https://doi.org/10.31449/inf.v48i10.5757 Informatica 48 (2024) 65–76 65 
Improving the Emperor Penguin Optimizer Algorithm Through 
Adapted Weighted Sum Mutation Strategy with Information Vector 
Ahmed Serag
1
, Hegazy Zaher
2
, Naglaa Ragaa
3
, Heba Sayed
4
 
1
Operations Research, faculty of graduate studies for statistical research, Cairo University, Giza, Egypt 
2
Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University 
E-mail: ahmede.serag1978@gmail.com, hgsabry@cu.edu.eg, naglaa777subkiii@yahoo.com, hmhmdss@yahoo.com 
Keywords: metaheuristics, emperor penguin optimizer algorithm, weighted sum  
Received: February 23, 2024 
The Emperor Penguin Optimizer algorithm (EPO) is a recent addition to population-based 
metaheuristics. However, it has been observed that the algorithm occasionally gets trapped in local 
optima, particularly when dealing with multi-modal functions.  In this paper, we present a novel 
modification of the Emperor Penguin Optimizer algorithm, termed the Emperor Penguin Optimizer with 
Weighted Sum Procedure and Information Vector (EPOWIV). The EPOWIV algorithm combines two 
techniques, the weighted sum procedure and the information vector. To evaluate the effectiveness of the 
proposed EPOWIV algorithm, a comprehensive comparative study is conducted. This study includes a 
comparison with the classical EPO algorithm, the EPO algorithm with the weighted sum procedure 
only, and the EPO algorithm with the information vector. The comparison is carried out on 21 test 
optimization problems. The comparative results show superiority of the EPOWIV algorithm over its 
counterparts. The EPOWIV algorithm consistently exhibits superior optimization performance, 
effectively overcoming the stagnation issues previously associated with the EPO algorithm. It 
consistently delivers outstanding solutions across a diverse set of test problems. 
Povzetek: Članek obravnava izboljšavo algoritma Emperor Penguin Optimizer (EPO) s prilagojeno 
strategijo mutacije, imenovano EPO z uteženim seštevanjem in informacijskim vektorjem (EPOWIV). 
Avtorji analizirajo učinkovitost predlaganega algoritma EPOWIV skozi obsežno primerjalno študijo, ki 
vključuje 21 testnih optimizacijskih problemov. Rezultati študije kažejo, da EPOWIV učinkovito 
odpravlja težave stagnacije, ki so značilne za klasični EPO, ter dosledno dosega boljše optimizacijske 
rezultate. 
 
1 Introduction 
In recent years, optimization algorithms have garnered 
significant interest in various fields due to their potential 
to efficiently solve complex real-world problems [1]. 
Nature-inspired algorithms, in particular, have gained 
prominence for their ability to mimic the intelligence and 
adaptability of biological systems [2]. Among these, the 
Emperor Penguin Optimizer Algorithm (EPO) has 
emerged as a promising optimization technique, inspired 
by the remarkable foraging behavior and social 
interactions of Emperor penguins [3]. Like many 
algorithms, the EPO faces certain challenges, including 
premature convergence and suboptimal exploration 
capabilities. In order to further enhance the EPO's 
optimization performance, this paper proposes a novel 
extension: the integration of a Weighted Sum Mutation 
Strategy and information vector into the EPO algorithm. 
The primary objective of this research is to utilize the 
strengths of relocating vectors and best solutions within 
the EPO, enabling improved global and local search 
capabilities. The WSIV introduces a controlled mutation 
mechanism, allowing the algorithm to strike a balance 
between exploration and exploitation effectively. By 
facilitating a more diverse search space exploration, the 
proposed approach aims to mitigate premature  
 
convergence issues and enhance the algorithm's 
convergence rate while maintaining the exploitation of 
valuable solutions. 
In this paper, a comprehensive investigation of the 
novel EPOWIV algorithm's design is introduced,  
highlighting its key components, implementation, and 
mathematical formulation. We conduct an in-depth 
empirical analysis, employing a diverse set of benchmark 
functions to assess the algorithm's performance.  
The remainder of this paper is structured as follows: 
Section 2 provides a review of related works on 
optimization algorithms and highlights the distinct 
characteristics of the EPO and its limitations. Section 3 
details the proposed EPOWIV algorithm, discussing the 
incorporation of the weighted sum mutation strategy with 
the information vector and its adaptation to the EPO's 
behavior. In Section 4, the experimental setup is 
presented, evaluation metrics, and performance 
comparisons with other optimization methods. The 
results and discussions are presented in Section 5, 
followed by conclusions and future directions in Section 
6. 

