#  $I{{}^{c}}_{SDE^{+}}$ \- An Indicator for Constrained Multi-objective
Optimization

Oladayo S. Ajani, Rammohan Mallipeddi
Department of Artificial Intelligence
Kyungpook National University
Daegu
<EMAIL_ADDRESS>mallipeddi.ram@gmail.com}email@email
&Sri Srinivasa Raju M
Department of Mathematics
National Institute of Technology, Silchar
Silchar
<EMAIL_ADDRESS>

###### Abstract

The effectiveness of Constrained Multi-Objective Evolutionary Algorithms
(CMOEAs) depends on their ability to reach the different feasible regions
during evolution, by exploiting the information present in infeasible
solutions, in addition to optimizing the several conflicting objectives. Over
the years, researchers have proposed several CMOEAs to handle CMOPs. However,
among the different CMOEAs proposed most of them are either decomposition-
based or Pareto-based, with little focus on indicator-based CMOEAs. In
literature, most indicator-based CMOEAs employ - a) traditional indicators
used to solve unconstrained multi-objective problems to find the indicator
values using objectives values and combine them with overall constraint
violation to solve Constrained Multi-objective Optimization Problem (CMOP) as
a single objective constraint problem, or b) consider each constraint or the
overall constraint violation as objective(s) in addition to the actual
objectives. In this paper, we propose an effective single-population
indicator-based CMOEA referred to as $I{{}^{c}}_{SDE^{+}}$ that can explore
the different feasible regions in the search space. $I{{}^{c}}_{SDE^{+}}$ is
an (I)ndicator, that is an efficient fusion of constraint violation (c),
shift-based density estimation (SDE) and sum of objectives (+). The
performance of CMOEA with $I{{}^{c}}_{SDE^{+}}$ is favorably compared against
9 state-of-the-art CMOEAs on 6 different benchmark suites with diverse
characteristics.

_K_ eywords Constraint Handling $\cdot$ Evolutionary Multi-objective
Optimization $\cdot$ Indicator-based Evolutionary Algorithm

##  1 Introduction

Optimization problems characterized by conflicting objectives and constraints
are referred to as Constrained Multi-objective Optimization Problems (CMOPs)
[1] as in (1)

$\left\\{\begin{matrix}Minimize:\quad\bm{f}(\bm{x})=\left[f_{1}(\bm{x}),f_{2}(\bm{x}),....,f_{m}(\bm{x})\right]\\\
\\\ subject\hskip 5.69046ptto:\quad\quad\begin{matrix}g_{i}(\bm{x})\leq
0,\hskip 1.42271pti=1,...,r\quad\quad\\\
\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h_{i}}(\bm{x})=0,\hskip
1.42271pti=r+1,...,q\quad\quad\\\ \quad\bm{x}=(x_{1},...,x_{n})\hskip
1.42271pt\in\hskip
1.42271pt\Omega\subset\mathbb{R}^{n}\end{matrix}\end{matrix}\right.$ (1)

where $\bm{f}(x)\in\mathbb{R}^{m}$ is an objective vector with _m_ objective
values, $\Omega$ denotes variable space, _r_ and (_q-r_) are the number of
inequality and equality constraints, respectively. Employing a small positive
value $\epsilon$ ($=10^{-6}$) [2], (_q-r_) equality constraints can be
transformed into inequality constraints (2). Then the overall Constraint
Violation (CV) of _x_ is given by (3)

$g_{i}(\bm{x})=|g_{i}(\bm{x})|-\epsilon\leq 0,\quad i=r+1,...q$ (2)

$CV(\bm{x})=\sum_{i=1}^{q}{max(0,g_{i}(\bm{x}))}$ (3)

A solution $\bm{x}$ is termed feasible if $CV(\bm{x})=0$; else infeasible.
Given two solutions $\bm{x}_{1}$ and $\bm{x}_{2}$, $\bm{x}_{1}$ is said to
dominate $\bm{x}_{2}$ (represented as
$\bm{x}_{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\prec}\hskip
1.42271pt\bm{x}_{2}$) iff $f_{j}(\bm{x}_{1})\leq f_{j}(\bm{x}_{2})$ for each
$i\in{j,...,m}$ and $f(\bm{x}_{1})\neq f(\bm{x}_{2})$. The terms associated
with a CMOP as shown in (1), are briefly defined as follows;

  * •

Unconstrained Pareto Set (UPS) and Unconstrained Pareto Front (UPF): For any
$\bm{x}^{\lozenge}\in$ UPS, $\nexists\bm{x}\in\Omega$ such that
$\bm{x}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\prec}\hskip
1.42271pt\bm{x}^{\lozenge}$. UPF$=\left\\{f(\bm{x})|\bm{x}\in UPS\right\\}$

  * •

Constrained Pareto Set (CPS) and Constrained Pareto Front (CPF): For any
$x^{\bigstar}\in$ CPS, $\nexists\bm{x}\in\Omega$ such that $CV(\bm{x})=0$ and
$\bm{x}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\prec}\hskip
1.42271pt\bm{x}^{\bigstar}$. CPF $=\left\\{f(\bm{x})|\bm{x}\in CPS\right\\}$

As shown in Fig. 1, constraints complicate the search process [3] by: a)
introducing discontinuities in search space, and/or b) altering relative
arrangement of UPF and CPF. Therefore, the search process should cross
multiple infeasible barriers (Fig. 1(a & b)) or navigate through multiple
narrow feasible regions (Fig. 1(c)) to reach the continuous and discrete CPF,
respectively. In addition, UPF and CPF can be separated (Fig. 1(a)) or
overlapped (Fig. 1(b)) or CPF $\subset$ UPF (Fig. 1(c)).

Figure 1: Characteristics of CMOPs showing feasible and infeasible regions,
relative location of CPF and UPF. LIRCMOP 8 and LIRCMOP 5 are problem
instances taken from LIRCMOP [4] while DASCMOP 4 is a problem instance taken
from DASCMOP [5]

Over the years, Constrained Multi-Objective Evolutionary Algorithms (CMOEAs)
have shown significant potential in solving CMOPs [6, 7, 8, 9, 10, 11, 12, 13,
14]. In general, the effectiveness of CMOEAs depend on their ability to reach
the different feasible regions during evolution, by exploiting the information
present in infeasible solutions. Depending on featured methodologies employed
to exploit the infeasible solutions, CMOEAs are classified as: fitness-based,
ranking-based, multi-staged and multi-population based CMOEAs [15, 9].

In fitness-based CMOEAs [16, 17], a CMOP is transformed into an unconstrained
problem by adding a penalty term to the actual objective values. In ranking-
based CMOEAs, the objectives and the constraints are handled independently
using Constrained Dominance Principle (CDP) or combining CDP with shift-based
density estimation [18] or stochastic ranking [19]. Although, fitness-based
and ranking-based CMOEAs are simple, their performance heavily depends on the
associated hyperparameters, which not appropriately set results in severe
performance degradation especially in CMOPs characterized by discontinuities
shown in Fig. 1. Because such CMOPs require that some level of priority be
given to infeasible solutions during evolution to overcome premature
convergence. To deal with such CMOPs, multi-staged and multi-population based
CMOEAs are gaining prominence recently. Multi-staged CMOEAs [11, 10],
generally divide the evolution process into different stages to achieve the
required balance between objectives and constraints. In other words, some
stages are designed to focus on the exploration of feasible regions while
others focus on enforcing diversity and uniformity in the solution set. For
example, in PPS [11] the population is first pushed towards UPF by neglecting
CV and pulled towards CPF by considering CV. Similarly, MOEADDAE [10] proposed
a mechanism to detect stagnation based on feasibility ratio and rate of change
of CV. Once stagnation is detected, weights associated with CV are modified to
assist the search in jumping out of local optima. The main drawback of multi-
staged CMOEAs is the uncertainty associated with the transition between
stages. In multi-population based CMOEAs [20], multiple populations co-evolve
by sharing information. However, they fail to achieve consistency in terms of
convergence and diversity in the main population. In CTAEA [7], two archives
are considered where one archive accounts for diversity while the other
accounts for convergence. Similarly, in CCMO [9], the first population solves
CMOP (1) and the second population optimizes the objectives neglecting the
constraints. Note that the diversity archive of CTAEA, second population of
CCMO and the push stage of PPS serve similar purpose of locating UPF. However,
the associated mechanisms of multi-staged and multi-population based CMOEAs
waste function evaluations through unnecessary exploration of infeasible
regions [10].

In literature, most CMOEAs are either decomposition-based or Pareto-based,
with minimal work on indicator-based CMOEAs. Because it is difficult to design
an indicator that can account for CV in addition to convergence and diversity.
In AR-MOEA [6], the Inverted Generation Distance (IGD) indicator [21] was
employed, where the associated set of reference points are adapted according
to the contribution of candidate solutions in an external archive. The
reference point adaptation depends on the characteristics of each CMOP and can
seriously degrade the CMOEA performance. In TIGE_2 [12], 3 separate
indicators, one each for convergence, diversity and CV are combined through
different ranking and balancing schemes to obtain an appropriate balance
between 3 goals. In [14], _m_ dimensional objective vectors corresponding to
solutions of CMOP (1) are transformed into scalar values by employing
indicators such as $I_{{\epsilon}^{+}}$ and hypervolume (HV) developed to
solve Unconstrained Multi-objective Optimization Problems (UMOPs). By
combining scalar values with CV, CMOP (1) can be solved as a single-objective
constraint problem using respective constraint handling methods such as
superiority of feasible and epsilon constraint. However, the framework fails
to provide diverse set of solutions corresponding to CPF. In ICMA [13], an
indicator was proposed where a cost value-based distance was introduced into
objective space, which is then combined with CV to evaluate the contribution
of each individual in exploring the promising areas. However, as the indicator
fails to enforce diversity by itself, weight vectors are employed to divide
the objective space into multiple regions. In addition, an archive is employed
to store feasible solutions produced during the evolution and is pruned using
environmental selection of [22].

