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D. Irawan and B. Naujoks
Although completely tunable, it would not make sense if all researchers use
different problems, all tuned by themselves, to compare algorithms’ performances.
To address this, the WFG also created nine standard test problems (collectively
termed as “WFG test suite”) so researchers can use this for comparing algorithms.
The nine standard test problems have predefined shape and transformation
functions. Users can still tune the number of objectives and variables to some extent
(there are still some requirement on the variables).
8.7 Summary
In this chapter we learned about the difference between single- , multi- , and
many-objective optimization problems. We also mentioned the challenges which
makes many-objective problems much more difficult to solve than multi-objective
problems. We also learned the basics of evolutionary algorithms and the building
blocks: recombination, mutation, and selection.
Furthermore, we learned how evolutionary algorithms can solve multi- and
many-objective optimization problems. Several well-known algorithms were
explained. For multi-objective problems, NSGA-II and SMS-EMOA can be used.
Both algorithms use non-dominated sorting as their primary selection operator.
Their secondary selection operator differs. NSGA-II uses crowding distance, while
SMS-EMOA uses S-metric selection. Another difference is that NSGA-II uses
(μ + μ) selection scheme, while SMS-EMOA uses the steady-state (μ + 1) scheme.
For many-objective problems, MOEA/D and NSGA-III can be used. The first
algorithm, MOEA/D, decomposes the many-objective problem into many singleobjective problems. The second algorithm, NSGA-III, is an algorithm based
on NSGA-II. The difference lies in the secondary selection method: instead of
crowding distance, NSGA-III uses reference points which represent the preferences
and weights of different objective functions. Surrogate models which are used in
multi- and many-objective problems were also mentioned in this chapter.
In the last section, benchmarking problems for comparing algorithms are
explained. These benchmarking problems are predefined functions which are
intended to test how well an algorithm can solve problems with different
characteristics and difficulties. Some test problems are designed to be scalable,
meaning that the number of objective functions and variables can be changed. This
feature is especially valuable for researches in many-objective optimization.
References
1. R.B. Agrawal, K. Deb, R.B. Agrawal, Simulated binary crossover for continuous search space.
Complex Systems 9,115–148 (1994)
2. N. Andreasson, A. Evgrafov, M. Patriksson, E. Gustavsson, M. Onnheim, Introduction to
Continuous Optimization, 2nd edn. (Studentlitteratur AB, Lund, 2013)
D. Irawan and B. Naujoks
Although completely tunable, it would not make sense if all researchers use
different problems, all tuned by themselves, to compare algorithms’ performances.
To address this, the WFG also created nine standard test problems (collectively
termed as “WFG test suite”) so researchers can use this for comparing algorithms.
The nine standard test problems have predefined shape and transformation
functions. Users can still tune the number of objectives and variables to some extent
(there are still some requirement on the variables).
8.7 Summary
In this chapter we learned about the difference between single- , multi- , and
many-objective optimization problems. We also mentioned the challenges which
makes many-objective problems much more difficult to solve than multi-objective
problems. We also learned the basics of evolutionary algorithms and the building
blocks: recombination, mutation, and selection.
Furthermore, we learned how evolutionary algorithms can solve multi- and
many-objective optimization problems. Several well-known algorithms were
explained. For multi-objective problems, NSGA-II and SMS-EMOA can be used.
Both algorithms use non-dominated sorting as their primary selection operator.
Their secondary selection operator differs. NSGA-II uses crowding distance, while
SMS-EMOA uses S-metric selection. Another difference is that NSGA-II uses
(μ + μ) selection scheme, while SMS-EMOA uses the steady-state (μ + 1) scheme.
For many-objective problems, MOEA/D and NSGA-III can be used. The first
algorithm, MOEA/D, decomposes the many-objective problem into many singleobjective problems. The second algorithm, NSGA-III, is an algorithm based
on NSGA-II. The difference lies in the secondary selection method: instead of
crowding distance, NSGA-III uses reference points which represent the preferences
and weights of different objective functions. Surrogate models which are used in
multi- and many-objective problems were also mentioned in this chapter.
In the last section, benchmarking problems for comparing algorithms are
explained. These benchmarking problems are predefined functions which are
intended to test how well an algorithm can solve problems with different
characteristics and difficulties. Some test problems are designed to be scalable,
meaning that the number of objective functions and variables can be changed. This
feature is especially valuable for researches in many-objective optimization.
References
1. R.B. Agrawal, K. Deb, R.B. Agrawal, Simulated binary crossover for continuous search space.
Complex Systems 9,115–148 (1994)
2. N. Andreasson, A. Evgrafov, M. Patriksson, E. Gustavsson, M. Onnheim, Introduction to
Continuous Optimization, 2nd edn. (Studentlitteratur AB, Lund, 2013)