8 An Introduction to Many-Objective Evolutionary Optimization
301
1
0
1
1
1
0
f3
f3
f2
1
f1
Fig. 8.25 Pareto front for DTLZ2 to DTLZ4 test problems in 3D objective space. The Pareto front
lies on the hyperplane
f 2 (x) = 1 (hypersphere with radius 1) and all objective values at positive.
This is also the real Pareto front for WFG4 to WFG9 (after normalization)
Because the Pareto fronts are easy to construct, it is also easy to obtain reference
points on the Pareto front to calculate performance metrics that requires references
on the Pareto front, such as the GD in Sect. 8.3.2.2.
8.6.2.2 WFG
The WFG test kit [26] is a relatively recent work on test problems done by the
Walking Fish Group. The WFG test kit has an interesting feature: on top of
scalability, the WFG suggests that the difficulties of test problems should also
be tunable. The WFG test kit allows users to tune the test problems by adding
transformation functions which should increase their difficulties.
The WFG test kit is made of some building blocks called the shape functions
and transformation functions. The shape function is the test functions’ base shape.
Each of the objectives can have one of the shape functions, i.e., all objectives can
have different shapes, but only use one shape function for each objective f i . The
transformation function will then transform (i.e., make changes to) the base shape
in form of bias, shift, and reduction.
301
1
0
1
1
1
0
f3
f3
f2
1
f1
Fig. 8.25 Pareto front for DTLZ2 to DTLZ4 test problems in 3D objective space. The Pareto front
lies on the hyperplane
f 2 (x) = 1 (hypersphere with radius 1) and all objective values at positive.
This is also the real Pareto front for WFG4 to WFG9 (after normalization)
Because the Pareto fronts are easy to construct, it is also easy to obtain reference
points on the Pareto front to calculate performance metrics that requires references
on the Pareto front, such as the GD in Sect. 8.3.2.2.
8.6.2.2 WFG
The WFG test kit [26] is a relatively recent work on test problems done by the
Walking Fish Group. The WFG test kit has an interesting feature: on top of
scalability, the WFG suggests that the difficulties of test problems should also
be tunable. The WFG test kit allows users to tune the test problems by adding
transformation functions which should increase their difficulties.
The WFG test kit is made of some building blocks called the shape functions
and transformation functions. The shape function is the test functions’ base shape.
Each of the objectives can have one of the shape functions, i.e., all objectives can
have different shapes, but only use one shape function for each objective f i . The
transformation function will then transform (i.e., make changes to) the base shape
in form of bias, shift, and reduction.