makeDTLZ4Function {smoof}R Documentation

DTLZ4 Function (family)

Description

Builds and returns the multi-objective DTLZ4 test problem. It is a slight modification of the DTLZ2 problems by introducing the parameter α\alpha. The parameter is used to map xixiα\mathbf{x}_i \rightarrow \mathbf{x}_i^{\alpha}.

The DTLZ4 test problem is defined as follows:

Minimize f1(x)=(1+g(xM))cos(x1απ/2)cos(x2απ/2)cos(xM2απ/2)cos(xM1απ/2),f_1(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \cos(x_2^\alpha\pi/2) \cdots \cos(x_{M-2}^\alpha\pi/2) \cos(x_{M-1}^\alpha\pi/2),

Minimize f2(x)=(1+g(xM))cos(x1απ/2)cos(x2απ/2)cos(xM2απ/2)sin(xM1απ/2),f_2(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \cos(x_2^\alpha\pi/2) \cdots \cos(x_{M-2}^\alpha\pi/2) \sin(x_{M-1}^\alpha\pi/2),

Minimize f3(x)=(1+g(xM))cos(x1απ/2)cos(x2απ/2)sin(xM2απ/2),f_3(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \cos(x_2^\alpha\pi/2) \cdots \sin(x_{M-2}^\alpha\pi/2),

\vdots\\

Minimize fM1(x)=(1+g(xM))cos(x1απ/2)sin(x2απ/2),f_{M-1}(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \sin(x_2^\alpha\pi/2),

Minimize fM(x)=(1+g(xM))sin(x1απ/2),f_{M}(\mathbf{x}) = (1+g(\mathbf{x}_M)) \sin(x_1^\alpha\pi/2),

with 0xi10 \leq x_i \leq 1, for i=1,2,,n,i=1,2,\dots,n,

where g(xM)=xixM(xi0.5)2g(\mathbf{x}_M) = \sum\limits_{x_i\in \mathbf{x}_M}(x_i-0.5)^2

Usage

makeDTLZ4Function(dimensions, n.objectives, alpha = 100)

Arguments

dimensions

[integer(1)]
Number of decision variables.

n.objectives

[integer(1)]
Number of objectives.

alpha

[numeric(1)]
Optional parameter. Default is 100, which is recommended by Deb et al.

Value

[smoof_multi_objective_function]

References

K. Deb and L. Thiele and M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zurich, 112, 2001


[Package smoof version 1.6.0.3 Index]