

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 1, 2020, 48-68
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Performances assessment of MOMA-Plus method on
multiobjective optimization problems

Alexandre Som1, Kounhinir Somé2∗, Abdoulaye Compaoré2, Blaise Somé1

1 Department of Mathematics, Laboratory of Numerical Analysis, Computer Science and
Biomathematics, Joseph Ki-Zerbo University, Ouagadougou, Burkina Faso.
2 Department of Mathematics, Laboratory of Fundamental and Applied Mathematics, Nor-
bert Zongo University, Koudougou, Burkina Faso.

Abstract. This work is devoted to evaluate the performances of the MOMA-Plus method in
solving multiobjective optimization problems. This assessment is doing on the complexity of its
algorithm, the convergence and the diversity of solutions in relation to the Pareto front. All these
parameters were evaluated on non-linear multiobjective test problems and obtained solutions are
compared with those provided by the NSGA-II method. This comparative study made it possible to
highlight the performances of MOMA-Plus method for solving non-linear multiobjective problems.

2020 Mathematics Subject Classifications: 90C29, 90C30, 49M30, 49M37

Key Words and Phrases: Multiobjective optimization, MOMA-Plus method, Performances
assessment

1. Introduction

Multi-objective optimization has been, for several decades, a discipline of Mathe-
matics that allows to solve optimization problems where several criteria are involved
simultaneously[1, 9, 13, 14]. Several researchers have developed methods or algorithms
to find compromise solutions which would be as close as possible to the best values of
criteria but also verify the constraints. Most of these methods can be classified in two
groups: the exact methods and metaheuristics. Metaheuristics have been developed to
overcome the shortcomings of direct or traditional methods. Indeed, on some kinds of
problems, direct or traditional methods are unable either to find Pareto optimal solutions,
or to converge quickly towards the Pareto front, or to give a good distribution of solu-
tions around the Pareto front. Therefore, metaheuristics are developed to overcome these
shortcomings.
In literature, we can find several metaheuristics among which the best known and the

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3581

Email addresses: irenotesom89@gmail.com (A. Som), kounhinir.some@unz.bf (K. Somé),
abdoulaye.compaore@unz.bf (A. Compaoré), blaisesomeouaga1@gmail.com (B. Somé)

http://www.ejpam.com 48 c© 2020 EJPAM All rights reserved.


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 49

most used are: simulated annealing[10], tabu search[11], genetic algorithms[9], NSGA-II
algorithm[1], MOMA method[15, 19, 22], MOMA-Plus method[23], and more.

In this article, we will focus on the MOMA-Plus method. Indeed, MOMA (Multi-
Objective Metaheuristic based on Alienor method) have been developed by K. Somé et
al.[15, 22] and is based on an Alienor transformation[4]. It uses scalarization and penal-
ization techniques to transform the multiobjective problem with constraints into a single
objective problems without constraints and with only one variable. It has provided sat-
isfying results[15, 19, 22] and even it has been improved. This improvement has given
rise to the MOMA-Plus method[23]. The MOMA-Plus method uses the Nelder-Mead[16]
algorithm instead of Operator Preserving Optimality (OPO) to find the optimum in a
discretized domain. The idea of associating the Nelder-Mead algorithm is the fact that
there is a dependence on the choice of an parameter θ0 in using the OPO [19, 23] in the
MOMA algorithm that influences MOMA efficiency.

Note that the MOMA-Plus method has been designed to solve deterministic multi-
objective continuous variable optimization problems. But later, some authors adapted it
to solve other types of optimization problems. We can quote: A. Compaoré et al[17, 20]
for fuzzy optimization problems, and J. Poda et al[7, 8] for combinatorial optimization
problems. However, a study of performances of the MOMA-Plus method has not been
done yet, hence the purpose of this work.

Indeed, in this article, we intend to evaluate the performance of the MOMA-Plus
method on some test problems. For this purpose, a comparative study of the obtained
results will be done on these test problems using the NSGA-II method developed by K.
Deb et al[1]. Therefore, we provide a complete study on the performances of the MOMA-
Plus method because in all of the former works on MOMA-Plus, no study has been done
about the complexity of the algorithm, the convergence and the diversity of solutions on
the Pareto front.

