T. Santos, and S. Xavier,

“PBI-BFS-MaOA: A Many-Objective Evolutionary Algorithm with PBI-Based Boundary-Front Selection”,

Latin-American Journal of Computing (LAJC), vol. 13, no. 2, 2026.

PBI-BFS-MaOA: A Many-

Objective Evolutionary

Algorithm with PBI-

Based Boundary-Front

Selection

ARTICLE HISTORY

Received 8 March 2026

Accepted 26 May 2026

Published 7 July 2026

Thiago Santos

Federal University of Ouro Preto (UFOP)

Associate Professor

Ph.D. in Mathematics

Brazil

santostf@ufop.edu.br

ORCID: 0000-0002-2435-2786

Sebastião Xavier

Federal University of Ouro Preto (UFOP)

Associate Professor

Ph.D. in Mathematics

Brazil

semarx@ufop.edu.br

ORCID: 0009-0004-2765-0764

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

This work is licensed under a Creative Commons

Attribution-NonCommercial-ShareAlike 4.0 International License.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July - December 2026

PBI-BFS-MaOA: A Many-Objective Evolutionary

Algorithm with PBI-Based Boundary-Front

Selection

Thiago Santos

Federal University of Ouro Preto (UFOP)

Associate Professor

Ph.D. in Mathematics

Brazil

santostf@ufop.edu.br

Sebasti

ao Xavier

Federal University of Ouro Preto (UFOP)

Associate Professor

Ph.D. in Mathematics

Brazil

semarx@ufop.edu.br

Abstract—Reference-guided many-objective evolutionary al-

gorithms often lose selection pressure when Pareto dominance

becomes scarce and the ﬁnal accepted front must be truncated.

We propose PBI-BFS-MaOA, a many-objective evolutionary

algorithm that preserves Pareto ranking for feasible solutions

and modiﬁes only the survival decision on the boundary front.

The method combines cumulative ideal–nadir normalization,

penalty-based boundary intersection association, active-niche

ﬁltering, and occupancy-aware survivor insertion. These opera-

tions are activated where convergence and directional coverage

must be decided simultaneously. We evaluate the algorithm

on DTLZ1–DTLZ4 and WFG1–WFG4 with M ∈{5, 8, 10}

objectives, using the averaged Hausdorff distance ∆

, Wilcoxon

signed-rank tests, Friedman rank analysis, and runtime mea-

surements. PBI-BFS-MaOA obtains the best mean ∆

in 13

of 24 benchmark instances, with its strongest gains on high-

dimensional DTLZ cases and degenerate WFG3 instances, while

its runtime remains between NSGA-III and CMOEA-CD.

Keywords—many-objective optimization, environmental selec-

tion, reference directions, penalty-based boundary intersection

I. INTRODUCTION

Many-objective optimization is difﬁcult not only because

more criteria must be evaluated, but because Pareto domi-

nance rapidly loses its ability to separate candidate solutions.

Li et al. [1] identify this loss of discrimination as a central

obstacle once the number of objectives moves beyond the

classical two- or three-objective setting. Santos and Takahashi

[2] give a formal account of the same phenomenon: as

objective dimensionality increases, the probability that one

candidate dominates another decreases sharply. The popula-

tion then accumulates mutually non-dominated solutions, and

environmental selection must choose among candidates that

the initial dominance relation no longer orders with enough

contrast.

This loss of contrast changes the role of survival selec-

tion. After the clearly superior fronts have been accepted,

the remaining population slots are often disputed by an

overﬂowing boundary front. At that point, the algorithm

must preserve convergence while still maintaining directional

coverage. The problem is geometric as much as statistical. In

high-dimensional objective spaces, local density is harder to

estimate, front shapes are harder to interpret, and unsupported

reference directions may receive attention simply because the

selection rule lacks a sharper local signal. Pal et al. [3] discuss

a related difﬁculty from the perspective of objective reduc-

tion: deciding which information remains relevant becomes

itself a nontrivial design problem.

Several algorithmic families address this pressure from

different angles. Decomposition methods organize search

through scalar subproblems, as in MOEA/D [4], and

dominance-decomposition hybrids such as MOEA/DD [5]

show that Pareto ordering and directional structure can

coexist in the same survival mechanism. Reference-guided

algorithms follow a related geometric logic. NSGA-III [6],

θ-DEA [7], and RVEA [8] all use reference information to

recover discrimination when non-dominated sets become too

large. Their shared premise is clear: many-objective selection

needs more than front rank.

Directional guidance, however, is not neutral. Ishibuchi

et al. [9] show that decomposition-based performance is

strongly affected by Pareto-front shape. A reference structure

that works well on a regular front may become less reliable

on disconnected, biased, or degenerate geometries. Qiu et

al. [10], Liu et al. [11], Li et al. [12], and Wang et al.

[13] respond to this issue by adapting reference structures or

strengthening dominance with reference-vector information.

These contributions suggest that survival quality depends not

only on having directions, but on deciding which directions

are actually supported by the current population.

The present work focuses on that decision at a narrower

location: the boundary front. We do not replace Pareto rank-

ing, redeﬁne dominance globally, or introduce a multiarchive

search architecture. Instead, PBI-BFS-MaOA preserves the

usual Pareto scaffold for feasible solutions and intervenes

only when the ﬁrst overﬂowing front must be truncated. The

proposed survival rule combines cumulative ideal–nadir nor-

malization, penalty-based boundary intersection association,

active-niche ﬁltering, and occupancy-aware insertion. The

aim is to use geometric information exactly where ordinary

front ordering becomes underdetermined.

This local view has practical value in technological de-

cision problems where many objectives must be balanced

under a limited evaluation budget. Engineering design, energy

dispatch, logistics planning, portfolio allocation, scheduling,

and resource management often require simultaneous trade-

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

T. Santos, and S. Xavier,

“PBI-BFS-MaOA: A Many-Objective Evolutionary Algorithm with PBI-Based Boundary-Front Selection”,

Latin-American Journal of Computing (LAJC), vol. 13, no. 2, 2026.

offs among cost, reliability, risk, environmental impact, and

service quality. In such settings, wasting evaluations on

poorly supported directions can delay the discovery of usable

compromises. A boundary-front rule that protects the global

Pareto order while improving the last survival decision is

therefore relevant beyond benchmark optimization.

