8 An Introduction to Many-Objective Evolutionary Optimization
277
The choice of the scheme and mechanism usually differentiates the EAs. For
example: SMS-EMOA uses (μ + 1) scheme with non-dominated sorting and Smetric selection, NSGA-II uses (μ + μ) scheme with non-dominated sorting and
crowd-distance selection, and NSGA-III uses (μ + μ) non-dominated sorting and
reference-point distance. These operators will be discussed further in Sect. 8.3.3.
8.3 Multi-Objective Optimization
This section will discuss how to solve multi-objective optimization problems.
Several methods as well as some performance metrics to compare solutions will
be described.
8.3.1 Method Classifications Based on Preference-Imposing
Timing
In Sect. 8.1.2, it was mentioned that in multi- and many-objective problems, we are
concerned with the solutions in the Pareto set. This would imply that in a decision
making process, decision makers must choose the “best” design from the Pareto set
considering his/her preference on the trade-off between the objectives (the Pareto
front). The preference can be imposed before (a priori), after (a posteriori), or
progressively within the optimization loop.
8.3.1.1 A Priori Method
A priori methods simplify the problem by transforming the problems into one or
a series of single-objective optimization problems (SOP). Several methods that fall
into this category are described below.
Lexicographic Method
The lexicographic method considers an absolute importance order [16]. The method
is similar with the process of sorting words in dictionaries [28]:
• Sort by the first letter
• For the same first letter, then sort by the second letter
• Continue to the next letters until all items have different ranks or all letters in the
word are used
277
The choice of the scheme and mechanism usually differentiates the EAs. For
example: SMS-EMOA uses (μ + 1) scheme with non-dominated sorting and Smetric selection, NSGA-II uses (μ + μ) scheme with non-dominated sorting and
crowd-distance selection, and NSGA-III uses (μ + μ) non-dominated sorting and
reference-point distance. These operators will be discussed further in Sect. 8.3.3.
8.3 Multi-Objective Optimization
This section will discuss how to solve multi-objective optimization problems.
Several methods as well as some performance metrics to compare solutions will
be described.
8.3.1 Method Classifications Based on Preference-Imposing
Timing
In Sect. 8.1.2, it was mentioned that in multi- and many-objective problems, we are
concerned with the solutions in the Pareto set. This would imply that in a decision
making process, decision makers must choose the “best” design from the Pareto set
considering his/her preference on the trade-off between the objectives (the Pareto
front). The preference can be imposed before (a priori), after (a posteriori), or
progressively within the optimization loop.
8.3.1.1 A Priori Method
A priori methods simplify the problem by transforming the problems into one or
a series of single-objective optimization problems (SOP). Several methods that fall
into this category are described below.
Lexicographic Method
The lexicographic method considers an absolute importance order [16]. The method
is similar with the process of sorting words in dictionaries [28]:
• Sort by the first letter
• For the same first letter, then sort by the second letter
• Continue to the next letters until all items have different ranks or all letters in the
word are used