For a best presentation of our work, we will first describe the MOMA-Plus method.
Then, make a study of performances such as: calculation of metrics of convergence and
metric of diversity; and also the complexity of the MOMA-Plus algorithm. Finally, we will
calculate the performance index for each test problem and relative to methods MOMA-
Plus and NSGA-II. We will conclude our work with a conclusion.


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 50

2. MOMA-Plus method

Let’s consider n,m and p be natural integers. Let’s consider also the following multi-
objective optimization problem:

min f(x) =
(
f1(x), · · · , fp(x)

)T
s.t :

{
g(x) =

(
g1(x), · · · , gm(x)

)T
≤ 0;

x = (x1, · · · , xn) ∈ Rn;

(1)

where fj , j = 1, p are the objective functions, gi, i = 1,m are the constraints of the
problem and x = (x1, · · · , xn) are the decision variables.
Some definitions are necessary for the best understanding of this work.
Let S be the set of eligible solutions, i.e. S = {x ∈ Rn/g1(x) ≤ 0; ...; gm(x) ≤ 0}.

Definition 1. A solution x∗ ∈ S is called Pareto optimal if there is no other solution
x ∈ S such that fj(x) ≤ fj(x

∗), ∀j ∈ {1, .., p} and for a certain k ∈ {1, .., p} such that
fk(x) < fk(x

∗).

Definition 2. The ideal point is the vector z ∈ Rp whose components zj are obtained by
individually minimizing each objective function fj, under on all constraints. We have:

zj = min fj(x)

s.t :

{
g(x) = (g1(x), · · · , gm(x))T ≤ 0;
x = (x1, · · · , xn) ∈ Rn.

(2)

The steps of the MOMA-Plus method are as follow:

(i) aggregation of objective functions;

(ii) penalization constraints;

(iii) reduction of research space;

(iv) resolution in the reduced search space

(v) configuration of the solution to the initial space.

2.1. Aggregation of objective functions

The MOMA-Plus method uses an aggregation technique to transform the multiobjec-
tive optimization problem into a single-objective optimization problem. The aggregation
function used here is the weighted Tchebychev distance because all problems are non-
linear. It is defined by the following equation:

H(f(x), λ, z) = max
k
{λk|fk(x)− zk|}, k = 1, p. (3)


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 51

The coefficients λk, k = 1, p are weights of the objective functions with

p∑
k=1

λk = 1.

By applying (3) to the problem (1) we obtain:

min H(f(x), λ, z)

s.t :

{
g(x) = (g1(x), · · · , gm(x))T ≤ 0;
x = (x1, · · · , xn) ∈ Rn.

(4)

The problem (4) is mono-objective, therefore it can have a global optimum for a fixed λ.
The following theorems prove the equivalence between the solutions of the initial problem
and the aggregate problem.

Theorem 1. Any Pareto optimal solution of the problem (1) is an optimal solution for
the problem (4) and reciprocally.

Proof. Let x∗ be an Pareto optimal solution of the problem (1) and note that I =
{1, 2, · · · , p}. Then, there is no x ∈ S such as :

fj(x) ≤ fj(x∗),∀j ∈ I and k 6= j such as fj(x) < fj(x
∗).

Let’s suppose that x∗ is not an optimal solution for the problem (4). Then :

∃x ∈ S : H(f(x), λ, z) < H(f(x∗), λ, z).

That is equivalent to

∃x ∈ S : max
j∈I

{
λj |fj(x)− zj |

}
≤ max

j∈I

{
λj |fj(x∗)− zj |

}
⇒ ∃x ∈ S,∃l ∈ I : λl|fl(x)− zl| ≤ λl|fl(x∗)− zl|
⇒ ∃x ∈ S,∃l ∈ I : |fl(x)− zl| ≤ |fl(x∗)− zl|, because λ ≥ 0
⇒ ∃x ∈ S,∃l ∈ I : fl(x)− zl ≤ fl(x∗)− zl, because (fl(x)− zl ≥ 0,∀x ∈ S)
⇒ ∃x ∈ S,∃l ∈ I : fl(x) ≤ fl(x∗) that is absurd

Consequently, x∗ is an optimal solution of problem (4).
Now, let x∗ be an optimal solution of problem (4). Then

∀x ∈ S : H(f(x∗), λ, z) < H(f(x), λ, z).