We evaluate the proposed intervention on DTLZ and WFG

benchmark families with M ∈{5, 8, 10} objectives in the Py-

mooLab environment [14]. Performance is measured with the

averaged Hausdorff distance ∆

, Wilcoxon signed-rank tests,

Friedman rank analysis, and runtime measurements. NSGA-

III and CMOEA-CD serve as structurally distinct baselines:

the ﬁrst is the canonical reference-guided Pareto method, and

the second is a recent archive-cooperation approach. Section

II places the proposal in the literature, Section III deﬁnes the

survival rule, Section IV reports the numerical protocol and

results, and Section V concludes the paper.

II. R

ELATED WORK

The many-objective literature can be organized around a

common design question: once Pareto dominance no longer

separates most candidates, where should additional discrimi-

nation enter the evolutionary pipeline? Some methods modify

the global search decomposition, others adapt the reference

structure, and still others revise the dominance relation. The

present work belongs to the environmental-selection line, but

it is useful to position it against these alternatives [1].

Decomposition-based methods were among the earliest

scalable responses. Zhang and Li [4] distributed the search

across scalar subproblems, thereby reducing reliance on

global pairwise dominance. Li et al. [5] later combined dom-

inance and decomposition in MOEA/DD. This line of work

established a key principle for many-objective search: front

rank and directional scalarization can operate at different

resolutions. Pareto order can maintain the broad convergence

scaffold, while a directional mechanism can resolve local

competition among candidates that are otherwise difﬁcult to

separate.

Reference-guided methods express the same principle in

geometric form. Deb and Jain [6] used reference points to

guide environmental selection in NSGA-III, which remains

the most direct baseline for a modiﬁed reference-based sur-

vival stage. Yuan et al. [7] introduced angular information

through θ-DEA, and Cheng et al. [8] built RVEA around

reference-vector-guided survival. These algorithms differ in

implementation, but they share a central conclusion: once

non-dominated sets become too large, directional information

is needed to make survival decisions operational.

That conclusion must be qualiﬁed. Ishibuchi et al. [9] show

that decomposition-based algorithms are highly sensitive to

Pareto-front shape. A ﬁxed reference structure may be effec-

tive on regular fronts and less reliable on disconnected, bi-

ased, or degenerate fronts. Qiu et al. [10] improved objective-

space decomposition under this concern, while Liu et al. [11],

Li et al. [12], and Wang et al. [13] introduced self-guided,

redistributed, or reference-vector-based dominance mecha-

nisms. The common lesson is that the reference structure

should not be treated as automatically valid everywhere in

the objective space.

A second line preserves Pareto semantics while weakening

or extending its strict form. Zhu et al. [15] generalized

Pareto optimality for many-objective search. Tian et al.

[16] strengthened dominance by combining convergence and

diversity information, and Zhu et al. [17], [18] later devel-

oped generalized or relaxed dominance as a broader design

framework. These methods show that dominance can be

repaired, but they often act globally even though the strongest

ambiguity may occur at a speciﬁc survival stage.

Environmental-selection studies make that local ambiguity

explicit. Cheng et al. [19] argued that mating and envi-

ronmental selection should be designed jointly, because the

quality of selected parents is inseparable from the survival

pressure they face. Sharma and Shukla [20] studied line-

prioritized normalization and survivor choice, Myszkowski

and Laszczyk [21] investigated diversity-based selection un-

der constraints, and Liu et al. [22] treated environmental

selection through clustering. These contributions differ in

mechanics, but they agree on a practical point: the ﬁnal

accepted front is not a bookkeeping remainder. It is often

where convergence and spread are either preserved or lost.

CMOEA-CD [23] provides a recent contrast. Instead of

modifying one stage of selection, it uses three collabo-

rative archives: a forward-exploration archive, a diversity-

enhancement archive, and a feasibility-exploitation archive.

This architecture separates exploration, diversity recov-

ery, and feasible-solution intensiﬁcation across different

population-management channels. It is broader than the de-

sign studied here, and for that reason it is a useful comparator.

Our question is more restricted: can a focused intervention

in the boundary front recover selection pressure without

replacing the surrounding evolutionary framework?

PBI-BFS-MaOA is therefore positioned between reference-

guided selection and environmental-selection reﬁnement. It

keeps the Pareto-front order intact, associates only the critical

front with normalized PBI directions, ﬁlters directions that

lack support from the leading front when the front coverage

is clearly sparse, and inserts survivors with an occupancy-

aware rule. The contribution is intentionally local. Its value

is not that it redesigns the entire many-objective algorithm,

but that it targets the point at which the standard reference-

guided survival rule becomes least decisive.

III. P

ROPOSED METHOD

The proposed method is a generational many-objective

evolutionary algorithm whose main contribution lies in en-

vironmental selection rather than reproduction. Sampling,

mating, and variation follow a classical evolutionary back-

bone. The restricted question is whether survival selection

can recover discrimination inside the boundary front without

discarding the global Pareto scaffold. To do so, the method

retains front-based ordering where the population already

provides a clear rank structure and intervenes only when the

ﬁrst overﬂowing front must be trimmed. The implementation

then applies cumulative ideal–nadir normalization, PBI-based

association, active-niche ﬁltering, and occupancy-aware in-

sertion. The method is therefore a survival-stage reﬁnement

within the standard evolutionary loop, focused on the point

where dominance contrast is weakest in practice.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July - December 2026