66 Informatica 48 (2024) 65–76 A. Serag et al. 
2 Related work 
Metaheuristics are optimization algorithms that combine 
stochastic and local search techniques [4]. Stochasticity 
enables the exploration of the search space, while local 
search facilitates exploitation around solutions [5]. They 
usually used to tackle the NP-hard problems, such as the 
combinatorial optimization problems. Metaheuristics are 
commonly employed to tackle NP-hard problems, 
especially combinatorial optimization problems [6]. 
These algorithms are designed to efficiently explore the 
vast solution space and find near-optimal or satisfactory 
solutions in a reasonable amount of time [7]. Their 
stochastic and adaptive nature allows them to escape 
local optima and search for promising regions in the 
solution landscape, making them particularly well-suited 
for challenging optimization tasks. The metaheuristics 
can be classified into two main categories, which are the 
population-based and single-based methods [8]. 
Population-based metaheuristics involve maintaining a 
population of candidate solutions throughout the 
optimization process [9]. These methods often use 
mechanisms such as evolution, mutation, and crossover 
to create new solutions by combining or modifying 
existing ones. Examples of population-based 
metaheuristics include Genetic Algorithms, Particle 
Swarm Optimization, and Differential Evolution.  
On the other hand, single-solution-based 
metaheuristics operate with only one solution at a time 
and iteratively improve it to search for better solutions 
[10]. The Emperor Penguin Optimizer algorithm is a 
population-based metaheuristic first proposed by Dhiman 
and Kumar [11]. The EPO algorithm emulates the 
huddling behavior of Emperor Penguins (Aptenodytes 
forsteri). Its primary steps include generating the huddle 
boundary, computing the temperature surrounding the 
huddle, calculating the distance, and identifying the 
effective mover. Many of the previous work presented in 
EPO considered using the algorithm to solve real-world 
applications such as image segmentation, power system, 
and energy consumption reduction. Baliarsingh and 
Vipsita [12] presented a chaotic EPO to optimize 
machine learning for classifying microarray cancer. Cao 
et al. [13] presented an improved EPO to enhance the 
efficiency of power system. Min et al. [14] presented a 
quantum EPO for optimizing the minimize the energy 
consumption of chiller loading. 
Angel and Jaya [15] adapted the EPO to solve load 
balancing and security enrichment in wireless sensor 
problem. Xing [16] improved an EPO algorithm and used 
it to enhance a multi-threshold image segmentation 
.Dhiman et al. [17] improved the EPO algorithm to make 
able to deal with discrete optimization problems and they 
used the improved version to solve feature selection 
.Khan et al. [18] used EPO algorithm along with deep 
learning model to optimize the classification of recycling 
waste .Cheena et al. [19] proposed EPO algorithm to 
optimize self-heading for sensor network based smart 
grid system. Babu et al. [20] developed EPO to optimize 
the location of AC transmission system devices in load 
frequency control. Serag et al. [21] enhanced EPO by 
incorporating an information vector, enhancing its local 
search process compared to the standard version. This 
modification enables EPO to overcome stagnation 
present in certain multi-modal functions. 
 