Motivated by the need for simple yet efficient indicator for CMOEAs, this work
proposes $I^{c}_{SDE^{+}}$ which is an amalgamation of CV based on superiority
of feasible, Sum of OBjectives (SOB) and Shift-based Density Estimation (SDE).
Efficient fusion of the 3 components, enables CMOEA with $I^{c}_{SDE^{+}}$
achieve feasible solutions with better convergence and diversity. In addition,
CMOEA with $I^{c}_{SDE^{+}}$ is an efficient single population framework, an
alternative to complex multi-staged and/or multi-population approaches, to
handle CMOPs. Unlike some indicators specifically designed for CMOPs [13],
$I^{c}_{SDE^{+}}$ is an extension of $I_{SDE^{+}}$ developed to solve UMOPs.

The rest of this paper is structured as follows. In Section II, some
background for the current work is presented. The effectiveness of
$I^{c}_{SDE^{+}}$ and associated CMOEA are detailed in Section III. In Section
IV, the effectiveness of the proposed framework is experimentally demonstrated
and compared with 9 state-of-the-art CMOEAs on various benchmark problems.
Finally, Section V concludes the paper.

##  2 Background on Basic Components of $I^{c}_{SDE^{+}}$

###  2.1 Sum of Objectives (SOB)

Fitness assignment [23] involves the transformation of multi-objective space
into a scalar one to facilitate an easy comparison of solutions. It has been
demonstrated that Weighted Sum of OBjectives (WSOB) possess better convergence
characteristics compared to other methods such as maximum ranking, favor
relation and average ranking [23, 24]. SOB is a special case of WSOB where the
weights are set to 1 as in (4).

$SOB({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})=\sum_{j=1}^{m}f_{j}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})$
(4)

###  2.2 Shift-based Density Estimation (SDE)

In multi-objective optimization, SDE [25] evaluates the density of a solution
(_$\bm{x}$_) in population (_P_) of size _N_ by shifting the location of the
other solutions in _P_ , in the objective space. As SDE involves shifting
solutions in the objective space, it posses certain ability to impart
convergence. Therefore, an indicator referred to as $I_{SDE}$ given by (5) was
proposed [26].

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}I_{SDE}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})=\underset{\bm{y}\in
P,\bm{x}\neq\bm{y}}{min}\left\\{dist({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(\bm{x}}),\hat{f}({\bm{y}}_{i}))|1\leq
i\leq N-1\right\\}}$ (5)

where $dist()$ is the Euclidean distance. The shifted location
$\hat{f}({\bm{y}}_{i})$ of objective vector $f({\bm{y}}_{i})$ w.r.t to
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f({\bm{x}})}$
is defined as

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{f}_{j}({y}_{i})=\left\\{\begin{matrix}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f_{j}(\bm{x}})&if\quad
f_{j}(\bm{y}_{i})<{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f_{j}(\bm{x}})\\\
f_{j}(\bm{y}_{i})&otherwise\end{matrix}\right.j\in(1,...,m)}$ (6)

To further enhance convergence, SOB is fused into $I_{SDE}$ to propose
$I_{SDE^{+}}$ [3], an effective indicator for UMOPs.

$I_{SDE^{+}}(\bm{x})=\underset{\bm{y}\in\bm{P_{S(\bm{x}})},\bm{x}\neq\bm{y}}{min}\left\\{dist({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(\bm{x}}),\hat{f}({\bm{y}}_{i}))|1\leq
i\leq
N_{S({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}-1\right\\}$
(7)

where
${P_{S({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}}\in{P}$
and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{y}}\in{P_{S({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}}$
such that
$SOB({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{y}})<SOB({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})$.
$N_{S({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}$
is
$|P_{S({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}|$.

##  3 Proposed $I{{}^{c}}_{SDE^{+}}$ Indicator and CMOEA

###  3.1 $I^{c}_{SDE^{+}}$ indicator

In the proposed CMOEA framework, the purpose of the indicator is to assist the
selection of $N$ individuals from a set of $2N$ individuals that guarantees -
a) convergence, b) diversity and c) posses the ability to explore the
infeasible barriers. In other words, the indicator should possess the ability
to prioritize infeasible solutions that help CMOEA navigate the infeasible
barriers, while maintaining an appropriate balance between convergence and
diversity among the solutions.

In CMOEAs, the focus is to first seek feasible solutions, followed by
convergence and diversity. Therefore, to evaluate $I^{c}_{SDE^{+}}$,
population is sorted on CV, followed by SOB. In other words, feasible
solutions are prioritized followed by solutions with smaller SOB. The least
ranked solution is assigned an indicator value of one, the highest possible
value. Then, to evaluate $I{{}^{c}}_{SDE^{+}}$ of a given solution _p_ , only
the solutions in _P_ that are ranked lower compared to it are shifted (8)

$I^{c}_{SDE^{+}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})=\underset{\bm{y}\in
P_{SC(x)},\bm{x}\neq\bm{y}}{min}\left\\{dist({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(\bm{x}}),\hat{f}(_{y_{i}}))|1\leq
i\leq
N_{SC({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}-1\right\\}$
(8)

where
$P_{SC({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}\in
P$ and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{y}}\in
P_{SC({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}$
such that

$\left\\{\begin{matrix}CV({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{y}})<CV({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})\\\
SOB({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{y}})<SOB({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})\quad
if\quad
CV({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{y}})=CV({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})\end{matrix}\right.$
(9)

$N_{SC({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}$
is
$|P_{SC({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{x}})}|$.
By setting CV of solutions to zero, which is the case in UMOPs, (8)
degenerates into (7) and $I^{c}_{SDE^{+}}$ can be viewed as a generalized
version of $I_{SDE^{+}}$ [3]

Figure 2:  Objective space of LIRCMOP 8 [4] with 26 solutions divided into 5
sets and 3 stages to demonstrate the working principle of CMOEA with
$I^{c}_{SDE^{+}}$. $CV_{SET3}>CV_{SET4}=CV_{SET2}>CV_{SET1}=CV_{SET5}=0$.
$SOB_{SET5}<SOB_{SET4}<SOB_{SET3}<SOB_{SET2}<SOB_{SET1}$ Table 1: To
demonstrate the working principle of CMOEA with $I^{c}_{SDE^{+}}$, a group of
26 solutions plotted on the objective space of LIRCMOP 8 are categorized into
different sets and stages. Each ordered pair represent the rank and
$I^{c}_{SDE^{+}}$ values of corresponding solutions, respectively. Selected
solutions in each stage are highlighted

Solutions STAGE1 STAGE2 STAGE3 SET1 $\bm{x}_{1}$ [1, 1.000] [1, 1.000] [7,
0.000] $\bm{x}_{2}$ [7, 0.040] [7, 0.0375] [13, 0.000] $\bm{x}_{3}$ [6, 0.027]
[6, 0.0270] [12, 0.000] $\bm{x}_{4}$ [5, 0.068] [5, 0.0676] [11, 0.000]
$\bm{x}_{5}$ [4, 0.027] [4, 0.0270] [10, 0.000] $\bm{x}_{6}$ [2, 0.3108] [2,
0.2875] [8, 0.000] $\bm{x}_{7}$ [3, 0.1351] [3, 0.1250] [9, 0.000] SET2
$x_{8}$ [9, 0.122] [13, 0.000] [19, 0.000] $\bm{x}_{9}$ [11, 0.079] [15,
0.000] [21, 0.000] $\bm{x}_{10}$ [10, 0.014] [14, 0.000] [20, 0.000]
$\bm{x}_{11}$ [8, 0.098] [12, 0.000] [18, 0.000] SET3 $x_{12}$ [13, 0.108]
[17, 0.014] [23, 0.000] $\bm{x}_{13}$ [14, 0.014] [18, 0.014] [24, 0.000]
$\bm{x}_{14}$ [12, 0.081] [16, 0.041] [22, 0.000] $\bm{x}_{15}$ [15, 0.054]
[19, 0.013] [25, 0.000] $\bm{x}_{16}$ [5, 0.068] [20, 0.027] [26, 0.000] SET4
$x_{17}$ - [8, 0.150] [14, 0.019] $\bm{x}_{18}$ - [9, 0.163] [15, 0.000]
$\bm{x}_{19}$ - [10, 0.075] [16, 0.009] $\bm{x}_{20}$ - [11, 0.103] [17,
0.000] SET5 $x_{21}$ - - [6, 0.094] $\bm{x}_{22}$ - - [3, 0.085] $\bm{x}_{23}$
- - [1, 1.000] $\bm{x}_{24}$ - - [2, 0.085] $\bm{x}_{25}$ - - [4, 0.117]
$\bm{x}_{26}$ - - [5, 0.075]

###  3.2 Working Principle of CMOEA with $I^{c}_{SDE^{+}}$

In Fig. 2, objective space of LIRCMOP 8 with 26 solutions is considered to
explain the working principle of CMOEA with $I{{}^{c}}_{SDE^{+}}$ and
demonstrate its ability to navigate infeasible barriers (in grey). CMOEAs are
expected to navigate (in the direction of arrow) through multiple infeasible
regions, to reach the optimal CPF. To illustrate different scenarios during
evolution, we consider 3 stages where solutions are divided into 5 sets as in
Table I. STAGE1, STAGE2 and STAGE3 contain 16, 20 and 26 solutions,
respectively. In each stage, solutions are sorted first on CV followed by SOB
(9) and $I{{}^{c}}_{SDE^{+}}$ values are calculated (8). In each stage, the
order pair next to the solutions indicate the rank (8) and
$I{{}^{c}}_{SDE^{+}}$ value (9), respectively. In each stage, selected
solutions are highlighted (in bold). According to [10], CV across an
infeasible barrier first increases to a certain point (indicated by solid
line) and then decreases. Therefore, solutions on the dotted contour lines in
Fig. 2 have similar CV. In other words,
$CV_{SET3}>CV_{SET4}=CV_{SET2}>CV_{SET1}=CV_{SET5}=0$. However, SOB decreases
continuously in the direction of the arrow, i.e.,
$SOB_{SET5}<SOB_{SET4}<SOB_{SET3}<SOB_{SET2}<SOB_{SET1}$.