Algorithm 1 Framework of the implemented PBI-BFS-

MaOA

1: Input: population size N, reference directions W,

penalty parameter θ

2: Sample and evaluate the initial population S

3: Initialize z

min

and z

max

from S

4: Apply environmental selection to obtain S

and its tour-

nament ﬁtness

5: while the stopping criterion is not met do

6: Select parents by tournament using constraint violation

and current ﬁtness

7: Generate offspring Q

with a standard genetic varia-

tion operator

8: Form the merged population P

= S

∪Q

9: Update z

min

and z

max

from P

10: Apply environmental selection on P

11: Obtain the next population S

t+1

and the updated

tournament ﬁtness

12: end while

13: Extract the current approximation set by ﬁltering the

population for feasible nondominated solutions

A. Framework of the Implemented Method

Let S

denote the population at generation t, with |S

| =

N, and let W = {w

,...,w

} be the externally supplied

set of reference directions. The process begins by sampling

and evaluating an initial population, from which cumulative

ideal and nadir estimates are initialized. Each generation

then performs tournament-based parent selection, offspring

generation by a standard variation operator, parent–offspring

merging, and one environmental-selection call that returns

both the survivors and the rank-based ﬁtness values reused

in the next tournament selection. Algorithm 1 summarizes

the implemented architecture, and Fig. 1 presents the same

process from the viewpoint of the survival decision.

The boundary-front rule is active from the ﬁrst

environmental-selection call onward, rather than being intro-

duced as a late-stage correction. This matters because many-

objective runs may produce large non-dominated subsets well

before the ﬁnal generations. The method is therefore directed

at the M>3 regime, where the last accepted front can

dominate the survival outcome. The selection path moves

through feasibility ﬁltering, Pareto-front insertion, critical-

front association, active-niche checking, and occupancy-

aware insertion.

B. PBI-Based Boundary-Front Selection

Given the merged population P

, aggregated constraint

violation is computed for each candidate x

∈P

CV(x



max{0,g

)}. (1)

This value induces the feasible subset

= {x

∈P

: CV( x

)=0}. (2)

If |P

| <N, all feasible solutions are retained and the

remaining slots are ﬁlled by the least infeasible candidates

ranked by increasing CV. This branch is part of the imple-

mented survival routine, although the numerical analysis in

this paper uses unconstrained benchmark families. Its role is

to establish a clear priority order: feasibility is handled ﬁrst

when feasible points are scarce, and the directional boundary-

front rule is invoked only when enough feasible candidates

exist for truncation to become the dominant issue.

When |P

|≥N, selection proceeds on the feasible

subset. The algorithm performs non-dominated sorting on the

original objective vectors of P

and obtains ordered fronts

, F

,.... Complete fronts are copied into the next popu-

lation until the ﬁrst overﬂowing front F

ℓ

is reached. Thus,

the global convergence scaffold remains Pareto-ordered. The

directional rule does not reshufﬂe clearly superior fronts; it

acts only where Pareto ranking no longer determines the

remaining survivor set.

The candidates in the critical front are then evaluated

in normalized objective space. Let z

min

and z

max

denote

the cumulative ideal and nadir estimates updated across

generations. For an objective vector f

, the normalized vector

− z

min

max(z

max

− z

min

,ϵ)

, (3)

where ϵ is a componentwise safeguard against null spans.

Each reference direction is normalized to unit length, and

for every candidate–direction pair the algorithm computes

(i, k)=

⊤

(i, k)=



∥

− d

(i, k)

. (4)

The corresponding penalty-based boundary-intersection score

PBI(i, k)=d

(i, k)+θ

(i, k), (5)

where



θ, M ≤ 3,



M/3, M>3.

(6)

Cumulative ideal–nadir normalization keeps directional com-

parisons stable across generations instead of allowing tran-

sient objective spans to dominate the association step. The

PBI score separates an axial component d

from an orthog-

onal component d

, so each candidate is evaluated by both

progress along a direction and deviation from that direction.

The scaling in (6) increases the orthogonal penalty as M

grows, which is consistent with the stronger ambiguity ob-

served in higher-dimensional objective spaces. The associated

niche of x

is then

⋆

(i) = arg min

PBI(i, k). (7)

This association does not treat each direction as an indepen-

dent scalar subproblem in the MOEA/D sense [4]. It provides

a local geometric view only for candidates that have already

passed Pareto-front screening.

C. Active-Niche Filtering and Occupancy-Aware Insertion

Once the complete fronts have been accepted, let A

denote

survivors already inserted before truncating the boundary

front F

ℓ

. Current occupancy is calculated for every niche

= |{x

∈A

: k

⋆

(i)=k}|. (8)

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

T. Santos, and S. Xavier,

“PBI-BFS-MaOA: A Many-Objective Evolutionary Algorithm with PBI-Based Boundary-Front Selection”,

Latin-American Journal of Computing (LAJC), vol. 13, no. 2, 2026.

Initialize population

and evaluate

Update z

min

, z

max

and normalize objectives

Merge parents

and offspring

Enough feasible

solutions?

Keep feasible +

least infeasible

Non-dominated sorting

on feasible set

Insert complete fronts

until overﬂow

Critical front:

compute d

, best niche

Estimate niche occupancy

and active niches

Choose least occupied niche

and minimize d

+ θ

Next population

and updated ﬁtness

yes

Fig. 1: Conceptual ﬂow of the implemented PBI-BFS-MaOA survival mechanism

This occupancy count drives the boundary-front insertion

rule. Additionally, the niche set that the ﬁrst Pareto front

activates is extracted in the algorithm:

Ω

= {k

⋆

(i):x

∈F

}. (9)

If |Ω

| < 0.8K, the implementation treats the current leading

front as insufﬁciently spread over the reference set. In that

case, when the intersection is nonempty, only niches repre-

sented by F

ℓ

and active in Ω

are admissible. Otherwise,

all boundary-front niches remain admissible. The ﬁrst front

therefore acts as the support signal: if the best feasible

solutions occupy only a restricted subset of directions, the

algorithm avoids spending survivor slots on directions that

the current population does not substantively support.

