T}\in [0,1]^{n}$. $X_M$ represents the later $k=(n-M+1)$ variables of decision vector, where n and M denote the number of decision variables and the number of objectives, respectively. In our experiments, $k=10$ is set.

DTLZ3:

$$\begin{aligned}&f_1 (X)=\left( {1+g(X_M )} \right) \cos \left( {0.5x_1 \pi } \right) \cdots \cos \left( {0.5x_{M-1} \pi } \right) \\&f_2 (X)=\left( {1+g(X_M )} \right) \cos \left( {0.5x_1 \pi } \right) \cdots \sin \left( {0.5x_{M-1} \pi } \right) \\&\cdots \\&f_{M-1} (X)=\left( {1+g(X_M )} \right) \cos \left( {0.5x_1 \pi } \right) \sin \left( {0.5x_2 \pi } \right) \\&f_M (X)=\left( {1+g(X_M )} \right) \sin \left( {0.5x_1 \pi } \right) \\ \end{aligned}$$

where $g(X_M )=100\left[ \left| {X_M } \right| +\sum _{x_i \in X_M } \left( \left( {x_i -0.5} \right) ^{2}-\cos \left( {20\pi \left( {x_i -0.5} \right) } \right) \right) \right]$, and $X=(x_1 ,\ldots ,x_n )^{T}\in [0,1]^{n}$. $X_M$ represents the later $k=(n-M+1)$ variables of decision vector, where n and M denote the number of decision variables and the number of objectives, respectively. In our experiments, $k=10$ is set.