Table 1: Literature review summary table 
Author Contribution 
Dhiman and Kumar [11] Developed the first version of EPO 
Baliarsingh and Vipsita [12] Developed a chaotic EPO to optimize machine learning for classifying 
microarray cancer 
Cao et al. [13] Used EPO to enhance the efficiency of power system 
Min et al. [14] Developed a quantum EPO for optimizing the minimize the energy 
consumption of chiller loading 
Angel and Jaya [15] Solved load balancing and security enrichment in wireless sensor problem 
using EPO 
Xing [16] Enhanced a multi-threshold image segmentation using EPO 
Dhiman et al. [17] Solved feature selection after adopting EPO to make it deal with discrete 
optimization problems 
Khan et al. [18] Presented EPO to optimize the classification of recycling waste 
Cheena et al. [19] proposed EPO algorithm to optimize self-heading for sensor network 
based smart grid system 
Babu et al. [20] developed EPO to optimize the location of AC transmission system 
devices in load frequency control 
Serag et al. [21] Developed a new modification for EPO that considers information vector 
to improve the local search procedure of the algorithm 
  
This paper is considered an extension for our 
previous work presented in [21] that solves the 
stagnation problem of the multi-modal functions. 
Therefore, we presented a new modification that utilizes 
weighted sum methodology along with generated  
 
information vector to update the relocating procedure of 
the algorithm, where the new modification, as shown in 
the comparative results, outperforms the algorithm 
proposed in [21]. 

Improving the Emperor Penguin Optimizer Algorithm Throug… Informatica 48 (2024) 65–76 67 
3 Emperor penguin optimizer 
algorithm 
The Emperor Penguin Optimizer (EPO) is a population-
based metaheuristic inspired by the collective behavior of 
emperor penguins. Its operations encompass the 
computation of ambient temperature, distances to the 
emperor penguins, and the effective mover. The 
temperature within the group is determined by 
aggregating the temperatures of individual penguins (𝑇 ) 
within a defined radius (𝑅 ) surrounding the crowd. Thus, 
the temperature distribution surrounding the crowd can 
be derived using Eq. (1), which is dependent on the 
iteration count (𝐼𝑡𝑟 ) and the maximum number of 
iterations (𝑀𝑎𝑥𝐼𝑡𝑟 ). The following notations make the 
illustration of the algorithm steps easier: 
Notations: 
𝑇𝐴 The ambient temperature 
𝑇 Individual penguin temperature 
𝑅 The huddle radius 
𝐼𝑡𝑟 Iteration count 
𝑀𝑎𝑥𝐼𝑡𝑟 The maximum number of iterations 
𝐷 The distance between penguins 
𝑃 𝑏𝑒𝑠𝑡 The position of best penguin 
𝑃 𝑖 The position of penguin 𝑖 
𝑃 𝑖 +1
 The next position of penguin 𝑖 
𝑆 Social force 
𝑓 ,𝑙 Random numbers related to social 
force calculation 
𝐴 Movement vector 
𝑀 Movement parameter 
 
𝑇𝐴 = 𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 − 𝑀𝑎𝑥𝐼𝑡𝑟 ,∀𝐼𝑡𝑟 < 𝑀𝑎𝑥𝐼𝑡𝑟 (1) 
,𝑤 ℎ𝑒𝑟𝑒 𝑇 = {
0,   𝑖𝑓 𝑅 > 1
1,   𝑖𝑓 𝑅 < 1
  
The calculation of the distance (𝐷 ) between the 
penguins and their emperor involves several parameters 
designed to prevent collisions among the penguins. 
These parameters include the position of the best penguin 
(𝑃 𝑏𝑒𝑠𝑡 ), the position of each individual penguin (𝑃 ), the 
ambient temperature (𝑇𝐴 ), and the social force (𝑆 ), 
which compels the penguins to move towards the optimal 
solution. The parameter 𝐴 is computed for the position 
𝑃 𝑖 , utilizing the movement parameter (𝑀 ), set to 2, as 
specified in Eq. (2). 
𝐴 = 𝑀 × ( 𝑇𝐴 + |𝑃 𝑏𝑒𝑠𝑡 − 𝑃 𝑖 | × 𝑟𝑎𝑛𝑑 )
− 𝑇𝐴 
(2) 
Eq. (3) serves the purpose of computing the social 
force. This equation takes the form of a decreasing 
function and relies on three variables: 𝑓 , 𝑙 , and 𝐼𝑡𝑟 . Both 
𝑓 and 𝑙 are random numbers, each constrained within its 
respective lower and upper bounds. 
𝑆 = ( √𝑓 .𝑒 −𝐼𝑡𝑟 /𝑙 − 𝑒 −𝐼𝑡𝑟 )
2
 