In STAGE1, according to (9), solutions in SET1 are ranked ahead of SET2 and
SET2 ahead of SET3 because $CV_{SET1}<CV_{SET2}<CV_{SET3}$. In STAGE1, 8 out
of 16 solutions selected based on $I^{c}_{SDE^{+}}$ (in bold) are prioritized
from different layers (Fig. 2) or sets (Table I) with a scope for exploration.
In other words, even when a feasible region is found and the population
contains enough feasible solutions (entire SET1), preference is given to
infeasible solutions (from SET2 and SET3). In other words, there is a certain
portion of the population that keeps exploring the potential regions, in
addition to preserving the feasible solutions found. This is because when
sorted on CV, feasible solutions with large objective values (SET1) are ranked
ahead of infeasible solutions with smaller objectives (SET2 and SET3).
According to (8), solutions of SET1, when projected onto SET2, does not effect
$I{{}^{c}}_{SDE^{+}}$ values of solutions in SET2. Similarly, SET2 projected
onto SET3, does not effect $I{{}^{c}}_{SDE^{+}}$ values of SET3. In other
words, $I^{c}_{SDE^{+}}$ value of a solution is effected by the solutions
within the same set or having similar CV. Therefore, diverse solutions from
across different sets are selected.

In STAGE2, 4 more solutions (SET4) are added below the solid line (Fig. 2). In
STAGE2, from the 10 of 20 solutions selected, it can be observed that no
solutions from SET2 are selected while only one solution from SET3 is
selected. This is because $CV_{SET4}=CV_{SET2}<CV{SET3}$ and
$SOB_{SET4}<SOB_{SET3}<SOB_{SET2}$. Therefore, solutions in SET4 are ranked
ahead of SET2 and SET3. Because $SOB_{SET4}<SOB_{SET3}<SOB_{SET2}$, solutions
of SET4 when projected on solutions of SET2 and SET3, significantly decrease
$I^{c}_{SDE^{+}}$ values of SET2 and SET3. However, solutions in SET1 are
ranked ahead of SET4 because $CV_{SET1}<CV_{SET4}$ but the do not effect the
$I^{c}_{SDE^{+}}$ values of SET4 because $SOB_{SET4}<SOB_{SET1}$. Therefore,
solutions from SET1 and SET4 are given preference. In other words, some of the
feasible solutions obtained till that point of evolution (SET1) are preserved
while certain part of the population contains solutions (SET4) that helps in
exploration.

In STAGE3, 6 feasible solutions in between the infeasible barriers (SET5) are
added (Fig. 2). Solutions in SET5 have smaller CV and smaller objective values
than SET1, SET2, SET3 and SET4. To select 13 out 26 solutions, 8 highlighted
solutions with non-zero $I^{c}_{SDE^{+}}$ values selected first, and the
remaining solutions are randomly picked. Therefore, once the feasible
solutions with better SOB exist, then evolution moves forward as the feasible
solutions with better SOB replace the feasible solutions with larger SOB.

Therefore, $I^{c}_{SDE^{+}}$ enables CMOEA to explore the infeasible regions
even when the population contains sufficient number of feasible solutions. In
other words, the efficient fusion of 3 basic components of $I^{c}_{SDE^{+}}$
\- CV, SOB and SDE, enable them complement each other in balancing both
exploration and exploitation within a single population framework.

On the contrary, certain portion of the population in the proposed framework
keeps exploring the infeasible region even after reaching the optimal CPF
(Fig. 1(a)), as is the case with most CMOEAs in the literature. Therefore, the
number of feasible solutions in the population at any given time would be less
the desired number, _N_. However, once the search reaches the optimal CPF,
over the iterations as the population evolves, different combinations of
diverse sets are produced on the optimal CPF as well in the infeasible regions
below. In other words, CMOEA with $I^{c}_{SDE^{+}}$ tries to approximate the
optimal CPF with a set of solutions that is less than _N_.

###  3.3 Framework of CMOEA with $I^{c}_{SDE^{+}}$

Algorithm 1 presents the general structure of CMOEA with $I^{c}_{SDE^{+}}$.
After the initial parameter setting (Line 1), starting with a uniformly
initialized population (_P_) of size _N_ (Line 2), $I^{c}_{SDE^{+}}$ values
are evaluated (Line 3). Until a predefined stopping criterion is met,
operations such as mating selection, variation, indicator evaluation and
environmental selection are iterated (Lines 4-9). Finally, the population
(_P_) is returned.

1 Input: $N$

2$\bm{P}\leftarrow\textit{{InitializePopulation}}(N)$

3 $\bm{I^{c}_{SDE^{+}}}\leftarrow\textit{{IndicatorEvaluation}}($P$)$

4

5while  _not done_ do

6 $\bm{P^{{}^{\prime}}}\leftarrow\textit{{MatingSelection}}$ ($\bm{P}$, $N$,
$I^{c}_{SDE^{+}}$)

7 $\bm{Q}\leftarrow\bm{P}\cup\textit{{Variation}}$ ($\bm{P^{{}^{\prime}}},N$)

8 $\bm{I^{c}_{SDE^{+}}}\leftarrow\textit{{IndicatorEvaluation}}$ ($\bm{Q}$)

9
$[\bm{P,I^{c}_{SDE^{+}}}]\leftarrow\textit{{EnvironmentalSelection}}$($\bm{Q}$,
N, $\bm{I^{c}_{SDE^{+}}}$)

10 end while

Output: $\bm{P}$

Algorithm 1 Framework of CMOEA with $\bm{I^{c}_{SDE^{+}}}$

1 Input:$P,N$, $\bm{I^{c}_{SDE^{+}}}$

2 $\bm{P}\leftarrow\phi$

3 while  _$|\bm{P^{\prime}}| <N$_ do

4 select two individuals $x$ and $y$ randomly from $\bm{P}$

5 if  _$I^{c}_{SDE+}(\bm{x}) >I^{c}_{SDE+}(\bm{y})$_ then

6 $\bm{P^{\prime}}\leftarrow$ $\bm{P^{\prime}}$ $\cup(\bm{x})$

7 else

8 $\bm{P^{\prime}}\leftarrow\bm{P^{\prime}}\cup(\bm{y})$

9 end if

10

11 end while

Output: $\bm{P^{\prime}}$

Algorithm 2 $\bm{I^{c}_{SDE^{+}}}$ based Mating Selection

1 Input: $Q$, $N$, $\bm{I^{c}_{SDE^{+}}}$

2 sort solutions in $Q$ in descending order of $\bm{I^{c}_{SDE^{+}}}$

3 $[\bm{P,I^{c}_{SDE^{+}}}]\leftarrow$ $N$ solutions with large indicator
values are selected and ties are resolved randomly

Output: $\bm{P,I^{c}_{SDE^{+}}}$

Algorithm 3 $\bm{I^{c}_{SDE^{+}}}$ based Environmental Selection

As in Algorithm 2, during mating selection, based on $I^{c}_{SDE^{+}}$
promising solutions from the immediate population are selected through binary
tournament selection. In binary tournament selection, out of the two randomly
selected solutions, the solution with highest $I^{c}_{SDE^{+}}$ value is
preferred (lines 4–9). The solutions selected during mating selection are used
to produce new solutions using combination of variation operators such as - a)
simulated binary crossover and polynomial mutation, or b) Differential
Evolution (DE) operator and polynomial mutation [10]. Finally, during
environmental selection (Algorithm 3), _N_ individuals with the highest
$I^{c}_{SDE^{+}}$ values are picked from the union of the current population
and offspring population produced through mating and variation operators.

##  4 Experimental Setup, Results and Discussion

The performance of CMOEA with $I^{c}_{SDE^{+}}$ (in short $I^{c}_{SDE^{+}}$)
is evaluated on 6 different benchmark suites namely; CF [27], RWCMOP [28],
LIRCMOP [4], DASCMOP [5], MW [29] and CDTLZ [30]. Similar to MOEADDAE [10],
$I^{c}_{SDE^{+}}$ also employs DE operator and polynomial mutation to optimize
CF, RWCMOP, LIRCMOP and DASCMOP test suites, while GA simulated binary
crossover and polynomial mutation are employed to optimize MW and CDTLZ test
suites. The performance of CMOEA with $I^{c}_{SDE^{+}}$ is compared with 9
state-of-the-art CMOEAs namely - AR-MOEA [6], CTAEA [7], CMOEAD[8], CCMO [9],
MOEADDAE [10], PPS [11], TiGE_2 [12], ICMA [13], HypECDP [14]. CMOEAD is
traditional decomposition-based algorithm combined with superiority of
feasible to solve CMOPs and can be used as baseline. CTAEA and CCMO are multi-
population CMOEAs. PPS and MOEADDAE are multi-stage CMOEAs. AR-MOEA, TIGE_2,
ICMA, and HypECDP are indicator based CMOEAs. To evaluate the performance of
CMOEAs, the hypervolume (HV) indicator is used. The simulations are carried
out in the PLATEMO [31] framework and employed the standard parameters and
protocols.

Each CMOEA is run 30 times on each problem instance. The results in terms of
HV and statistical comparison based on the Wilcoxon signed rank test are
presented in Tables 2 \- 7. Specifically, Tables 2, 3, 4, 5, 6 and 7 summarize
the results on CF [27], CDTLZ [30], DASCMOP [5], LIRCMOP [4], MW [29] and
RWCMOP [28], respectively. In each table, the mean and standard deviation
values of the best preforming CMOEA (i.e. highest mean HV) for each problem
instance are highlighted. In addition, the symbols $-$,$+$, and $=$ next to HV
values indicate that the performance of $I^{c}_{SDE^{+}}$ is superior,
inferior and similar to the corresponding CMOEA.