The 0. 8K threshold is a conservative coverage gate rather

than a tuned benchmark-speciﬁc parameter. It activates ﬁl-

tering only when at least 20% of the reference directions

are unsupported by the leading front. This choice separates

broadly covered fronts from fronts with visibly sparse direc-

tional support while avoiding the overly aggressive behavior

that would arise from ﬁltering whenever a small number

of directions is missing. The value is also easy to interpret

operationally: active-niche ﬁltering is used only when the ﬁrst

front no longer represents most of the reference scaffold.

The last rule for insertion acts one survivor at a time. Every

step identiﬁes the least-occupied admissible niches, breaks

ties randomly, and selects one niche. Let C

⊆F

ℓ

denote the

candidates now assigned to the chosen niche k. The selected

candidate is

⋆

= arg min

∈C

(i, k

⋆

(i)) + θ

(i, k

⋆

(i))) . (10)

This rule couples two pressures that in many-objective sur-

vival are often in tension. Poor niche occupancy allows

for angular spread throughout the reference structure, and

the within-niche PBI minimization favors candidates with

better local convergence relative to the selected direction. The

design is deliberately local. It does not change the ranking

relation for the entire population, nor does it try to ﬁnd a

new global density model. It resolves the speciﬁc ambiguity

created by the critical front after the better-ranked fronts have

already been accepted.

After insertion, survivors receive rank-based ﬁtness values

according to their Pareto-front index. These values, together

with constraint violation, are reused by tournament selection.

Offspring generated by the variation operator are then merged

back into the population, and the same survival rule is applied

again. The empirical behavior reported in this paper should

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July - December 2026

therefore be read as the effect of a repeated boundary-front

decision, not as a one-time tie-breaking operation.

The additional computational cost is concentrated in the

critical-front association and insertion stage. Let n

= |P

be the number of feasible candidates, b = |F

ℓ

| the size of

the boundary front, K the number of reference directions,

M the number of objectives, and r the number of remain-

ing survivor slots. Non-dominated sorting over the feasible

subset follows the usual front-ranking cost. The proposed

modiﬁcation adds O(bKM) operations for normalized PBI

association, O(n

+ K) operations to compute current niche

occupancy and active niches, and up to O(r(b + K)) oper-

ations for iterative occupancy-aware insertion under a direct

implementation. Since r ≤ b, the added survival-stage cost

is bounded by O(bKM + b

+ bK + n

). This is higher than

the simplest NSGA-III niching pass, but it remains localized

to the boundary front and is consistent with the observed

runtime proﬁle: slower than NSGA-III, yet substantially

cheaper than the broader multiarchive management used by

CMOEA-CD.

D. Design Highlights and Current Scope

The proposed approach has four active design choices.

First, it keeps Pareto-front ordering as the global convergence

scaffold and reserves directional scalarization for the moment

at which that scaffold no longer determines the survivor set.

Second, it uses cumulative ideal–nadir normalization to keep

PBI association numerically stable across the run without

excessive sensitivity to transient population spans. Third, it

scales the orthogonal PBI penalty by



M/3 at M>3,

directly targeting the many-objective regime for which the

method is designed. Fourth, it combines active-niche ﬁltering

and occupancy-aware insertion, making the boundary-front

decision dependent on both local scalarization values and the

current elite support of the reference structure.

These design choices also deﬁne the limits of the present

claim. The constructor exposes the value σ

, and the source

ﬁle includes a helper for niche-penalty calculation, but in the

current implementation that helper does not take part in the

active survival path. The method is therefore not a fuzzy-

dominance framework, a radial-repulsion mechanism, or a

multiarchive strategy. The contribution, as evidenced in the

code and by the benchmarks, is narrower and more speciﬁc:

a Pareto-ordered many-objective evolutionary approach with

normalized PBI-based boundary-front selection, conditional

active-niche ﬁltering, and occupancy-aware survivor inser-

tion.

I V. N

UMERICAL SIMULATION AND ANALYSIS

A. Benchmark Test Problems

For the numerical study, we use two benchmark groups,

namely DTLZ1–DTLZ4 [24] and WFG1–WFG4 [25], with

objective counts M ∈{5, 8, 10}. These settings place each

experiment in the many-objective regime for which the pro-

posed survival rule was developed. The benchmark families

are deliberately complementary. DTLZ provides canonical

and relatively interpretable front structures, while WFG intro-

duces stronger distortions in shape, modality, deceptiveness,

and degeneracy. Taken together, the two suites test whether

the mechanism works only on regular fronts or also under

geometries where reference-guided truncation is more difﬁ-

cult.

The DTLZ family isolates complementary sources of dif-

ﬁculty. DTLZ1 has a linear Pareto front with strong multi-

modality, making it useful for testing whether the boundary-

front rule remains stable under competing local attractors.

DTLZ2 has a smooth spherical front and provides a cleaner

directional-coverage test. DTLZ3 retains the DTLZ2 geome-

try but adds severe multimodality, making it a difﬁcult con-

vergence test without changing the fundamental front shape.

DTLZ4 biases the mapping toward extreme regions and is

relevant for methods that depend on directional balance.

The WFG suite directs the evaluation toward more irregu-

lar geometries. WFG1 adopts bias and mixed shape transfor-

mations, while WFG2 generates disconnected and deceptive

structures. WFG3 produces degenerate fronts, and WFG4

adds strong multimodality. WFG3 is particularly relevant

because degeneracy is the scenario in which active-niche

ﬁltering should have its clearest effect.

We vary the number of objectives from 5 to 8 and 10 to

test whether the local survival logic becomes more relevant

as the selection stage grows more crowded.

B. Algorithms Comparison and Experimental Settings

The benchmark compares PBI-BFS-MaOA with two struc-

turally relevant baselines: NSGA-III [6] and CMOEA-CD

[23]. NSGA-III is the direct baseline because it combines

Pareto fronts with reference directions and performs en-

vironmental selection through reference-guided niching. It

is therefore the natural comparator for testing whether a

new boundary-front rule improves a reference-guided sur-

vival protocol without changing the broader evolutionary

paradigm.

