makeZDT6Function function

ZDT6 Function

Builds and returns the two-objective ZDT6 test problem. For $m$ objective it is defined as follows [REMOVE_ME]$f(\mathbf{x}) = \left(f_1(\mathbf{x}), f_2(\mathbf{x})\right) [REMOVE_ME_2]$

with [REMOVE_ME]$f_1(\mathbf{x}) = 1 - \exp(-4\mathbf{x}_1)\sin^6(6\pi\mathbf{x}_1), f_2(\mathbf{x}) = g(\mathbf{x}) h(f_1(\mathbf{x}_1), g(\mathbf{x})) [REMOVE_ME_2]$

where [REMOVE_ME]$g(\mathbf{x}) = 1 + 9 \left(\frac{\sum_{i = 2}^{m}\mathbf{x}_i}{m - 1}\right)^{0.25}, h(f_1, g) = 1 - \left(\frac{f_1(\mathbf{x})}{g(\mathbf{x})}\right)^2 [REMOVE_ME_2]$

and $\mathbf{x}_i \in [0,1], i = 1, \ldots, m$. This function introduced two difficulities (see reference): 1. the density of solutions decreases with the closeness to the Pareto-optimal front and 2. the Pareto-optimal solutions are nonuniformly distributed along the front.

makeZDT6Function(dimensions)

Arguments

• dimensions: [integer(1)]

  Number of decision variables.

Returns

[smoof_multi_objective_function]

Description

Builds and returns the two-objective ZDT6 test problem. For $m$ objective it is defined as follows

with

where

and $\mathbf{x}_i \in [0,1], i = 1, \ldots, m$. This function introduced two difficulities (see reference): 1. the density of solutions decreases with the closeness to the Pareto-optimal front and 2. the Pareto-optimal solutions are nonuniformly distributed along the front.

References

E. Zitzler, K. Deb, and L. Thiele. Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation, 8(2):173-195, 2000