The performance of baseline CMOEAD is not consistent across the different test
suites showing significantly degraded results compared to $I^{c}_{SDE^{+}}$.
In other words, $I^{c}_{SDE^{+}}$ is superior or comparable to CMOEAD on
87.01% of the test instances. In problem instances such as CF10, LIRCMOP5 and
LIRCMOP6, CMOEAD fails to find feasible solutions in all the 30 runs unlike
the other CMOEAs.

In comparison with multi-stage based CMOEAs, the performance of
$I^{c}_{SDE^{+}}$ is superior or comparable to MOEADDAE and PPS on 70.13% and
71.43% of the test instances, respectively. However, the superior performance
of multi-stage based CMOEAs compared to $I^{c}_{SDE^{+}}$ can be observed on
LIRCMOP test suite where PPS and MOEADDAE perform better than
$I^{c}_{SDE^{+}}$ on 10 test instances each. The elevated performance of
multi-stage based CMOEAs on LIRCMOP test suite can be attributed to the
alternating exploration and exploitation stages based on the employed
switching mechanism. In most of the test instances, this process is expected
to waste the function evaluations. However, the mechanism appropriately suits
the characteristics of LIRCMOP test suite.

In comparison with multi-population based CMOEAs, $I^{c}_{SDE^{+}}$ performs
better or comparable on 77.92% and 66.23% of test instances. While multi-
population based CMOEAs fails on most test suites, they demonstrate
competitive performance of DASCMOP and MW test suites. The superior
performance of multi-population based CMOEAs can be attributed to their design
where one of the population always explores with the aim of finding the
Unconstrained Pareto Front (UCF) while the other explores the Constrained
Pareto Front (CPF). In CMOPs such as DASCMOP where in most of problem
instances CPF $\subset$ UPF, the population exploring the UPF enables the
CMOEA navigate the infeasible barriers. However, the use of multi-populations
results in the wastage of function evaluations.

Among the indicator-based CMOEAs, the performance of TIGE_2 is significantly
inferior compared to $I^{c}_{SDE^{+}}$. In other words, $I^{c}_{SDE^{+}}$
performs better or comparable on 92.21% of the test instances. Compared to
ARMOEA and HypECDP the performance of $I^{c}_{SDE^{+}}$ is better or
comparable on 74.03% and 77.92% of the test instances. However, both ARMOEA
and HypECDP demonstrate superior performance compared to $I^{c}_{SDE^{+}}$ on
RWCMOP test suite by performing better on 14 and 16 test instances,
respectively out of 25 test instances. In addition, the performance of
$I^{c}_{SDE^{+}}$ in comparison with ICMA where a specialized indicator was
developed for constrained multi-objective problems is better or comparable in
77.92% of the test instances. The improved performance of ICMA compared to
$I^{c}_{SDE^{+}}$ can be observed on CF test suite.

From the analysis, it is evident the performance of state-of-the-art CMOEAs is
not consistent across the different test suites considered. In other words,
alleviate performance on test suite compared to $I^{c}_{SDE^{+}}$ is
compensated by degraded performance on the other test suites. However, the
performance of $I^{c}_{SDE^{+}}$ is consistently ranked among the top 3
performing CMOEAs on each test suite. The superior performance of
$I^{c}_{SDE^{+}}$ can be attributed to the complementary nature of the 3
components - constraint violation, sum of objectives and shift-based density
estimation. In addition, $I^{c}_{SDE^{+}}$ enables the development of single
population-based CMOEA framework facilitating the efficient use of function
evaluations.

Table 2: Performance Comparison of $I{{}^{c}}_{SDE^{+}}$ against state-of-the-
art CMOEAs in terms of HV (mean and standard deviation) as well as Wilcoxon’s
signed-rank test on CF test suite

Problem N M D FEs ARMOEA CTAEA CMOEAD CCMO MOEADDAE PPS TiGE_2 ICMA HypECDP
cISDE CF1 100 2 10 300000 5.5577e-1 (1.35e-3) - 5.2057e-1 (4.78e-3) -
5.5495e-1 (3.14e-3) - 5.6316e-1 (2.94e-4) - 5.6454e-1 (3.04e-4) - 5.6493e-1
(3.91e-4) = 4.9069e-1 (1.24e-2) - 5.6170e-1 (1.86e-3) - 5.5566e-1 (1.71e-3) -
5.6477e-1 (1.51e-4) CF2 100 2 10 300000 6.1717e-1 (2.76e-2) - 6.4862e-1
(1.24e-2) - 5.7526e-1 (2.88e-2) - 6.6141e-1 (7.73e-3) - 6.4395e-1 (2.46e-2) -
6.7709e-1 (1.88e-3) + 5.5169e-1 (4.14e-2) - 6.7447e-1 (1.25e-3) + 6.4796e-1
(2.18e-2) - 6.7000e-1 (6.64e-3) CF3 100 2 10 300000 1.7355e-1 (5.05e-2) -
2.0321e-1 (3.71e-2) = 1.4995e-1 (5.21e-2) - 2.1032e-1 (3.18e-2) = 1.5889e-1
(5.13e-2) - 1.8836e-1 (5.31e-2) = 1.0272e-1 (5.62e-2) - 2.5495e-1 (3.55e-2) +
1.8692e-1 (3.95e-2) - 2.1433e-1 (3.71e-2) CF4 100 2 10 300000 4.2673e-1
(2.35e-2) - 4.3068e-1 (3.12e-2) - 3.8908e-1 (3.92e-2) - 4.5354e-1 (2.95e-2) -
3.7334e-1 (6.62e-2) - 4.8584e-1 (4.27e-2) = 3.0289e-1 (5.97e-2) - 5.1620e-1
(3.11e-3) + 4.2496e-1 (3.61e-2) - 4.6880e-1 (1.63e-2) CF5 100 2 10 300000
2.1344e-1 (6.65e-2) - 2.5057e-1 (8.81e-2) - 2.3831e-1 (7.99e-2) - 2.7530e-1
(7.23e-2) = 2.1248e-1 (7.30e-2) - 3.4156e-1 (4.50e-2) + 2.4056e-1 (7.13e-2) -
3.6204e-1 (6.34e-2) + 3.0012e-1 (5.60e-2) = 3.1160e-1 (5.34e-2) CF6 100 2 10
300000 6.4143e-1 (1.54e-2) - 6.3617e-1 (1.88e-2) - 6.0690e-1 (2.92e-2) -
6.6355e-1 (1.14e-2) - 6.6872e-1 (1.46e-2) - 6.5600e-1 (1.37e-2) - 5.7653e-1
(3.25e-2) - 6.8619e-1 (2.76e-3) + 6.4935e-1 (1.49e-2) - 6.8133e-1 (1.22e-2)
CF7 100 2 10 300000 4.5906e-1 (6.19e-2) = 4.4986e-1 (8.76e-2) = 4.2541e-1
(1.03e-1) - 4.8488e-1 (7.64e-2) = 3.3679e-1 (1.32e-1) - 4.4027e-1 (1.04e-1) =
3.8496e-1 (6.87e-2) - 6.1806e-1 (1.32e-2) + 4.2788e-1 (1.02e-1) - 4.7125e-1
(1.11e-1) CF8 150 3 10 300000 4.0315e-1 (2.82e-2) - 3.2177e-1 (1.58e-2) -
3.7416e-1 (3.16e-2) - 3.5867e-1 (1.02e-1) - 4.2471e-1 (5.19e-2) - 3.9209e-1
(1.78e-2) - 8.7528e-2 (2.65e-2) - 4.4114e-1 (1.34e-2) - 9.8389e-2 (5.12e-2) -
5.0122e-1 (1.97e-2) CF9 150 3 10 300000 4.4186e-1 (2.58e-2) - 4.1332e-1
(1.59e-2) - 4.2686e-1 (3.12e-2) - 4.4742e-1 (4.24e-2) - 4.6729e-1 (3.33e-3) -
4.7919e-1 (1.97e-2) - 1.2176e-1 (4.63e-2) - 5.2011e-1 (4.04e-3) - 2.3151e-1
(3.46e-2) - 5.3216e-1 (1.09e-2) CF10 150 3 10 300000 2.1601e-1 (1.46e-1) =
1.9397e-1 (5.70e-2) - 0.0000e+0 (0.00e+0) - 1.7229e-1 (4.92e-2) - 1.6024e-1
(8.76e-2) - 2.6441e-1 (7.12e-2) - 9.8521e-2 (1.50e-2) - 3.8068e-1 (4.29e-2) =
9.1731e-2 (0.00e+0) = 3.4766e-1 (9.26e-2) +/-/= 0/8/2 0/8/2 0/10/0 0/7/3
0/10/0 2/4/4 0/10/0 6/3/1 0/8/2

Table 3: Performance Comparison of $I{{}^{c}}_{SDE^{+}}$ against state-of-the-
art CMOEAs in terms of HV (mean and standard deviation) as well as Wilcoxon’s
signed-rank test on CDTLZ test suite

