IEEE Computational Intelligence Magazine - May 2023 - 21

and multi-objective vehicle routing problems with time windows
(MOVRPTWs). Tian et al. [12] proposed a DRL-based
adaptive operator selection technique for MOPs, in which the
decision vectors and candidate operators are regarded as states
and actions, respectively. They employed the Deep-Q-Learning
technique to determine the best operator for a decision
vector. Zhou et al. [33] established an RL-based framework
that could learn the dynamic environment and determine the
appropriate prediction-based strategy for dynamic MOPs.
It should be noted that this paper only reviews the related
research explicitly proposed for handling MOPs. Additionally,
there is no research on applying RL or DRL techniques in the
CMOP field to the best ofour knowledge.
C. EMTfor MOPs
In recent years, researchers have been focusing on applying
EMT [34], [35] to handling MOPs. Liang et al. [36] proposed an
algorithm that contains a subspace alignment using a mapping
matrix obtained by subspace learning to reduce negative knowledge
transfer and a self-adaptive differential evolution (DE) operator
with a trial vector generation strategy to generate promising
solutions. Liu et al. [37] utilized EMT to solve large-scale MOPs
(LMOPs). They established a discriminative reconstruction network
(DRN)model (containing an encoder, a decoder, and a classifier)
trained by the obtained non-dominated solutions to transfer
the solutions of source LMOPs to the target LMOP to assist its
optimization and learn a dimensional-reduced Pareto-optimal
subspace of the target LMOP. To overcome the drawback most
studies barely take into account, in which some problems cannot
converge under restrictive computational budgets with the traditional
EMT framework, Wei et al. [38] proposed a generalized
In the proposed EMT framework, an
arbitrary number of algorithmic
strategies that possess different pros
can be embedded.
resource allocation framework that adopts a normalized attainment
function to quantify convergence status and a multi-step nonlinear
regression to evaluate the stability ofperformance. Then, the conventional
resource allocation algorithm is refined using the mentioned
techniques to learn the characteristics ofMOPs.
Several approaches have been proposed in the CMOP field.
Qiao et al. [39] proposed anEMT framework forCMOPs. They
considered the whole parent or offspring solution set as knowledge
and established an EMT framework containing multiple
tasks for one to all constraint conditions. They further proposed a
dynamic auxiliary task-based EMT framework [40] for CMOPs
that employs an " -constrained technique to adjust the auxiliary
task through different constraint relaxation levels. Jiao et al. [41]
proposed a multiform optimization framework to solve a
CMOP task with an auxiliary CMOP task using a relaxation factor
" for searching in different sizes ofthe feasible space derived
from the original CMOP task. All these works adopt fixed algorithmic
strategies in their EMT framework. Although [40]
and [41] comprised a dynamic auxiliary task, they both employed
a relaxation factor to achieve dynamic constraint relaxation,
rather than the dynamic adjustment of algorithmic strategies.
Under this circumstance, these methods might encounter limitations
as introduced in Section I.
Table I summarizes the features, advantages, and limitations
of the existing CMOEAs. It shows that different
TABLE I A summary of the features, advantages, and limitations of the existing CMOEAs.
CATEGORY
ALGORITHMS
Pareto dominance-based1. CCMO [6]
2. BiCo [7]
3.C-TAEA [20]
4. CMOEA-MS [21]
5. DSPCMDE [22]
6. URCMO [23]
7. ToM [24]
Constraint
relaxation-based
1. c-DPEA [9]
2. PPS [8]
3. MOEA/D-DAE [25]
4. CRS-DE [26]
5. DC-NSGA-III [27]
Even-search based
1. ICMA [10]
2. CCEA [11]
3. MOEA/D-DMepsilon [28]
4. CAEAD [29]
ADVANTAGES
Suitable for CMOPs whose CPF is the same
or part of the UPF; Fast convergence speed
(approximation to the CPF and UPF).
LIMITATIONS
Ineffective if the CPF is far from the UPF; Perform
poorly when feasible regions are small.
Suitable for CMOPs whose CPF is near to the
UPF and partially coincides with the UPF;
Explore feasible regions (even small nones)
near the UPF.
Parameter-sensitive to problems; Design of
details are problem-dependent.
Can handle different types of CMOPs; Suitable
for CMOPs whose CPF is not near to the UPF.
Large biased search regions; Hard to maintain
diversity especially when CPF is irregular.
MAY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 21
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