Let’s suppose that x∗ is not an Pareto optimal solution for the problem (1). Then :

∃x ∈ S, ∀j ∈ I, fj(x) < fj(x
∗)

⇒ ∃x ∈ S,∀j ∈ I, fj(x)− zj < fj(x
∗)− zj

⇒ ∃x ∈ S,∀j ∈ I, λj |fj(x)− zj | < λj |fj(x∗)− zj |
⇒ ∃x ∈ S,∀j ∈ I,max

j∈I

{
λj |fj(x)− zj |

}
< max

j∈I

{
λj |fj(x∗)− zj |

}
⇒ ∃x : H(f(x), λ, z) < H(f(x∗), λ, z), that is absurd.

Consequently, x∗ is an Pareto optimal solution of problem (1) �


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 52

2.2. Penalization

This step consists in transforming the problem (4) into an optimization problem with-
out constraints. The used penalty function derives from the Lagrangian function and is
defined by[21]:

L(x) = H(f(x), λ, z) + η
m∑
i=1

(gi(x) + |gi(x)|). (5)

η is the defined penalty coefficient such as :

η ≥ M −H(f(x), λ, z)∑
i=1,m

gi
with M = max

x∈S
H(f(x), λ, z)

.
By using the function (5), the problem (4) becomes a single objective optimization

problem without constraint given by the following formulation:{
Glob.minL(x)

x ∈ D;
(6)

with D, a subset of Rn defined by the boundaries of the variables. The following theorem
characterizes the global optimum of the problem (6).

Theorem 2. Let x∗ ∈ S be a point that realizes the global minimum of the problem (6)
then x∗ is a point that realizes global minimum of the problem (4).

Proof. Let’s suppose that x∗ is a point that realizes the global minimum of the problem
(6) then ∀x ∈ S : L(x∗) ≤ L(x) that means that:

∀x ∈ S, L(x∗) ≤ L(x),

then

H(f(x∗), λ, z) + η
m∑
k=1

(gk(x
∗) + |gk(x∗)|) < H(f(x), λ, z) + η

m∑
k=1

(gk(x) + |gk(x)|).

By the definition of the set S we have gi(x) ≤ 0⇒ gi(x) + |gi(x)| = 0 therefore we have:

m∑
i=1

(
gi(x) + |gi(x)|

)
= 0

Consequently,
H(f(x∗), λ, z) ≤ H(f(x), λ, z)

Hence x∗ is a point that also realizes the global minimum of the problem (4).�


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 53

2.3. Reduction of research space

Definition 3. The Alienor transformation is any transformation that reduces a function
of several variables to a function of a single variable using the α−dense curves which
corresponds to a reduction of the search space.

The α−dense curves are studied in [6] and the interested reader will be able to consult
it. The Alienor transformation that we will use in this work is that of Konfé-Cherruault[4].
It is given by the following equation:

xi = hi(θ) =
1

2

(
(bi − ai) cos(ωiθ + φi) + ai + bi

)
, i = 1, · · · , n. (7)

In the equation (7), ωi and φi are slowly increasing sequences and θ ∈ [0; θmax] with

θmax =
(b− a)θ1 + (b+ a)

2
and θ1 =

2π − φ1
ω1

.

Several types of Alienor transformations exist in the literature and the interested read-
ers will be able to consult [2, 3, 5, 6, 12].

By applying the equation (7) to the variables of the problem (6), we obtain the opti-
mization problem with single objective without constraint and with only one variable by
the following formulation: {

Glob.min F (θ)

θ ∈ [0; θmax] .
(8)

Theorem 3. Any minimum of the problem (6) can be approached by a minimum of the
problem (8).

Proof. For the proof, see [5].

The resolution of the problem (8) is to find the θ value that minimizes the function F .

2.4. Resolution in reduced space

The MOMA-Plus method is applied in a discrete interval to which the Nelder-Mead
algorithm is applied, more precisely in the neighborhood of the discrete points, as shown
in the following figure :

Figure 1: Research space discretization procedure


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 54

This process is repeated until the coverage of the whole domain. Nelder-Mead’s method
known as fminsearch in Matlab, is very effective for optimization of a single variable[16].
To maximize the chances of obtaining the global optimum, the research domain has been
discretized in nested domains with center xi and the search for a local solution is realized
next to neighborhood of each point. It is among these solutions that the overall optimum
will be chosen.