Problem N M D FEs ARMOEA CTAEA CMOEAD CCMO MOEADDAE PPS TiGE_2 ICMA HypECDP
cISDE C1_DTLZ1 92 3 7 46000 8.3350e-1 (6.93e-3) + 8.3363e-1 (5.79e-3) +
8.3443e-1 (7.30e-3) + 8.3282e-1 (6.99e-3) + 8.3577e-1 (6.35e-3) + 7.9849e-1
(2.14e-2) - 2.7192e-1 (1.14e-1) - 8.1799e-1 (7.41e-3) - 2.8445e-1 (9.39e-2) -
8.2892e-1 (4.48e-3) C1_DTLZ3 92 3 12 92000 7.4927e-2 (1.81e-1) - 4.4962e-1
(1.86e-1) - 2.2183e-1 (2.75e-1) - 5.5813e-1 (1.51e-3) - 5.5601e-1 (1.60e-3) -
4.3894e-1 (1.75e-1) - 0.0000e+0 (0.00e+0) - 3.4868e-1 (2.05e-1) - 2.4047e-2
(7.38e-2) - 5.5974e-1 (1.29e-3) C2_DTLZ2 92 3 12 23000 5.1146e-1 (1.85e-3) +
5.0317e-1 (3.16e-3) = 5.1474e-1 (2.56e-4) + 5.1368e-1 (1.90e-3) + 5.1533e-1
(1.22e-3) + 4.7943e-1 (1.31e-2) - 4.0809e-1 (2.55e-2) - 4.8224e-1 (3.83e-3) -
4.9536e-1 (3.06e-3) - 5.0214e-1 (3.02e-3) C3_DTLZ1 92 3 7 69000 2.5251e-1
(5.95e-2) - 2.7051e-1 (4.94e-2) - 2.8657e-1 (6.42e-2) = 2.9213e-1 (1.04e-2) -
2.9909e-1 (1.21e-2) - 2.6706e-1 (5.81e-2) - 9.0213e-3 (1.91e-2) - 2.5865e-1
(1.60e-2) - 1.3504e-2 (2.68e-2) - 3.1650e-1 (9.61e-3) C3_DTLZ4 92 3 12 69000
7.4292e-1 (1.05e-1) - 7.8459e-1 (1.20e-3) - 7.8651e-1 (4.88e-2) - 7.7946e-1
(4.58e-2) - 3.8013e-1 (1.69e-1) - 7.5409e-1 (3.25e-2) - 7.6175e-1 (3.72e-3) -
7.3759e-1 (6.79e-3) - 6.5890e-1 (1.28e-1) - 7.9352e-1 (4.45e-4) +/-/= 2/3/0
1/3/1 2/2/1 2/3/0 2/3/0 0/5/0 0/5/0 0/5/0 0/5/0

Table 4:  Performance Comparison of $I{{}^{c}}_{SDE^{+}}$ against state-of-
the-art CMOEAs in terms of HV (mean and standard deviation) as well as
Wilcoxon’s signed-rank test on DASCMOP test suite

Problem N M D FEs ARMOEA CTAEA CMOEAD CCMO MOEADDAE PPS TiGE_2 ICMA HypECDP
cISDE DASCMOP1 300 2 30 300000 4.8466e-3 (4.59e-3) - 1.6243e-1 (5.03e-4) -
6.0020e-3 (6.55e-3) - 6.3241e-3 (5.10e-3) - 2.8654e-2 (4.22e-2) - 1.9649e-1
(4.16e-3) - 1.5082e-1 (7.99e-3) - 1.9242e-1 (4.75e-3) - 7.3850e-3 (4.95e-3) -
2.0013e-1 (2.41e-3) DASCMOP2 300 2 30 300000 2.4770e-1 (6.80e-3) - 3.1053e-1
(1.12e-2) - 2.4167e-1 (9.44e-3) - 2.5830e-1 (5.49e-3) - 2.7170e-1 (1.80e-2) -
3.4978e-1 (4.15e-4) + 3.1445e-1 (5.99e-3) - 3.3973e-1 (2.37e-3) - 2.4883e-1
(6.63e-3) - 3.4872e-1 (7.39e-4) DASCMOP3 300 2 30 300000 2.0852e-1 (1.96e-4) -
2.5921e-1 (9.02e-3) - 1.9595e-1 (4.86e-2) - 2.1746e-1 (1.53e-2) - 2.3012e-1
(3.27e-2) - 3.0461e-1 (1.31e-2) - 2.5189e-1 (2.08e-2) - 3.0999e-1 (9.16e-4) -
2.1071e-1 (8.27e-3) - 3.1180e-1 (1.59e-4) DASCMOP4 300 2 30 300000 1.3511e-1
(5.86e-2) - 2.0090e-1 (1.74e-3) + 2.0390e-1 (1.79e-4) + 2.0244e-1 (3.21e-3) +
2.0327e-1 (3.45e-4) + 1.8800e-1 (1.40e-2) - 1.8929e-1 (5.66e-3) - 1.9634e-1
(1.25e-2) = 1.9887e-1 (1.37e-2) + 1.9713e-1 (1.39e-2) DASCMOP5 300 2 30 300000
2.8362e-1 (1.13e-1) - 3.4951e-1 (3.87e-4) + 3.5124e-1 (2.97e-4) + 3.5166e-1
(1.90e-4) + 3.5039e-1 (3.58e-4) + 3.4257e-1 (1.11e-2) = 3.4115e-1 (1.95e-3) -
3.4708e-1 (2.19e-3) = 3.4426e-1 (4.24e-2) - 3.4767e-1 (1.17e-3) DASCMOP6 300 2
30 300000 1.8426e-1 (9.12e-2) - 3.0911e-1 (8.78e-4) = 2.6626e-1 (7.89e-2) =
3.1080e-1 (4.84e-3) + 2.9200e-1 (2.31e-2) = 2.7689e-1 (3.46e-2) - 3.0782e-1
(2.21e-3) = 3.1020e-1 (1.06e-3) + 2.7406e-1 (7.95e-2) - 2.9833e-1 (2.88e-2)
DASCMOP7 300 3 30 300000 2.9258e-1 (2.89e-4) + 2.8662e-1 (1.27e-3) + 2.9177e-1
(4.98e-4) + 2.9239e-1 (5.06e-4) + 2.9061e-1 (4.17e-4) + 2.6876e-1 (2.72e-2) =
2.6469e-1 (2.87e-3) - 2.8160e-1 (4.63e-3) + 1.8877e-1 (2.08e-2) - 2.7286e-1
(1.20e-2) DASCMOP8 300 3 30 300000 2.1319e-1 (2.75e-4) + 2.0498e-1 (7.36e-4) +
2.1152e-1 (3.97e-4) + 2.1298e-1 (5.31e-4) + 1.9854e-1 (3.87e-3) + 1.6437e-1
(2.95e-2) - 1.9651e-1 (1.71e-3) + 1.9481e-1 (5.25e-3) = 1.1289e-1 (8.01e-3) -
1.9236e-1 (4.78e-3) DASCMOP9 300 3 30 300000 1.2613e-1 (1.10e-2) - 1.9484e-1
(7.05e-3) = 1.6269e-1 (4.82e-2) = 1.3619e-1 (7.36e-3) - 1.5411e-1 (3.31e-2) -
2.0006e-1 (1.89e-3) + 1.5290e-1 (2.09e-2) - 1.5751e-1 (9.76e-3) - 9.6044e-2
(5.15e-3) - 1.9556e-1 (4.76e-3) +/-/= 2/7/0 4/3/2 4/3/2 5/4/0 4/4/1 2/5/2
1/7/1 2/4/3 1/8/0

Table 5: Performance Comparison of $I{{}^{c}}_{SDE^{+}}$ against state-of-the-
art CMOEAs in terms of HV (mean and standard deviation) as well as Wilcoxon’s
signed-rank test on LIRCMOP test suite

