Builds and returns the multi-objective DTLZ7 test problem. This problem can be characterized by a disconnected Pareto-optimal front in the search space. This introduces a new challenge to evolutionary multi-objective optimizers, i.e., to maintain different subpopulations within the search space to cover the entire Pareto-optimal front.
The DTLZ7 test problem is defined as follows:
Minimize \(f_1(\mathbf{x}) = x_1,\)
Minimize \(f_2(\mathbf{x}) = x_2,\)
\(\vdots\\\)
Minimize \(f_{M-1}(\mathbf{x}) = x_{M-1},\)
Minimize \(f_{M}(\mathbf{x}) = (1+g(\mathbf{x}_M)) h(f_1,f_2,\cdots,f_{M-1}, g),\)
with \(0 \leq x_i \leq 1\), for \(i=1,2,\dots,n,\)
where \(g(\mathbf{x}_M) = 1 + \frac{9}{|\mathbf{x}_M|} \sum_{x_i\in\mathbf{x}_M} x_i\)
and \(h(f_1,f_2,\cdots,f_{M-1}, g) = M - \sum_{i=1}^{M-1}\left[\frac{f_i}{1+g}(1 + sin(3\pi f_i))\right]\)
makeDTLZ7Function(dimensions, n.objectives)[integer(1)]
Number of decision variables.
[integer(1)]
Number of objectives.
[smoof_multi_objective_function]
Attention: Within the succeeding work of Deb et al. (K. Deb and L. Thiele and M. Laumanns and E. Zitzler (2002). Scalable multi-objective optimization test problems, Proceedings of the IEEE Congress on Evolutionary Computation, pp. 825-830) this problem was called DTLZ6.
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