2.5. Configuration of the obtained solutions to the initial space

After the execution of Nelder-Mead’s algorithm, the last step of MOMA-Plus is the
configuration of the obtained solutions. Indeed, it is the transition from the optimal θ to
the variables xi using in the formulation (7). Note that this solution configuration provides
all the Pareto optimal solutions for the initial problem.

2.6. MOMA-Plus algorithm

The algorithm of the MOMA-Plus method is as follows :

Algorithm 1 Algorithm of the MOMA-Plus method

(i) For k from 1 to p do
f(x)←− maxλk|fk(x)− zk| (”scalarization”)
End For

(ii) g(x)←− g1(x) + |g1(x)|
For i from 2 to m do
g(x)←− gi(x) + |gi(x)|
End For
L(x)←− f(x) + η ∗ g(x)(”Penalization”)

(iii) For i from 1 to n do (”Space reduction”)
xi = hi(θ)
End For
f(θ)←− f(h1(θ), h2(θ), · · · , hn(θ))

(iv) θ ←− Nelder −Mead(F (θ)) (”Resolution in reducing space”)

(v) For i From 1 to n, (Configuration)
xi = hi(θ) (”Solutions configuration”)
End for

(vi) Display the solution x corresponding to λk value


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 55

3. Performances analysis

Performances analysis is done on the test problems defined below. This analysis is
mainly based on the complexity of computation time, convergence and diversity.

3.1. Complexity study

The MOMA-Plus algorithm starts with the scalarization where it is about a compar-
ison of p quantity λk(fk − zk), of which the complexity at worst is O(p2). Then, follows
the penalization, at this level the complexity is O(m) at worst with m the number of
constraints. The complexity of the transformation of decision variables into a single vari-
able is at worst equal to O(n). Let J + 1 be the size of formed simplex for applying
the Nelder-Mead algorithm where J is the size of search space. In our work, the search
space is one dimension so J = 1. Therefore, the complexity of Nelder-Mead algorithm
in MOMA-Plus method is constant. Then, as the search space is discretized into N + 1
points the complexity of Nelder-Mead is O(N). After finding the optimum by the Nelder-
Mead algorithm, the next step is to reconstruct the global and final solution. This step is
complex. So in one iteration, the complexity is:

O(p2) +O(m) +O(n) +O(N) +O(n).

If the size of the weight coefficients is K then the final complexity of the MOMA-Plus
algorithm is :

T = K.O(max{p2;m;n;N2}). (9)

This result is justified by the bellow theorem:

Theorem 4. Let f and g be two functions of the variable n ∈ N and with positive values.
Let’s consider two algorithms A and B of complexity respectively O(f(n)) and O(g(n)).
We have:

O(f(n)) +O(g(n)) = O(f(n) + g(n)) = O(max{f(n); g(n)}).

Proof. For the demonstration, see [18] �

The other method that we present in this work, and whose results will be compared.
With those of the MOMA-Plus method, is the NSGA-II method (Non-dominated Sorted
Genetic Algorithm II). It was developed by Deb and Srinivas[1]. It is a method that uses
the concept of elitism for which the best solutions are preserved and used through the
notion of Pareto dominance the distance of obstruction-[1] the crossover and mutation op-
erations for the creation of next generation. That makes the algorithm quickly convergence
towards the optimal solutions.

The complexity of the NSGA-II method is O(p.n2)[1]; where p is the size of the used
objective functions and n is the size of the input variables.
With regard to the MOMA-Plus complexity, we can confirm that the MOMA-Plus method
is effective because it belongs to the class of polynomial complexity algorithms.