Problem N M D FEs ARMOEA CTAEA CMOEAD CCMO MOEADDAE PPS TiGE_2 ICMA HypECDP
cISDE LIRCMOP1 300 2 30 300000 1.2513e-1 (7.10e-3) - 1.4622e-1 (2.80e-2) -
1.1732e-1 (8.94e-3) - 1.5035e-1 (1.67e-2) - 2.0660e-1 (1.75e-2) - 2.3707e-1
(8.23e-4) + 1.5997e-1 (7.29e-3) - 2.1830e-1 (4.22e-3) - 1.2142e-1 (6.30e-3) -
2.2942e-1 (1.47e-3) LIRCMOP2 300 2 30 300000 2.3935e-1 (9.53e-3) - 3.0861e-1
(1.52e-2) - 2.4003e-1 (1.21e-2) - 2.6328e-1 (1.74e-2) - 3.4396e-1 (1.24e-2) =
3.6021e-1 (5.76e-4) + 2.9146e-1 (7.29e-3) - 3.4320e-1 (5.13e-3) - 2.4085e-1
(1.13e-2) - 3.5243e-1 (1.24e-3) LIRCMOP3 300 2 30 300000 1.1153e-1 (9.17e-3) -
1.2257e-1 (2.19e-2) - 1.0395e-1 (1.27e-2) - 1.3651e-1 (1.29e-2) - 1.2277e-1
(2.90e-2) - 2.0547e-1 (1.03e-3) + 1.4044e-1 (8.61e-3) - 1.9084e-1 (6.23e-3) -
1.1087e-1 (9.86e-3) - 1.9667e-1 (4.00e-3) LIRCMOP4 300 2 30 300000 2.0534e-1
(1.32e-2) - 2.2988e-1 (3.23e-2) - 1.9652e-1 (1.67e-2) - 2.2951e-1 (1.73e-2) -
2.6063e-1 (2.75e-2) - 3.1515e-1 (5.81e-4) + 2.4491e-1 (1.02e-2) - 2.9913e-1
(6.04e-3) - 2.0725e-1 (1.29e-2) - 3.0401e-1 (4.38e-3) LIRCMOP5 300 2 30 300000
0.0000e+0 (0.00e+0) - 9.6995e-3 (3.69e-2) - 0.0000e+0 (0.00e+0) - 1.6395e-1
(1.94e-2) - 2.9371e-1 (2.65e-5) + 2.9351e-1 (6.91e-5) + 4.8731e-2 (6.31e-2) -
2.7258e-1 (5.41e-2) = 0.0000e+0 (0.00e+0) - 2.9142e-1 (6.69e-4) LIRCMOP6 300 2
30 300000 0.0000e+0 (0.00e+0) - 0.0000e+0 (0.00e+0) - 0.0000e+0 (0.00e+0) -
1.2147e-1 (1.23e-2) - 1.9896e-1 (4.05e-5) + 1.9885e-1 (5.52e-5) + 4.6648e-2
(4.19e-2) - 1.9857e-1 (2.21e-4) + 0.0000e+0 (0.00e+0) - 1.8063e-1 (3.18e-2)
LIRCMOP7 300 2 30 300000 2.3589e-1 (4.50e-2) - 2.4633e-1 (1.11e-2) - 8.1771e-2
(1.18e-1) - 2.5332e-1 (9.44e-3) - 2.9274e-1 (6.23e-4) + 2.9582e-1 (3.37e-4) +
2.3305e-1 (1.48e-2) - 2.5296e-1 (1.79e-2) - 2.2544e-1 (7.70e-2) - 2.9070e-1
(1.37e-3) LIRCMOP8 300 2 30 300000 1.3807e-1 (1.08e-1) - 1.5387e-1 (9.85e-2) -
2.9412e-2 (7.63e-2) - 2.4184e-1 (9.92e-3) - 2.9480e-1 (1.73e-4) + 2.9603e-1
(7.81e-5) + 2.0405e-1 (1.46e-2) - 2.6038e-1 (1.32e-2) - 1.1311e-1 (1.15e-1) -
2.8733e-1 (1.43e-2) LIRCMOP9 300 2 30 300000 1.3837e-1 (5.86e-2) - 4.2727e-1
(2.90e-2) - 2.1712e-1 (8.69e-2) - 4.1246e-1 (6.28e-2) - 5.6805e-1 (5.21e-5) +
4.6151e-1 (1.96e-2) - 3.6410e-1 (1.88e-2) - 5.0391e-1 (3.16e-2) - 1.7040e-1
(8.85e-2) - 5.5537e-1 (5.39e-3) LIRCMOP10 300 2 30 300000 1.0742e-1 (7.53e-2)
- 5.5099e-1 (6.53e-2) - 4.0143e-1 (1.98e-1) - 6.2571e-1 (2.05e-2) - 7.0920e-1
(5.60e-5) + 6.9034e-1 (3.67e-2) - 1.8806e-1 (2.64e-2) - 7.0770e-1 (2.97e-4) +
1.4066e-1 (1.37e-1) - 7.0165e-1 (2.72e-3) LIRCMOP11 300 2 30 300000 2.1468e-1
(7.60e-2) - 6.3668e-1 (1.04e-2) - 2.6733e-1 (9.94e-2) - 6.6293e-1 (1.59e-2) -
6.9395e-1 (2.80e-5) + 6.2118e-1 (7.06e-2) = 3.4816e-1 (5.63e-2) - 6.7837e-1
(1.58e-2) - 2.5140e-1 (6.63e-2) - 6.9307e-1 (1.11e-3) LIRCMOP12 300 2 30
300000 3.3402e-1 (8.47e-2) - 5.7665e-1 (8.54e-3) - 4.2111e-1 (6.65e-2) -
5.3951e-1 (4.52e-2) - 6.2034e-1 (2.00e-5) + 5.6832e-1 (9.79e-3) - 5.0397e-1
(1.13e-2) - 6.1107e-1 (1.07e-2) = 3.2430e-1 (1.08e-1) - 6.0485e-1 (2.82e-2)
LIRCMOP13 300 3 30 300000 5.9184e-4 (1.09e-4) - 5.7650e-1 (7.57e-4) +
5.5589e-4 (2.53e-5) - 5.7746e-1 (4.48e-4) + 5.7356e-1 (5.43e-4) + 5.5410e-1
(3.29e-3) + 2.7524e-1 (8.50e-2) - 5.7003e-1 (4.67e-4) + 9.5660e-5 (2.17e-4) -
5.2426e-1 (9.31e-3) LIRCMOP14 300 3 30 300000 1.6391e-3 (2.46e-4) - 5.7554e-1
(8.03e-4) + 1.5741e-3 (4.16e-5) - 5.7571e-1 (4.06e-4) + 5.7434e-1 (7.15e-4) +
5.6417e-1 (1.56e-3) + 3.5311e-1 (7.02e-2) - 5.7047e-1 (6.18e-4) + 3.0104e-4
(4.48e-4) - 5.2999e-1 (5.99e-3) +/-/= 0/14/0 2/12/0 0/14/0 2/12/0 10/3/1
10/3/1 0/14/0 4/8/2 0/14/0

Table 6:  Performance Comparison of $I{{}^{c}}_{SDE^{+}}$ against state-of-
the-art CMOEAs in terms of HV (mean and standard deviation) as well as
Wilcoxon’s signed-rank test on MW test suite

Problem N M D FEs ARMOEA CTAEA CMOEAD CCMO MOEADDAE PPS TiGE_2 ICMA HypECDP
cISDE MW1 100 2 15 60000 4.4856e-1 (9.49e-2) - 4.8884e-1 (3.10e-4) - 4.6529e-1
(3.30e-2) - 4.8979e-1 (2.07e-4) + 4.8617e-1 (8.14e-3) = 4.2282e-1 (1.06e-1) -
4.3400e-1 (4.88e-2) - 4.8653e-1 (7.47e-4) - 4.7941e-1 (2.41e-2) = 4.8910e-1
(2.73e-4) MW2 100 2 15 60000 5.4100e-1 (2.58e-2) - 5.6056e-1 (1.19e-2) =
5.4976e-1 (1.28e-2) - 5.5240e-1 (1.43e-2) = 4.5154e-1 (8.37e-2) - 3.7752e-1
(7.32e-2) - 5.2374e-1 (2.14e-2) - 5.6896e-1 (9.79e-3) + 5.4420e-1 (1.49e-2) -
5.5905e-1 (1.23e-2) MW3 100 2 15 60000 5.1871e-1 (1.00e-1) = 5.4420e-1
(5.98e-4) + 5.4159e-1 (5.57e-3) - 5.4380e-1 (7.23e-4) = 5.4237e-1 (8.24e-4) -
5.4284e-1 (7.91e-4) - 5.2102e-1 (2.19e-2) - 5.3844e-1 (1.36e-3) - 5.4058e-1
(1.78e-2) - 5.4390e-1 (5.96e-4) MW4 100 3 15 60000 8.3914e-1 (1.15e-2) +
8.3809e-1 (2.02e-4) = 8.3935e-1 (2.14e-3) + 8.4134e-1 (4.91e-4) + 8.2179e-1
(7.83e-2) - 7.4558e-1 (1.05e-1) - 7.9301e-1 (1.86e-2) - 8.2437e-1 (3.27e-3) -
5.0729e-1 (8.31e-2) - 8.3803e-1 (1.38e-3) MW5 100 2 15 60000 1.8737e-1
(1.00e-1) - 3.1313e-1 (4.88e-3) - 2.9813e-1 (6.75e-2) = 3.2188e-1 (7.47e-3) -
3.0203e-1 (3.39e-2) - 2.2070e-1 (9.99e-2) - 2.8289e-1 (1.17e-2) - 3.1642e-1
(2.19e-3) - 1.9675e-1 (1.11e-1) - 3.2302e-1 (5.48e-4) MW6 100 2 15 60000
2.8218e-1 (2.62e-2) - 3.1214e-1 (8.30e-3) = 3.0828e-1 (1.15e-2) = 2.9895e-1
(1.86e-2) - 1.6075e-1 (8.83e-2) - 9.6163e-2 (8.46e-2) - 2.3467e-1 (6.78e-2) -
3.1308e-1 (9.53e-3) = 2.7970e-1 (4.44e-2) - 3.1217e-1 (1.37e-2) MW7 100 2 15
60000 3.7871e-1 (6.37e-2) - 4.0866e-1 (8.15e-4) + 4.1084e-1 (4.65e-4) +
4.1213e-1 (5.01e-4) + 4.1198e-1 (4.57e-4) + 4.1164e-1 (4.44e-4) + 3.8789e-1
(4.68e-3) - 4.0131e-1 (1.66e-3) - 4.0023e-1 (4.24e-2) - 4.0655e-1 (1.42e-3)
MW8 100 3 15 60000 5.0828e-1 (6.42e-2) - 5.2023e-1 (1.51e-2) - 5.2644e-1
(1.45e-2) = 5.3396e-1 (1.50e-2) = 4.2902e-1 (8.40e-2) - 3.4873e-1 (9.89e-2) -
4.7769e-1 (1.94e-2) - 4.8027e-1 (1.19e-2) - 3.5384e-1 (4.25e-2) - 5.3305e-1
(1.08e-2) MW9 100 2 15 60000 3.1163e-1 (1.43e-1) - 3.9147e-1 (1.88e-3) -
3.6266e-1 (7.65e-2) - 3.8380e-1 (7.25e-2) - 3.3900e-1 (1.36e-1) = 2.6690e-1
(1.56e-1) - 3.0002e-1 (1.20e-1) - 3.3866e-1 (1.89e-2) - 3.6798e-1 (7.21e-2) -
3.9431e-1 (2.89e-3) MW10 100 2 15 60000 3.4800e-1 (8.63e-2) - 4.4135e-1
(1.10e-2) + 3.9908e-1 (3.62e-2) = 4.0897e-1 (2.24e-2) = 2.9257e-1 (9.74e-2) -
1.8890e-1 (8.87e-2) - 3.9396e-1 (3.06e-2) - 4.2691e-1 (1.79e-2) + 3.4146e-1
(9.38e-2) - 4.1623e-1 (1.90e-2) MW11 100 2 15 60000 2.7879e-1 (3.15e-2) -
4.4150e-1 (1.24e-3) - 3.6304e-1 (8.36e-2) - 4.4594e-1 (2.68e-3) + 4.4382e-1
(1.03e-3) - 4.4726e-1 (2.11e-4) + 4.3178e-1 (2.51e-3) - 4.2631e-1 (6.35e-3) -
3.3158e-1 (8.26e-2) - 4.4448e-1 (5.79e-4) MW12 100 2 15 60000 5.1226e-1
(1.72e-1) - 6.0061e-1 (5.66e-4) - 5.8204e-1 (1.03e-1) = 5.6445e-1 (1.46e-1) -
4.8384e-1 (2.34e-1) - 4.3122e-1 (2.39e-1) - 4.3403e-1 (2.45e-1) - 5.8479e-1
(3.66e-3) - 5.3158e-1 (1.85e-1) - 6.0397e-1 (3.97e-4) MW13 100 2 15 60000
3.5649e-1 (8.50e-2) - 4.6086e-1 (1.03e-2) + 4.3085e-1 (3.87e-2) - 4.4505e-1
(1.80e-2) = 3.3373e-1 (9.57e-2) - 2.9515e-1 (1.03e-1) - 3.1744e-1 (4.16e-2) -
4.4756e-1 (1.08e-2) = 3.9451e-1 (7.33e-2) - 4.5051e-1 (1.24e-2) MW14 100 3 15
60000 4.6785e-1 (1.03e-2) + 4.6523e-1 (3.29e-3) = 4.4032e-1 (3.09e-3) -
4.7173e-1 (2.31e-3) + 4.7246e-1 (6.98e-3) + 4.4052e-1 (8.19e-3) - 4.4814e-1
(5.27e-3) - 4.5449e-1 (3.19e-3) - 4.4858e-1 (5.23e-3) - 4.6430e-1 (6.80e-3)
+/-/= 2/11/1 4/6/4 2/7/5 5/4/5 2/10/2 2/12/0 0/14/0 2/10/2 0/13/1