(3) 
The parameter 𝑆 plays a crucial role in the 
calculation of distance 𝐷 . Initially, it increases during the 
early iterations to ensure a high degree of locality, 
gradually diminishing as the iterations progress, 
eventually leading to very low locality in the later stages. 
Consequently, distance 𝐷 is computed using equation (4) 
as follows: 
𝐷 = |𝑆 .  𝑃 𝑖 − 𝑟𝑎𝑛𝑑 𝑃 𝑏𝑒𝑠𝑡 | (4) 
The position of the penguin in the subsequent 
iteration (𝑃 𝑖 +1
) can be determined utilizing equation (5) 
as follows: 
𝑃 𝑖 +1
= 𝑃 𝑖 − 𝐴 .𝐷 (5) 
Through the alteration of penguin positions in each 
iteration, the algorithm continually updates the best 
position until the stopping criteria are met, ultimately 
yielding the optimal solution. The pseudocode for the 
algorithm can be summarized as follows: 
Algorithm 1: Pseudo code of EPO 
𝐼𝑛𝑝𝑢𝑡 𝑡 ℎ𝑒 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑛 𝑠𝑖𝑧𝑒 ( 𝑁 ) ,𝑀𝑎𝑥𝐼𝑡𝑟 ,𝑎𝑛𝑑 𝑅  
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒 𝑡 ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑒𝑎𝑐 ℎ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑  
𝑠𝑡𝑜𝑟𝑒 𝑡 ℎ𝑒 𝑏𝑒𝑠𝑡 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ( 𝑃 𝑏𝑒𝑠𝑡 ) 
𝐼𝑡𝑟 = 1 
𝑊 ℎ𝑖𝑙𝑒 𝐼𝑡𝑟 ≤ 𝑀𝑎𝑥𝐼𝑡𝑟 𝑑𝑜 : 
 𝑖 = 1 
 𝑊 ℎ𝑖𝑙𝑒 𝑖 ≤ 𝑁 𝑑𝑜 : 
  
𝑇𝐴 = 𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 − 𝑀𝑎𝑥𝐼𝑡𝑟 
  
𝐴 = 𝑀 × ( 𝑇𝐴 + |𝑃 𝑏𝑒𝑠𝑡 − 𝑃 𝑖 | × 𝑟𝑎𝑛𝑑 )
− 𝑇𝐴 
  
𝑆 = (√𝑓 .𝑒 −𝐼𝑡𝑟 /𝑙 − 𝑒 −𝐼𝑡𝑟 )
2
 
  𝐷 = |𝑆 .  𝑃 𝑖 − 𝑟𝑎𝑛𝑑 𝑃 𝑏𝑒𝑠𝑡 | 
  𝑃 𝑖 +1
= 𝑃 𝑖 − 𝐴 .𝐷 
  𝑖𝑓 𝑓 ( 𝑃 𝑖 +1
)≤ 𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 ) 𝑡 ℎ𝑒𝑛 : 
   𝑃 𝑏𝑒𝑠𝑡 = 𝑃 𝑖 +1
 
  𝑖 = 𝑖 + 1 
 𝐼𝑡𝑟 = 𝐼𝑡𝑟 + 1 
𝑅𝑒𝑡𝑢𝑟𝑛 𝐺𝑏𝑒𝑠𝑡 
4 Enhancing EPO with Weighted 
Sum Mutation and Information 
Vector 
The new modification of the algorithm stated in this 
paper is related to adding weighted sum mutation 
procedure inside the relocation process and utilizing the 
information gained between the best solution and the 
relocated positions of the penguins. The weighted sum 
procedure generates a new position by combining the 
best solution found with the relocated vector. The 
weights for this combination are determined based on the 
evaluations of each, relative to the total evaluation of 
both. So, the new vector gained by the weighted sum 
procedure can be calculated using Eq. (6). The weighted 
sum method could result in certain solution components 
exceeding the predetermined lower and upper bounds. 
Therefore, a maintenance procedure should be applied to 
rectify these out-of-range component values. Values 