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 56

3.2. Test problems and numerical experiments

The problems below have been solved by K. Deb in [1], where he is doing a comparative
study between the NSGA-II and some algorithms like SPEA†, PAES‡, etc. The result of
this study was that the NSGA-II method provided satisfactory results compared to the
other methods.
In this work, we propose a comparative study of performances through the convergence
and diversity metrics of the MOMA-Plus and NSGA-II methods. The problems we have
used are recorded in the table below :

Table 1: multiobjective problems

Min-Ex: SCH:
min f1(x1, x2) = x1

min f2(x1, x2) =
1 + x2

x1
0.1 6 x1 6 1

0 6 x2 6 5


min f1(x) = x2

min f2(x) = (x− 2)2

−5 6 x 6 5

PLN1: PLN2:

min f1(x) = x1

min f2(x) = g(x)×
(
1− (

f1(x)

g(x)
)2
)

g(x) = 1 +
9

n− 1
×

n∑
i=2

xi

x = (x1, x2, ..., xn) ∈ [0.1]n



min f1(x) = x1

min f2(x) = g(x)× h(x)

g(x) = 1 +
9

n− 1
×

n∑
i=2

xi

h(x) = 1−

√
f1(x)

g(x)
−
f1(x)

g(x)
× sin(10πf1(x))

x = (x1, x2, ..., xn) ∈ [0.1]n

PLN3: PLN4:

min f1(x) = x1

min f2(x) = g(x)×
(
1−

√
f1(x)

g

)
g(x) = 1 +

9

n− 1
×

n∑
i=2

xi

x = (x1, x2, ..., xn) ∈ [0.1]n



min f1(x) = x1

min f2(x) = g

(
1−

√
f1(x)

g

)
avec g(x) = 1 + 10 (n− 1) +

n∑
i=1

(xi − 10cos(4πxi))

xi ∈ [0; 1]

POL: VNT:

min f1(x) = 1 + (A1 −B1)2 + (A2 −B2)2

min f2(x) = (x1 + 3)2 + (x2 + 1)2

A1 = 0, 5sin(1)− 2cos(1) + sin(2)− 1, 5cos(2)

A2 = 1, 5sin(1)− cos(1) + 2sin(2)− 0, 5cos(2)

B1 = 0, 5sin(x1)− 2cos(x1) + sin(x2)− 1, 5cos(x2)

B2 = 1, 5sin(x1)− cos(x1) + 2sin(x2)− 0, 5cos(x2)

x1, x2 ∈ [−π;π]



min f1(x) = 0.5(x21 + x22) + sin(x21 + x22)

min f2(x) =
3x12x2 + 4

8
+

(x1 − x2 + 1)2

27
+ 15

min f3(x) =
1

x21 + x22 + 1
− 1, 1exp(−(x21 + x22))

x1, x2 ∈ [−3; 3]

KUR:
min f1(x) =

2∑
i=1

[−10exp(−0.2
√
x2i + x2i+1)]

min f2(x) =
3∑

i=1
[|xi|0.8 + 5sin(x3i )]

xi ∈ [−5; 5], i = 1, 2, 3

†Pareto Archived Evolution Strategy
‡Strength Pareto Evolutionary Algorithm


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 57

Note that the domain of the decision variables of the multi-modal problem (PNL4)
has changed in comparing to the initial search space.

3.3. Graphical results MOMA-Plus/NSGA-II

In this section we present the results of the simulations of the multiobjective problems.
These simulations are made up of Pareto’s analytical front, the different fronts resulting
from MOMA-Plus and NSGA-II methods. For the simulations, we used MATLAB R2013b
as simulation and programming software.

Remark 1. For the MOMA-Plus method, the parameters are P= size of the objective
functions, n= size of the input variables, m= number of constraints, N= size of the dis-
cretization.
For the NSGA-II method we will note: GEN=number of generations, POP=population
size, PF= Pareto fraction, MUT=type of mutation, CRS=type of crossing, ST-GEN=
Stall Generation, the crossing parameter that has been used for these problems is ” crossover-
scattered ”.

3.3.1. MIN-EX Problem

The resolution parameters of MIN-EX problem are:

• for the NSGA-II method:

Table 2: Parameters for NSGA-II

GEN POP PF MUT ST-GEN Crossover

NSGA-II 250 250 0,5 uniform 300 Scattered

• for the MOMA-Plus method:

Table 3: Parameters for MOMA-Plus

p n m N

MOMA-Plus 2 2 0 100

• The fronts are defined below:


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 58

Figure 2: MOMA-Plus Pareto front

Figure 3: NSGA-II Pareto Front

3.3.2. SCH problem

The resolution parameters of SCH problem are:

• for the NSGA-II method:

Table 4: Parameters for NSGA-II method

GEN POP PF MUT ST-GEN Crossover

NSGA-II défaut 100 0,5 uniform 300 Scattered

• for the MOMA-Plus method:

Table 5: Parameters for MOMA-plus method

p n m N

MOMA-Plus 2 2 0 100

• The fronts are defined below:


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 59

Figure 4: MOMA-Plus Pareto Front
Figure 5: NSGA-II Pareto Front

3.3.3. PLN1 problem

The resolution parameters of PLN1 problem are:

• for the NSGA-II method :

Table 6: Parameters for NSGA-II method

GEN POP PF MUT ST-GEN Crossover

NSGA-II 250 400 0,5 uniform 300 Scattered

• for the MOMA-Plus method:

Table 7: Parameters for MOMA-Plus

p n m N

MOMA-Plus 2 30 0 100

• The fronts are represented below:


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 60

Figure 6: MOMA-Plus Pareto Front
Figure 7: NSGA-II Pareto Front

3.3.4. PLN2 problem

The resolution parameters of PLN2 problem are:

• for the MOMA-Plus method:

Table 8: Parameters for MOMA-Plus method

p n m N

MOMA-Plus 2 30 0 100

• for NSGA-II parameters:

Table 9: Parameters for NSGA-II

GEN POP PF MUT ST-GEN Crossover

NSGA-II 250 250 0,5 uniform 300 Scattered

• The fronts are represented below:


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 61

Figure 8: MOMA-Plus Pareto front
Figure 9: NSGA-II Pareto front

3.3.5. PLN3 problem

The resolution parameters of PLN4 problem are:

• for the MOMA-Plus method:

Table 10: Parameters for MOMA-Plus

p n m N

MOMA-Plus 2 30 0 100

• for the NSGA-II method:

Table 11: Parameters for NSGA-II

GEN POP PF MUT ST-GEN Crossover

NSGA-II 250 400 0,5 uniform 300 Scattered

• The fronts are represented below:


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 62

Figure 10: MOMA-Plus Pareto front

Figure 11: NSGA-II Pareto front

3.4. PLN4 problem

The resolution parameters of PLN4 problem are

• for MOMA-Plus method:

Table 12: Parameters for MOMA-Plus

p n m N

MOMA-Plus 2 30 0 100

• for NSGA-II method:

Table 13: Parameters for NSGA-II

GEN POP PF MUT ST-GEN Crossover

NSGA-II 10100 250 0,3 uniform 1500 Scattered

• The fronts are represented below:


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 63

Figure 12: MOMA-Plus Pareto Front

Figure 13: NSGA-II Pareto Front

Remark 2. The problems that we have solved until here admit continuous optimal analytic
Pareto fronts. So, we have represented in same figure the analytic Pareto front and this is
provided by MOMA-Plus or NSGA-II method. That will be not possible for the following
problems.

3.4.1. POL Problem

The set of Pareto optimal solution of POL problem is not continuous. Therefore, there is
not an analytic front for it. So, we can not represent the two fronts in the same figure.
However, we will represent only the obtained fronts from MOMA-plus and NSGA-II.

The resolution parameters of POL problem are:

• for MOMA-Plus method:

Table 14: Parameters for MOMA-Plus

p n m N

MOMA-Plus 2 2 0 100

• for NSGA-II method:

Table 15: Parameters for NSGA-II

GEN POP PF MUT ST-GEN Crossover

NSGA-II 250 150 0.5 uniform 300 Scattered


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 64

• The fronts are represented below:

Figure 14: Front Pareto MOMA-Plus
Figure 15: Front Pareto NSGA-II

4. Calculation of performance indicators

4.1. Calculation of numerical execution time

In this section we present the time taken by the computer to execute the programs in
oder to find the optimal solutions. The characteristic of the used computer are:

• Mark: DELL;

• Processor: INTEL(R) Core(TM) i5-3340M CPU @2.70GHZ 2.70GHZ;

• RAM: 8 GO

• System: 64 bits

the obtained results of the calculation time, in seconds, are recorded in the following table:

Table 16: Numerical calculation time table

MIN-EX SCH PLN1 PLN2 PLN3 PLN4

MOMA-Plus 24,353167 17,875150 115,167287 161,009449 95,343133 127,083839

NSGA-II 9,121357 3,804150 16,211981 11.027945 5,351849 27,864951

We find that NSGA-II is faster than MOMA-Plus.