Table 7: Performance Comparison of $I{{}^{c}}_{SDE^{+}}$ against state-of-the-
art CMOEAs in terms of HV (mean and standard deviation) as well as Wilcoxon’s
signed-rank test on RWCMOP test suite

Problem N M D FEs ARMOEA CTAEA CMOEAD CCMO MOEADDAE PPS TiGE_2 ICMA HypECDP
cISDE RWCMOP1 80 2 4 20000 6.0661e-1 (7.69e-4) + 6.0358e-1 (1.20e-3) +
1.0893e-1 (1.04e-4) - 6.0369e-1 (7.84e-4) + 5.5680e-1 (1.67e-2) - 4.8070e-1
(5.85e-2) - 5.1628e-1 (2.32e-2) - 9.8773e-2 (2.51e-2) - 6.0922e-1 (1.75e-4) +
5.9815e-1 (1.61e-3) RWCMOP2 80 2 5 20000 1.8904e-1 (1.47e-1) - 1.0888e-1
(1.52e-1) - 2.8555e-1 (1.40e-1) + 2.3415e-1 (1.56e-1) = 6.2865e-4 (3.01e-3) -
3.9137e-1 (1.01e-3) + 1.0551e-1 (1.28e-1) - 3.9731e-1 (2.11e-3) + 2.3798e-1
(1.34e-1) = 2.7352e-1 (7.01e-2) RWCMOP3 80 2 3 20000 8.9781e-1 (6.69e-4) -
8.6608e-1 (2.05e-2) - 1.2293e-1 (3.55e-2) - 8.9740e-1 (1.10e-3) - 9.0094e-1
(1.80e-4) + 8.3937e-1 (1.57e-2) - 7.3557e-1 (1.29e-1) - 9.1945e-2 (1.57e-3) -
2.5116e-1 (2.78e-1) - 9.0001e-1 (3.79e-4) RWCMOP4 80 2 4 20000 8.5398e-1
(7.96e-3) + 8.5128e-1 (8.78e-3) + 1.3982e-2 (2.99e-2) - 8.5261e-1 (7.47e-3) +
8.5699e-1 (1.44e-3) + 8.4129e-1 (1.72e-2) + 4.5797e-1 (1.64e-1) - 2.6353e-1
(3.97e-2) - 8.6033e-1 (2.86e-3) + 8.3719e-1 (8.55e-3) RWCMOP5 80 2 4 20000
4.3314e-1 (1.16e-3) + 4.2897e-1 (4.36e-3) = 4.1822e-1 (9.29e-3) - 4.3256e-1
(1.65e-3) + 4.2473e-1 (4.27e-3) - 4.3266e-1 (5.49e-4) + 3.9801e-1 (7.39e-3) -
3.8887e-1 (2.03e-2) - 4.3389e-1 (1.79e-3) + 4.3154e-1 (9.27e-4) RWCMOP6 80 2 7
20000 2.7678e-1 (9.80e-5) + 2.2018e-1 (6.18e-2) - 2.7632e-1 (2.22e-4) +
2.7669e-1 (3.12e-4) + 2.7666e-1 (1.42e-4) + 2.7664e-1 (2.14e-4) + 2.7133e-1
(1.67e-3) - 2.6241e-1 (2.86e-3) - 2.7725e-1 (9.59e-5) + 2.7449e-1 (3.52e-4)
RWCMOP7 80 2 4 20000 4.8351e-1 (5.48e-4) - 4.8149e-1 (5.11e-3) - 4.8017e-1
(7.48e-4) - 4.8407e-1 (7.61e-5) = 4.8302e-1 (3.68e-4) - 4.8299e-1 (4.36e-4) -
4.6850e-1 (1.51e-2) - 4.7903e-1 (5.40e-3) - 4.8457e-1 (2.70e-5) + 4.8403e-1
(5.21e-4) RWCMOP8 105 3 7 26250 2.6079e-2 (7.14e-5) + 2.5765e-2 (1.10e-3) =
9.6240e-3 (7.31e-4) - 2.5926e-2 (8.41e-5) + 2.5756e-2 (1.02e-4) - 2.4160e-2
(8.94e-4) - 2.0471e-2 (2.60e-4) - 2.3757e-2 (3.94e-4) - 2.6114e-2 (6.98e-5) +
2.5819e-2 (5.54e-4) RWCMOP9 80 2 4 20000 4.0956e-1 (8.11e-5) + 4.0761e-1
(1.01e-3) - 5.3069e-2 (5.07e-5) - 4.0865e-1 (2.22e-4) - 3.8236e-1 (4.69e-3) -
3.8245e-1 (8.62e-3) - 3.2232e-1 (1.26e-2) - 1.3417e-1 (8.36e-4) - 4.0963e-1
(5.17e-4) + 4.0916e-1 (2.33e-4) RWCMOP10 80 2 2 20000 8.4052e-1 (2.19e-3) -
8.4091e-1 (3.70e-3) - 7.9512e-2 (3.94e-4) - 8.3857e-1 (3.36e-3) - 8.4692e-1
(1.72e-4) - 8.4682e-1 (1.89e-4) - 8.3978e-1 (2.52e-3) - 7.2210e-1 (8.99e-2) -
8.4683e-1 (8.10e-4) = 8.4729e-1 (1.14e-4) RWCMOP11 212 5 3 53000 9.4658e-2
(2.49e-3) - 9.9651e-2 (4.08e-4) - 5.9562e-2 (1.01e-3) - 9.7550e-2 (6.75e-4) -
1.0102e-1 (3.57e-4) - 1.0034e-1 (3.29e-4) - 9.8074e-2 (8.08e-4) - 7.9270e-2
(2.98e-3) - 1.0042e-1 (8.31e-4) - 1.0213e-1 (2.36e-4) RWCMOP12 80 2 4 20000
5.5860e-1 (1.64e-3) + 5.4098e-1 (1.10e-2) = 7.0200e-2 (1.81e-2) - 5.5216e-1
(4.81e-3) + 5.4794e-1 (6.92e-3) = 5.3273e-1 (2.54e-2) - 5.4799e-1 (2.43e-3) =
0.0000e+0 (0.00e+0) - 5.6167e-1 (2.88e-4) + 5.4663e-1 (6.38e-3) RWCMOP13 105 3
7 26250 9.0299e-2 (7.58e-5) + 8.6679e-2 (4.27e-3) - 8.9835e-2 (1.80e-3) -
8.8736e-2 (1.89e-4) - 9.0250e-2 (6.06e-5) + 8.9525e-2 (3.65e-4) - 8.6741e-2
(7.11e-4) - 8.4126e-2 (1.59e-3) - 9.0262e-2 (7.46e-5) + 8.9986e-2 (1.19e-4)
RWCMOP14 80 2 5 20000 6.1681e-1 (1.27e-3) + 6.1657e-1 (1.12e-3) + 1.1076e-1
(2.23e-2) - 6.1404e-1 (1.75e-3) + 6.1037e-1 (3.93e-3) - 5.7034e-1 (2.02e-2) -
3.0629e-1 (1.30e-1) - 5.1941e-1 (3.11e-2) - 6.1863e-1 (1.05e-3) + 6.1238e-1
(2.19e-3) RWCMOP15 80 2 3 20000 5.4057e-1 (3.71e-3) + 5.3445e-1 (1.60e-2) +
6.6011e-2 (8.19e-6) - 5.3768e-1 (3.88e-3) + 3.8793e-1 (8.03e-2) - 5.3056e-1
(5.83e-3) = 5.0865e-1 (1.06e-2) - 8.3703e-2 (5.63e-3) - 5.4167e-1 (8.25e-4) +
5.2379e-1 (1.47e-2) RWCMOP16 80 2 2 20000 7.6245e-1 (1.44e-5) - 7.6139e-1
(7.47e-4) - 7.9117e-2 (8.58e-5) - 7.6156e-1 (3.09e-4) - 7.6081e-1 (2.34e-3) -
7.6161e-1 (9.67e-4) - 7.4203e-1 (7.16e-3) - 8.9209e-2 (1.41e-17) - 7.6392e-1
(1.80e-5) + 7.6274e-1 (5.27e-4) RWCMOP17 105 3 6 26250 2.6790e-1 (1.32e-2) +
2.4680e-1 (3.08e-2) + 2.3219e-1 (6.08e-2) = 2.7743e-1 (3.09e-2) + 2.1541e-1
(6.06e-2) = 2.2474e-1 (6.00e-2) = 2.4858e-1 (5.48e-2) = 2.4245e-1 (2.53e-2) =
2.6470e-1 (6.61e-2) + 2.3174e-1 (3.62e-2) RWCMOP18 80 2 3 20000 4.0507e-2
(1.39e-6) + 4.0265e-2 (1.25e-4) - 4.0262e-2 (2.71e-5) - 4.0494e-2 (3.92e-6) +
4.0465e-2 (1.17e-5) + 4.0438e-2 (2.15e-5) + 3.9215e-2 (5.16e-4) - 4.0440e-2
(2.69e-5) + 4.0509e-2 (1.00e-6) + 4.0365e-2 (7.61e-5) RWCMOP19 105 3 10 26250
3.1663e-1 (1.15e-2) - 1.3887e-1 (6.07e-2) - 1.8323e-1 (3.75e-2) - 3.2135e-1
(9.09e-3) - 2.7673e-1 (3.28e-2) - 3.1075e-1 (2.33e-2) - 3.1874e-1 (1.17e-2) -
1.6750e-1 (8.86e-2) - 3.2962e-1 (7.46e-3) - 3.4772e-1 (4.49e-3) RWCMOP20 80 2
4 20000 0.0000e+0 (0.00e+0) = 0.0000e+0 (0.00e+0) = 0.0000e+0 (0.00e+0) =
0.0000e+0 (0.00e+0) = 0.0000e+0 (0.00e+0) = 0.0000e+0 (0.00e+0) = 0.0000e+0
(0.00e+0) = 0.0000e+0 (0.00e+0) = 0.0000e+0 (0.00e+0) = 0.0000e+0 (0.00e+0)
RWCMOP21 80 2 6 20000 3.1699e-2 (1.09e-4) + 3.1597e-2 (6.78e-5) - 2.9323e-2
(2.41e-6) - 3.1728e-2 (5.49e-5) + 3.1529e-2 (8.83e-5) - 3.1611e-2 (2.02e-5) -
2.1361e-2 (2.45e-3) - 3.0881e-2 (6.11e-5) - 3.1761e-2 (2.80e-6) + 3.1640e-2
(7.55e-5) RWCMOP22 80 2 9 20000 0.0000e+0 (0.00e+0) - 0.0000e+0 (0.00e+0) -
0.0000e+0 (0.00e+0) - 0.0000e+0 (0.00e+0) - 8.5726e-1 (0.00e+0) = 8.0258e-1
(2.23e-1) = 0.0000e+0 (0.00e+0) - 5.7605e-1 (3.76e-1) = 0.0000e+0 (0.00e+0) -
8.2955e-1 (2.27e-1) RWCMOP23 80 2 6 20000 2.9213e-1 (1.52e-1) - 0.0000e+0
(0.00e+0) - 3.0561e-1 (1.18e-1) - 4.0835e-1 (1.31e-1) - 0.0000e+0 (0.00e+0) -
8.9106e-1 (5.64e-2) - 4.5356e-1 (2.65e-1) - 9.9856e-1 (4.52e-16) = 2.5223e-1
(1.20e-1) - 9.9856e-1 (4.52e-16) RWCMOP24 105 3 9 26250 0.0000e+0 (0.00e+0) -
1.0061e+0 (0.00e+0) 0.0000e+0 (0.00e+0) - 0.0000e+0 (0.00e+0) - 0.0000e+0
(0.00e+0) - 8.2141e-1 (2.98e-1) 9.9996e-1 (0.00e+0) 1.0000e+0 (0.00e+0)
0.0000e+0 (0.00e+0) - 0.0000e+0 (0.00e+0) - RWCMOP25 80 2 2 20000 2.4110e-1
(1.44e-5) + 2.3976e-1 (3.02e-3) = 2.3683e-1 (2.15e-4) - 2.4117e-1 (1.38e-5) +
2.4085e-1 (8.73e-5) = 2.4075e-1 (7.40e-5) = 1.9504e-1 (1.80e-2) - 2.3523e-1
(5.56e-4) - 2.4118e-1 (1.21e-5) + 2.4078e-1 (1.83e-4) +/-/= 14/9/2 6/14/5
2/20/3 12/9/4 5/14/6 6/14/5 1/21/3 3/18/4 16/5/4