CMOEA-CD is included for a different reason. Instead of

relying on a single environmental-selection regime, it orga-

nizes a forward-exploration archive, a diversity-enhancement

archive, and a feasibility-exploitation archive. Although

CMOEA-CD was proposed for constrained multiobjective

optimization, it remains informative here because it repre-

sents a recent and technically sophisticated alternative to

single-stage survivor selection. The comparison therefore

contrasts a local boundary-front intervention with a broader

archive-cooperation design.

All algorithms are run in the PymooLab framework [14].

For every problem instance, the population size is ﬁxed

at 100, the maximum number of function evaluations is

ﬁxed at 50,000, and 30 independent runs are performed.

The benchmark harness uses independent random seeds and

records the indicator values and summary statistics used in

the ﬁnal performance evaluation.

NSGA-III and PBI-BFS-MaOA use the same reference-

direction generation logic, which keeps the comparison fo-

cused on the survival operator rather than on different di-

rectional sets. For each instance, we report the mean and

standard deviation of the performance indicator, the winning

algorithm, the percentage gain of the winner over the closest

comparator, and the pairwise Wilcoxon decision marker. This

structure supports three readings: instance-wise comparison

through the table, suite-wise comparison through Friedman

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

T. Santos, and S. Xavier,

“PBI-BFS-MaOA: A Many-Objective Evolutionary Algorithm with PBI-Based Boundary-Front Selection”,

Latin-American Journal of Computing (LAJC), vol. 13, no. 2, 2026.

ranks, and computational-cost comparison through runtime

statistics.

C. Experimental Results on Benchmark

The main performance indicator is ∆

, the averaged Haus-

dorff distance between the approximation set returned by an

algorithm and a reference Pareto front [26]. Let A denote

the approximation set and let P

∗

denote the reference Pareto

front. For a point u and a ﬁnite set S, deﬁne

d(u, S)=min

s∈S

∥u − s∥

. (11)

The p-averaged generational distance and inverted gener-

ational distance are written as

(A, P

∗



|A|



a∈A

d(a, P

∗

)



1/p

, (12)

IGD

(A, P

∗





∗



∗

∈P

∗

d(p

∗

,A)





1/p

. (13)

Following Sch

utze et al. [26], the averaged Hausdorff

distance is then

∆

(A, P

∗

) = max {GD

(A, P

∗

), IGD

(A, P

∗

)} , (14)

Lower ∆

values indicate better performance because both

convergence toward the reference front and spread along that

front are penalized. The indicator is particularly relevant here

because the proposed survival rule must negotiate conver-

gence and diversity at the same stage.

A survival rule can reduce crowding while damaging

local convergence, or improve convergence while collapsing

directional coverage. ∆

is sensitive to both failure modes

and is therefore more stringent than a convergence-only

indicator.

For each test instance, Table I reports the mean ∆

value

and sample standard deviation over 30 independent runs.

Wilcoxon signed-rank tests are computed at signiﬁcance level

α =0.05 for pairwise comparisons, and Friedman tests are

applied to compare average ranks across the DTLZ and WFG

subsets.

The table is arranged as a single consolidated ﬂoating

environment so that the full benchmark can be inspected at

once. DTLZ and WFG are visually separated because the

interpretation depends strongly on the distinction between

canonical and distorted front geometries.

Table I shows that PBI-BFS-MaOA obtains the best mean

∆

value in 13 of the 24 benchmark instances. NSGA-

III is best in 7 cases, and CMOEA-CD is best in 4. The

distribution of wins is more informative than the raw total.

At M =5, PBI-BFS-MaOA leads in only 2 of 8 cases,

whereas at M =8and M = 10 it leads in 6/8 and

5/8 cases, respectively. This pattern agrees with the design

motivation: as the number of objectives increases, the over-

ﬂowing front becomes larger and less sharply separated under

standard reference-guided selection, creating more room for

a boundary-front intervention to help.

The DTLZ suite provides the clearest evidence in favor

of the proposed approach. The Friedman test over the 12

DTLZ instances returns average ranks of 1.417 for PBI-BFS-

MaOA, 1.833 for NSGA-III, and 2.750 for CMOEA-CD,

with χ

= 11.167 and p =3.76 × 10

−3

. Pairwise Wilcoxon

testing indicates a signiﬁcant advantage of PBI-BFS-MaOA

over CMOEA-CD (p =1.46 × 10

−3

), while the difference

from NSGA-III does not cross the 5% signiﬁcance threshold

(p =6.40 × 10

−2

). Thus, on DTLZ, the proposed method

clearly separates from CMOEA-CD and shows a repeated,

although not statistically decisive, advantage over NSGA-III.

The strongest margins occur when multimodality and

boundary-front pressure appear together. On DTLZ3 with

M = 10, PBI-BFS-MaOA obtains ∆

=1.52, compared

with 21.9 for NSGA-III and 75.6 for CMOEA-CD. On

DTLZ3 with M =8, the corresponding values are 1.54,

14.3, and 19. 8. For DTLZ1 with M =8and M = 10,

the proposed method reaches 1.09 × 10

−1

and 1.46 × 10

−1

compared with 2.00 × 10

−1

and 4.71 × 10

−1

for NSGA-III

and much larger values for CMOEA-CD. These are precisely

the cases in which many candidates can share the same front

rank while differing substantially in directional plausibility.

The evidence is not uniform across all landscapes. NSGA-

III remains marginally best on DTLZ1 with M =5and

on DTLZ2 with M =5and M = 10, while CMOEA-CD

obtains the best mean on DTLZ4 with M =5. The WFG

subset is also more balanced. The Friedman test does not

reject comparable performance among the three algorithms

on this subset (χ

=2.000, p =3.68× 10

−1

), and PBI-BFS-

MaOA and NSGA-III have the same average rank, 1.833.

This result is consistent with the fact that WFG includes

disconnected, deceptive, degenerate, and highly multimodal

structures, where different failure modes can dominate.