4.2. Study of the convergence metric

The convergence metric we use here is defined by the following formula[9]:

γ =

√
N∑
i=1

d2i

N
(10)


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 65

In this metric N is the size of the obtained solution by using MOMA-Plus or NSGA-II
solutions. The below table gives the value of N for each method on each problem. di is
the Euclidean distance between the obtained solution i and that of the nearest analytical
front.

Table 17: Size of the solutions obtained

N MIN-EX SCH PLN1 PLN2 PLN3 PLN4

MOMA-Plus 56 56 56 56 56 56

NSGA-II 83 56 200 125 200 75

This metric corresponds to the performance of the method, especially its ability to
converge towards the Pareto optimal analytical front. Thus, a high-performance and
effective method is one whose γ value is closed to zero. However, the calculation of this
metric involves two respective fronts: the given front by the used method and the Pareto
optimal analytical front.
The obtained results of the calculation of the convergence metric are recorded in the table
below :

Table 18: Metric γ values

γ MIN-EX SCH PLN1 PLN2 PLN3 PLN4

MOMA-Plus 0,0691 0,0053 0,0042 0,0599 0,0137 0,1154

NSGA-II 0,0321 0,0056 0,0025 0,0124 0,0175 0,0274

With regard to these obtained results, we notice that the values of the convergence
metric provided by the two methods are all closed to zero. It would mean that the MOMA-
Plus method is effective. In addition, we can see that the MOMA-Plus method approaches
Pareto optimal solutions are better than the NSGA-II method on problems (SCH) and
(PLN3).

4.3. Study of the diversity metric

The metric of diversity that we use here is defined by the following formula[9] :

∆ =

M∑
m=1

dem +
N−1∑
i=1
|di − d|

M∑
m=1

dem + (N − 1)d

(11)

In the relationship: di is the Euclidean distance between two solutions closed to the Pareto
front provided by the used method; d is the average of these distances; dem is the distance
between the extreme solutions of all the solutions on the Pareto front to the analytical


A. Som, K. Somé, A. Compaoré, B. Somé / Eur. J. Pure Appl. Math, 13 (1) (2020), 48-68 66

front.

The metric of diversity corresponds to the distribution of solutions on the Pareto front.
Note that a good distribution is the one whose ∆ value is closed to zero or even equal to
zero. The calculation of the metric of diversity of the MOMA-Plus and NSGA-II method
are recorded in the following table :

Table 19: Metric ∆ values

∆ MIN-EX SCH PLN1 PLN2 PLN3 PLN4

MOMA-Plus 1,1833 0,5537 0,0309 0,9818 0,3498 0,9835

NSGA-II 0,0290 0,0183 0,0319 0,0293 1,0023 0,9710

Here, we also see that MOMA-Plus method is better than NSGA-II method on prob-
lems (PLN1) and (PNL3.)

4.4. Particular Cases

The problems POL, VNT and KUR are discontinuous fronts and we have not an
analytic Pareto front. This makes it difficult to study convergence and diversity. Therefore,
a possible comparative study is difficult. Nevertheless, a study of convergence and diversity
has been done by combining the two fronts given by MOMA-Plus and NSGA-II that is
given us by the below table:

Table 20: joint table of performances

POL VNT KUR

γ 0,7645 0,2429 0,3699

∆ 0,7341 0.7856 0,7973

With regard to the obtained results, we can see that the solutions provided by the
MOMA-Plus and NSGA-II methods are very closed.

4.5. Results analysis

These two methods have given the Pareto optimal solutions in few time with good
distribution and diversity on the used test problems. There are some performances metric
where MOMA-Plus is better than NSGA-II on some test problems. So, MOMA-Plus can
be considered for the best resolution of multiobjective optimization problems.


REFERENCES 67

5. Conclusion

The results of this study using the MOMA-Plus method are satisfactory in view of
the comparison made with the NSGA-II method. Therefore, the MOMA-Plus method
can be taken into account among the reference metaheuristics in terms of its ability to
solve various problems types of multiobjective problems and also its qualities to quickly
converge towards the optimal solutions. Nevertheless, it is desirable that improvements in
the performance of MOMA-Plus are preciously doing on executing time.

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