((a)) CF

((b)) RWCMOP

((c)) LIRCMOP

((d)) DASCMOP

((e)) MW

((f)) CDTLZ

Figure 3: Radar plots showing the average rank of CMOEAs on each test suite
based on their HV values with Friedman Ranking in the legend

The average ranks corresponding to each CMOEA on different test suites based
on the mean HV values (radar plots) and Friedman ranking (legend) are
presented in Fig.3. As CMOEAs are ranked in ascending order of HV values,
CMOEA with largest HV is given a larger rank. Therefore, in the radar plots of
Fig. 3, CMOEA covering larger volume is considered to be better. During
ranking, CMOEAs that have equal mean HV values, the average rank is
considered. Table 8 presents the Friedman ranking results on each test suite
as well as the average Friedman ranks with scores. In Table 8, the values
corresponding the best performance are highlighted. All the statistical tests
conducted in this work were performed with a p-value of 0.05.

Table 8: Friedman ranking results of CMOEAs on each test suite as well as the
average Friedman ranks with scores

Algorithms Benchmark Test suites Average Rank Scores CF RWCOP LIRCMOP DASCMOP
MW CDTLZ AR-MOEA 4.90 7.26 2.25 3.33 3.79 4.80 4.388 7 CTAEA 4.7 4.86 5.46 7
.00 8.07 7 .00 6.182 4 CMOEAD 2.80 2.82 2.04 5.33 5.71 7.6 0 4.383 8 CCMO 6.60
6.74 6.21 6.67 8.29 7.60 7.018 2 MOEADDAE 4.60 5.54 8.64 6.11 4.85 7.60 6.223
3 PPS 7.70 5.62 8.21 5.67 2.86 4.20 5.710 6 TiGE 1.6 3.72 4.64 4.67 3.00 2.00
3.272 10 ICMA 9.3 3.44 7.64 6.44 6.43 3.8 6.175 5 HypECDP 4.00 8.26 1.96 2.89
3.71 2.40 3.870 9 $I^{c}_{SDE^{+}}$ 8.80 6.74 7.93 6.89 8.29 8.00 7.775 1

CMOEAD demonstrates degraded performance on all the test suites, except CDTLZ.
This can be attributed to premature convergence due to over-emphasis on
feasible solutions.

Generally, multi-stage based CMOEAs demonstrate superior performance on
LIRCMOP as they focus on CPF and UPF, separately in different stages. However,
MOEADDAE demonstrates competitive performance on CDTLZ whereas PPS fails. On
the other hand, PPS demonstrates competitive performance on CF whereas
MOEADDAE fails. The performance difference can be attributed to the
characteristics of different stages involved and the switching mechanism
employed. However, the performance of MOEADDAE and PPS degrades considerably
on MW, with average performance on RWCMOP and DASCMOP.

The performance of 2 multi-population based CMOEAs follow similar pattern as
in both the cases the goal of the 2 populations is same with only difference
in the mechanisms employed. As one of the population is designed to find the
UPF, multi-population approaches demonstrate better performance on DASCMOP
that feature broken fronts where CPF $\subset$ UPF. In addition, their
performance is competitive to the best performing CMOEA on MW. However, their
performance is average on test suites such as CF, RWCMOP, LIRCMOP and CTDLZ.
Among the 2 multi-population based CMOEAs, CCMO seems to have advantage over
CTAEA due to roles assigned to different populations and how they interact.

Among the state-of-the-art indicator-based CMOEAs, TIGE_2 fails on all the
test instances due to its inability to balance the 3 employed indicators.
HypECDP and AR-MOEA also fail in all test suites, except on RWCMOP, where they
are ranked 1 and 2. On the other hand, ICMA demonstrates superior performance
on CF and an average performance on LIRCMOP and DASCMOP. However, ICMA
demonstrates degreaded performance on RWCMOP, MW and CDTLZ. The inconsistent
and degraded performance of state-of-the-art indicator-based CMOEAs can be
attributed to the inability of the indicators - single or fusion of multiple
indicators, in enforcing the required diversity and convergence pressure
during the evolution process.

From the results, it is evident that none of the state-of-the-art CMOEAs show
consistent performance on different benchmark suites. Elevated performance on
one is followed by degraded performance on the others. However,
$I^{c}_{SDE^{+}}$ demonstrates consistent performance as it is always one of
the best 3 performing CMOEAs on all the test suites. In other words, it ranked
2, 3, 3, 2, 1 and 1 on CF, RWCMOP, LIRCMOP, DASCMOP, MW and CDTLZ,
respectively. For qualitative analysis, Fig. 4 presents the Pareto Fronts
(PFs) corresponding to the final population obtained by Top 2 ranked CMOEAs.
From Fig. 4, the consistent performance of $I^{c}_{SDE^{+}}$ in terms of
convergence and spread compared to the CCMO [9] and MOEADDAE [10] is evident.
The consistent performance of $I^{c}_{SDE^{+}}$ is due to its ability to
efficiently handle the different discontinuities and explore all the feasible
regions in the search space, due to the efficient fusion of the 3 components
that complement each other.

((a)) LIRCMOP 5

((b)) DASCMOP 1

((c)) MW 12

Figure 4: Performance comparison of Top 3 ranked CMOEAs in terms of final
Pareto Fronts (PFs) on a) LIRCMOP5, b) DASCMOP1 and c) MW12

##  5 Conclusions and Future Directions

A simple and efficient $I^{c}_{SDE^{+}}$ is proposed for CMOPs. The
effectiveness of CMOEA with $I^{c}_{SDE^{+}}$, a single population-based
framework, is demonstrated in comparison to the state-of-the-art CMOEAs.
Formulated as a generalized indicator, $I^{c}_{SDE^{+}}$ can be extended for
unconstrained multi-objective optimization by simply setting constraint
violation to zero. Even though CMOEA with $I^{c}_{SDE^{+}}$ demonstrates
significant performance, it fails to obtain corner solutions when solving
complex CMOPs. This might be due to the use of SOB that concentrates the
search around certain portions of the CPF, neglecting the corners. Instead of
SOB, employing WSOB and adapting the weights during the evolution can help
CMOEA cover the entire CPF.

## Acknowledgments

This work was supported by the Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (2021R1I1A3049810).

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