Even so, the WFG results support the speciﬁc role of

active-niche ﬁltering. PBI-BFS-MaOA obtains the best mean

value on all three WFG1 instances and on WFG3 for M =8

and M = 10. On WFG3 with M = 10, it obtains ∆

=7.23,

compared with 14.2 for NSGA-III and 15.2 for CMOEA-

CD. On WFG3 with M =8, the corresponding values

are 2.60, 7.14, and 8.31. Since WFG3 is degenerate, these

cases are aligned with the intended effect of active-niche

ﬁltering: when only a subset of directions is supported by the

leading front, limiting insertion to active niches can prevent

the algorithm from allocating survivors to weakly justiﬁed

directions.

Runtime results support a balanced interpretation. Across

instances, NSGA-III requires 10.69 s on average, PBI-BFS-

MaOA requires 14.73 s, and CMOEA-CD requires 50. 28

s. The proposed method is therefore approximately 37.8%

slower than NSGA-III, but 70.7% faster than CMOEA-CD.

Wilcoxon testing indicates that these runtime differences are

systematic (p<1.2 × 10

−7

in each pairwise comparison).

Across all 24 instances, the global Friedman test still favors

PBI-BFS-MaOA, with average ranks of 1.625, 1.

833, and

2.542 for PBI-BFS-MaOA, NSGA-III, and CMOEA-CD,

respectively (χ

= 11.083, p =3.92 × 10

−3

Taken together, the results position PBI-BFS-MaOA as a

quality–cost compromise. It is costlier than NSGA-III, sub-

stantially cheaper than CMOEA-CD, and frequently stronger

in high-dimensional or degenerate settings where boundary-

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July - December 2026

TABLE I. Statistical results obtained by NSGA-III, CMOEA-CD, and PBI-BFS-MaOA on the DTLZ and WFG problems

(∆

)

Problem M NSGA-III CMOEA-CD PBI-BFS-MaOA Best Gain (%)

DTLZ Suite

DTLZ1

5 6.63e-02 ± 2.8e-04

2.73e-01 ± 3.9e-01

−

6.75e-02 ± 7.0e-04

−

NSGA-III 1.75

8 2.00e-01 ± 1.7e-01

−

1.41e+00 ± 1.4e+00

−

1.09e-01 ± 1.2e-03

PBI-BFS-MaOA 45.19

10 4.71e-01 ± 7.6e-01

−

4.44e+00 ± 3.7e+00

−

1.46e-01 ± 1.2e-02

PBI-BFS-MaOA 68.91

DTLZ2

5 1.99e-01 ± 4.5e-05

2.24e-01 ± 3.9e-03

−

1.99e-01 ± 4.7e-04

−

NSGA-III 0.17

8 3.37e-01 ± 1.6e-04

−

4.92e-01 ± 2.0e-02

−

3.35e-01 ± 7.4e-04

PBI-BFS-MaOA 0.48

10 3.97e-01 ± 3.2e-04

7.06e-01 ± 6.7e-02

−

4.03e-01 ± 1.8e-03

−

NSGA-III 1.32

DTLZ3

5 1.73e+00 ± 1.6e+00

−

1.52e+00 ± 1.5e+00

−

3.79e-01 ± 5.4e-01

PBI-BFS-MaOA 75.09

8 1.43e+01 ± 7.3e+00

−

1.98e+01 ± 1.1e+01

−

1.54e+00 ± 2.4e+00

PBI-BFS-MaOA 89.21

10 2.19e+01 ± 7.9e+00

−

7.56e+01 ± 6.6e+01

−

1.52e+00 ± 1.5e+00

PBI-BFS-MaOA 93.07

DTLZ4

5 2.42e-01 ± 1.1e-01

2.17e-01 ± 3.8e-03

2.61e-01 ± 1.2e-01

CMOEA-CD 16.80

8 3.39e-01 ± 1.6e-02

−

4.51e-01 ± 9.7e-03

−

3.36e-01 ± 8.2e-04

PBI-BFS-MaOA 1.12

10 4.07e-01 ± 2.3e-02

5.56e-01 ± 1.7e-02

−

4.00e-01 ± 2.3e-03

PBI-BFS-MaOA 1.78

WFG Suite

WFG1

5 1.73e+00 ± 1.4e-02

−

1.94e+00 ± 1.4e-02

−

1.69e+00 ± 1.8e-02

PBI-BFS-MaOA 2.69

8 2.47e+00 ± 6.3e-02

−

2.87e+00 ± 2.1e-02

−

2.32e+00 ± 4.6e-02

PBI-BFS-MaOA 6.01

10 2.90e+00 ± 7.3e-02

−

3.36e+00 ± 4.0e-02

−

2.65e+00 ± 7.7e-02

PBI-BFS-MaOA 8.54

WFG2

5 7.47e-01 ± 2.3e-01

8.31e-01 ± 7.6e-02

8.64e-01 ± 2.9e-01

NSGA-III 13.56

8 1.93e+00 ± 6.5e-01

2.24e+00 ± 9.6e-02

−

2.26e+00 ± 6.4e-01

NSGA-III 14.73

10 2.70e+00 ± 7.3e-01

3.00e+00 ± 1.5e-01

−

2.73e+00 ± 8.3e-01

NSGA-III 1.09

WFG3

5 1.67e+00 ± 2.4e-01

2.67e+00 ± 3.8e-02

−

1.79e+00 ± 1.2e-01

−

NSGA-III 6.60

8 7.14e+00 ± 8.5e-01

−

8.31e+00 ± 1.5e-01

−

2.60e+00 ± 2.6e-01

PBI-BFS-MaOA 63.53

10 1.42e+01 ± 9.8e-01

−

1.52e+01 ± 1.8e-01

−

7.23e+00 ± 5.4e-01

PBI-BFS-MaOA 49.15

WFG4

5 1.72e+00 ± 8.7e-03

−

1.50e+00 ± 3.9e-02

1.74e+00 ± 6.8e-03

−

CMOEA-CD 13.65

8 5.32e+00 ± 5.0e-02

−

4.80e+00 ± 8.9e-02

5.30e+00 ± 1.3e-02

−

CMOEA-CD 9.45

10 7.73e+00 ± 1.2e-01

−

7.46e+00 ± 1.1e-01

7.54e+00 ± 3.2e-02

−

CMOEA-CD 1.14

front competition is prominent.