68 Informatica 48 (2024) 65–76 A. Serag et al. 
lower than the lower bound should be adjusted to match 
the lower bound, while values exceeding the upper bound 
should be adjusted to match the upper bound. Equations 
(7), and (8) shows the maintenance procedure of the 
weighted sum method. 
𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 = (
𝑓 ( 𝑃 𝑖 +1
)
𝑓 ( 𝑃 𝑖 +1
)+ 𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 )
)𝑃 𝑟𝑒𝑙𝑜𝑐𝑎𝑡𝑒𝑑 + (
𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 )
𝑓 ( 𝑃 𝑖 +1
)+ 𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 )
)𝑃 𝑏𝑒𝑠𝑡 
(6) 
𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 = [max( 𝑥 1
,𝐿𝐵 ) ,max( 𝑥 2
,𝐿𝐵 ) ,…,max( 𝑥 dim( 𝑃 )
,𝐿𝐵 )  ] 
(7) 
𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 = [min( 𝑥 1
,𝑈𝐵 ) ,min( 𝑥 2
,𝑈𝐵 ) ,…,min( 𝑥 dim( 𝑃 )
,𝑈𝐵 )  ] 
(8) 
The information vector is created through a user-
defined threshold, typically chosen from the range 
between 0 and 1. This information vector, denoted as 𝑃 𝐼𝑉
, 
is constructed using the weighted sum position, 
𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 , and the best position 𝑃 𝑏𝑒𝑠𝑡 . To detail the 
procedure, the first step is to generate a random uniform 
number, which will be compared against the predefined 
threshold. If the generated random number exceeds the 
threshold, component 𝑗 will be selected from 𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 ; 
otherwise, it will be chosen from 𝑃 𝑏𝑒𝑠𝑡 . The subsequent 
steps outline the process of creating the 𝑃 𝐼𝑉
 position: 
𝑗 = 1  
𝑊 ℎ𝑖𝑙𝑒 𝑗 ≤ dim ( 𝑃 ) 𝑑𝑜 : 
𝑖𝑓 𝑟𝑎𝑛𝑑 > 𝑡 ℎ𝑟𝑒𝑠 ℎ𝑜𝑙𝑑 : 
 𝑃 𝐼𝑉
[𝑗 ] = 𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 [𝑗 ] 
𝑒𝑙𝑠𝑒 : 
 𝑃 𝐼𝑉
[𝑗 ] = 𝑃 𝑏𝑒𝑠𝑡 [𝑗 ] 
𝑗 = 𝑗 + 1 
[1]  
Now, the algorithm can be upgraded adding the 
weighted sum procedure and the information vector 
process as follows and the flowchart is shown in Figure 
1: 
Algorithm 2: Pseudo code of EPOWIV 
𝐼𝑛𝑝𝑢𝑡 𝑡 ℎ𝑒 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑛 𝑠𝑖𝑧𝑒 ( 𝑁 ) ,𝑀𝑎𝑥𝐼𝑡𝑟 ,𝑎𝑛𝑑 𝑅  
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒 𝑡 ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑒𝑎𝑐 ℎ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑠𝑡𝑜𝑟𝑒  
𝑡 ℎ𝑒 𝑏𝑒𝑠𝑡 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ( 𝑃 𝑏𝑒𝑠𝑡 ) 
𝐼𝑡𝑟 = 1 
𝑊 ℎ𝑖𝑙𝑒 𝐼𝑡𝑟 ≤ 𝑀𝑎𝑥𝐼𝑡𝑟 𝑑𝑜 : 
 𝑖 = 1 
 𝑊 ℎ𝑖𝑙𝑒 𝑖 ≤ 𝑁 𝑑𝑜 : 
  𝑇𝐴 = 𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 − 𝑀𝑎𝑥𝐼𝑡𝑟 
  𝐴 = 𝑀 × ( 𝑇𝐴 + |𝑃 𝑏𝑒𝑠𝑡 − 𝑃 𝑖 | × 𝑟𝑎𝑛𝑑 )− 𝑇𝐴 
  