V. C

ONCLUSION AND FUTURE WORK

This paper addressed many-objective optimization from a

deliberately local standpoint. Instead of replacing the global

Pareto scaffold, PBI-BFS-MaOA strengthens the decision

made inside the ﬁnal overﬂowing front. The method pre-

serves Pareto ranking at the population level and introduces

cumulative ideal–nadir normalization, PBI-based association,

active-niche restriction, and occupancy-aware insertion only

where standard survivor selection becomes least discrimina-

tive. Its contribution is therefore a boundary-front survival

mechanism, not a new decomposition framework, a new

generalized-dominance relation, or a multiarchive architec-

ture.

The experimental evidence supports this design as a com-

petitive alternative. Across the 24 benchmark instances, PBI-

BFS-MaOA obtains the best overall Friedman rank and

performs especially well on difﬁcult DTLZ cases and on

WFG3 as the number of objectives increases from 5 to 8

and 10. The strongest results occur in the more crowded

conﬁgurations, where boundary-front competition is expected

to be most severe. The WFG results are more mixed: NSGA-

III remains strongest on WFG2, CMOEA-CD retains an

advantage on WFG4, and the Friedman test on the WFG

subset does not indicate a statistically signiﬁcant separation

among the three methods. Runtime places PBI-BFS-MaOA

between the two baselines, so its practical value lies in a

better quality–cost compromise than CMOEA-CD, although

it does not match the lower computational cost of NSGA-III.

These ﬁndings suggest a practical recommendation for

high-dimensional optimization tasks in engineering design,

logistics, scheduling, resource allocation, and related decision

systems: when a reference-guided algorithm repeatedly faces

large boundary fronts, a localized PBI-based truncation rule

can improve survivor quality without requiring a full redesign

of the evolutionary framework. The method should not be

read as a universal replacement for established reference-

guided or archive-based approaches. It is better understood

as a robust survival alternative for crowded or degenerate

many-objective fronts.

Future work should follow three directions. First, a ded-

icated ablation study should separate the individual ef-

fects of dimensional scaling, active-niche ﬁltering, and

occupancy-aware insertion. Second, the feasibility branch

deﬁned through constraint violation should be evaluated on

constrained DTLZ/WFG variants and engineering design

benchmarks, since the present numerical study is limited

to unconstrained problems. Third, sensitivity tests around

the active-niche coverage threshold should be carried out

to determine whether the conservative 0.8K gate remains

appropriate across broader front geometries and population

sizes.

CKNOWLEDGMENT

The authors would like to thank METISBR: A Brazil-

ian research group dedicated to Multi-Objective and

Many-Objective Optimization (MaOPs) (https://github.com/

METISBR), for the valuable discussions and the entire team’s

support during the development of this paper.

EFERENCES

[1] B. Li, J. Li, K. Tang, and X. Yao, “Many-objective evolutionary

algorithms: A survey,” ACM Computing Surveys, vol. 48, no. 1, pp.

1–35, 2015.

[2] T. Santos and R. H. C. Takahashi, “On the performance degradation of

dominance-based evolutionary algorithms in many-objective optimiza-

tion,” IEEE Transactions on Evolutionary Computation, vol. 22, no. 1,

pp. 19–31, 2018.

[3] M. Pal, S. Saha, and S. Bandyopadhyay, “Decor: Differential evo-

lution using clustering based objective reduction for many-objective

optimization,” Information Sciences, 2018.

[4] Q. Zhang and H. Li, “Moea/d: A multiobjective evolutionary algo-

rithm based on decomposition,” IEEE Transactions on Evolutionary

Computation, vol. 11, no. 6, pp. 712–731, 2007.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

https://doi.org/10.33333/lajc.vol13n2.04

T. Santos, and S. Xavier,

“PBI-BFS-MaOA: A Many-Objective Evolutionary Algorithm with PBI-Based Boundary-Front Selection”,

Latin-American Journal of Computing (LAJC), vol. 13, no. 2, 2026.

[5] K. Li, K. Deb, Q. Zhang, and S. Kwong, “An evolutionary many-

objective optimization algorithm based on dominance and decomposi-

tion,” IEEE Transactions on Evolutionary Computation, vol. 19, no. 5,

pp. 694–716, 2015.

[6] K. Deb and H. Jain, “An evolutionary many-objective optimization

algorithm using reference-point-based nondominated sorting approach,

part i: Solving problems with box constraints,” IEEE Transactions on

Evolutionary Computation, vol. 18, no. 4, pp. 577–601, 2014.

[7] Y. Yuan, H. Xu, B. Wang, and X. Yao, “A new dominance relation-

based evolutionary algorithm for many-objective optimization,” IEEE

Transactions on Evolutionary Computation, vol. 20, no. 1, pp. 16–37,

2016.

[8] R. Cheng, Y. Jin, M. Olhofer, and B. Sendhoff, “A reference vector

guided evolutionary algorithm for many-objective optimization,” IEEE

Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 773–

791, 2016.

[9] H. Ishibuchi, Y. Setoguchi, H. Masuda, and Y. Nojima, “Performance

of decomposition-based many-objective algorithms strongly depends

on pareto front shapes,” IEEE Transactions on Evolutionary Compu-

tation, vol. 21, no. 2, pp. 169–190, 2017.

[10] W. Qiu, J. Zhu, G. Wu, M. Fan, and P. N. Suganthan, “Evolutionary

many-objective algorithm based on fractional dominance relation and

improved objective space decomposition strategy,” Swarm and Evolu-

tionary Computation, 2021.

[11] S. Liu, Q. Lin, K.-C. Wong, C. A. C. Coello, J. Li, Z. Ming, and

J. Zhang, “A self-guided reference vector strategy for many-objective

optimization,” IEEE Transactions on Cybernetics, vol. 52, no. 2, pp.

1164–1178, 2022.

[12] W. Li, Y. Chen, Y. Dong, and Y. Huang, “A solution potential-based

adaptation reference vector evolutionary algorithm for many-objective

optimization,” Swarm and Evolutionary Computation, vol. 85, p.

101451, 2024.

[13] S. Wang, H. Wang, Z. Wei, F. Wang, Q. Zhu, J. Zhao, and Z. Cui, “A

pareto dominance relation based on reference vectors for evolutionary

many-objective optimization,” Applied Soft Computing, vol. 152, p.

111505, 2024.

[14] T. Santos and S. Xavier, “Pymoolab: An open-source visual analyt-

ics framework for multi-objective optimization using llm-based code

generation and mcdm,” 2026, preprint available at https://arxiv.org/abs/

2603.01345.

[15] C. Zhu, L. Xu, and E. D. Goodman, “Generalization of pareto-

optimality for many-objective evolutionary optimization,” IEEE Trans-

actions on Evolutionary Computation, vol. 20, no. 2, pp. 299–315,

2016.

[16] Y. Tian, R. Cheng, X. Zhang, Y. Su, and Y. Jin, “A strengthened dom-

inance relation considering convergence and diversity for evolution-

ary many-objective optimization,” IEEE Transactions on Evolutionary

Computation, vol. 23, no. 2, pp. 331–345, 2019.

[17] S. Zhu, L. Xu, E. Goodman, K. Deb, and Z. Lu, “A general framework

for enhancing relaxed pareto dominance methods in evolutionary

many-objective optimization,” Memetic Computing, vol. 14, pp. 289–

308, 2022.

[18] S. Zhu, L. Zeng, and M. Cui, “Symmetrical generalized pareto domi-

nance and adjusted reference vector cooperative evolutionary algorithm

for many-objective optimization,” Symmetry, vol. 16, no. 11, pp. 1–22,

2024.

[19] J. Cheng, G. G. Yen, and G. Zhang, “A many-objective evolutionary

algorithm with enhanced mating and environmental selections,” IEEE

Transactions on Evolutionary Computation, vol. 19, no. 4, pp. 592–

605, 2015.

[20] D. Sharma and P. K. Shukla, “Line-prioritized environmental selec-

tion and normalization scheme for many-objective optimization using

reference-lines-based framework,” Swarm and Evolutionary Computa-

tion, 2019.

[21] P. B. Myszkowski and M. Laszczyk, “Diversity based selection for

many-objective evolutionary optimisation problems with constraints,”

Information Sciences, 2021.

[22] S. Liu, J. Zheng, Q. Lin, and K. C. Tan, “Evolutionary multi and many-

objective optimization via clustering for environmental selection,”

Information Sciences, vol. 578, pp. 930–949, 2021.

[23] Z. Liu, F. Han, Q. Ling, H. Han, and J. Jiang, “Constraint-pareto domi-

nance and diversity enhancement strategy-based evolutionary algorithm

for solving constrained multiobjective optimization problems,” IEEE

Transactions on Evolutionary Computation, vol. 29, no. 6, pp. 2771–

2784, 2025.

[24] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable test prob-

lems for evolutionary multiobjective optimization,” in Evolutionary

Multiobjective Optimization: Theoretical Advances and Applications.

London, U.K.: Springer, 2005, pp. 105–145.

[25] S. Huband, L. Barone, L. While, and P. Hingston, “A scalable multi-

objective test problem toolkit,” Lecture Notes in Computer Science,

vol. 4193, pp. 280–295, 2006.

[26] O. Sch

utze, X. Esquivel, A. Lara, and C. A. C. Coello, “Using the

averaged hausdorff distance as a performance measure in evolution-

ary multiobjective optimization,” IEEE Transactions on Evolutionary

Computation, vol. 16, no. 4, pp. 504–522, 2012.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2026

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XIII, Issue 2, July 2026

AUTHORS

Thiago Santos is an Associate Professor at the Federal University

of Ouro Preto (UFOP), Ouro Preto, Brazil. He holds a Ph.D. in

Mathematics and coordinates both the Mathematics Education

Research Group (GEEMA) and the Applied Mathematics Group

of the Department of Mathematics. His research spans multi-

objective optimization, evolutionary computation, and mathematics

education, with special attention to the interaction between rigorous

mathematical modeling and computational methods. In optimization

and computational intelligence, his work focuses on the design

and analysis of metaheuristic algorithms able to handle several

conﬂicting objectives simultaneously, with applications in engineering

and applied sciences. He also develops research in mathematics

education, addressing pedagogical innovation, curriculum

organization, and the conceptual barriers faced by students in

advanced mathematical reasoning. Through this integrated agenda,

he contributes to the theoretical foundations of optimization

methods while supporting more eective approaches to university-

level mathematics teaching. He is a founding member of the METISBR

research group on multi-objective and many-objective optimization.

Thiago Santos

T. Santos, and S. Xavier,

“PBI-BFS-MaOA: A Many-Objective Evolutionary Algorithm with PBI-Based Boundary-Front Selection”,

Latin-American Journal of Computing (LAJC), vol. 13, no. 2, 2026.

Sebastiao Xavier is an Associate Professor at the Federal University

of Ouro Preto (UFOP), Brazil. He received his B.S., M.Sc., and Ph.D.

in Mathematics from the Federal University of Minas Gerais (UFMG),

developing specialized expertise in dynamical systems and real

foliations. His academic career includes extensive teaching experience

from basic education to graduate-level mathematics, together

with sustained participation in the institutional development of

mathematics programs at UFOP. Through the Mathematics Education

Research Group (GEEMA), he has contributed to the training of future

educators and to discussions on mathematical formation. His current

scientiﬁc work is centered on optimization, especially multiobjective

optimization and evolutionary strategies. In this area, he is interested

in connecting rigorous theoretical foundations with computational

procedures that can support practical decision-making. His research

proﬁle combines pure mathematics, applied optimization, educational

engagement, and academic service. He is a member of the METISBR

research group on multi-objective and many-objective optimization.

Sebastião Xavier