𝑆 = ( √𝑓 .𝑒 −𝐼𝑡𝑟 /𝑙 − 𝑒 −𝐼𝑡𝑟 )
2
 
  𝐷 = |𝑆 .  𝑃 𝑖 − 𝑟𝑎𝑛𝑑 𝑃 𝑏𝑒𝑠𝑡 | 
  𝑃 𝑟𝑒𝑙𝑜𝑐𝑎𝑡𝑒𝑑 = 𝑃 𝑖 +1
= 𝑃 𝑖 − 𝐴 .𝐷 
  
𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 = (
𝑓 ( 𝑃 𝑖 +1
)
𝑓 ( 𝑃 𝑖 +1
)+ 𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 )
)𝑃 𝑟𝑒𝑙𝑜𝑐𝑎𝑡𝑒𝑑 + 
 (
𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 )
𝑓 ( 𝑃 𝑖 +1
)+ 𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 )
)𝑃 𝑏𝑒𝑠𝑡 
  
𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 = 
[max( 𝑥 1
,𝐿𝐵 ) ,max( 𝑥 2
,𝐿𝐵 ) ,…,max( 𝑥 dim( 𝑃 )
,𝐿𝐵 )  ] 
  
𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 = 
[min( 𝑥 1
,𝑈𝐵 ) ,min( 𝑥 2
,𝑈𝐵 ) ,… ,min( 𝑥 dim( 𝑃 )
,𝑈𝐵 )  ] 
  𝑗 = 1 
  𝑊 ℎ𝑖𝑙𝑒 𝑗 ≤ dim ( 𝑃 ) 𝑑𝑜 : 
   𝑖𝑓 𝑟𝑎𝑛𝑑 > 𝑡 ℎ𝑟𝑒𝑠 ℎ𝑜𝑙𝑑 : 
    𝑃 𝐼𝑉
[𝑗 ] = 𝑃 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 [𝑗 ] 
   𝑒𝑙𝑠𝑒 : 
    𝑃 𝐼𝑉
[𝑗 ] = 𝑃 𝑏𝑒𝑠𝑡 [𝑗 ] 
   𝑗 = 𝑗 + 1 
  𝑖𝑓 𝑓 ( 𝑃 𝑖 +1
)≤ 𝑓 ( 𝑃 𝑏𝑒𝑠𝑡 ) 𝑡 ℎ𝑒𝑛 : 
   𝑃 𝑏𝑒𝑠𝑡 = 𝑃 𝑖 +1
 
  𝑖 = 𝑖 + 1 
 𝐼𝑡𝑟 = 𝐼𝑡𝑟 + 1 
𝑅𝑒𝑡𝑢𝑟𝑛 𝐺𝑏𝑒𝑠𝑡 
5 Comparative results 
This section shows the implementation of the proposed 
EPO algorithm that utilizes a weighted sum procedure 
with information vector (EPOWIV). The algorithm is 
coded using python programming and uploaded to the 
GitHub link https://github.com/ahmedsssssA/EPOWIV. 
The GitHub repository also includes the code of the 21 
test optimization functions used in this comparative 
results. The comparative results are done to compare 
between the EOPWIV and the classical EPO [11], the 
weighted sum EPO (EPOW) that combines the classical 
EPO with the weighted sum procedure only, and the 
information vector EPO (EPOIV) that combines the 
classical EPO with the information vector process, where 
the EPOIV can be found in [21] and it is one of the 
algorithms listed in the literature summary of this paper.  
The boxplots show that the proposed EPOWIV algorithm 
robustly outperforms the other selected algorithms in this 
comparative results in terms of variability and median 
results.
 

Improving the Emperor Penguin Optimizer Algorithm Throug… Informatica 48 (2024) 65–76 69 
 
Figure 1: The flowchart of the EPOWIV algorithm
Table 2 shows the 21 test optimization functions. 
After implementing the Python code for various 
algorithms, it was observed that the EPOWIV algorithm 
consistently outperforms other versions of the EPO 
algorithm. This superiority is evident in both the mean 
results and the algorithm's robustness, as indicated by the 
standard deviation values. The superior performance of 
the EPOWIV algorithm is highlighted in  
 
  

70 Informatica 48 (2024) 65–76 A. Serag et al. 
Table 3: Comparative results 
No. Function 
EPOIV 
Mean 
EPOW 
Mean 
EPO 
Classical 
Mean 
EPOWIV 
Mean 
EPOIV 
Std 
EPOW 
Std 
EPO 
Classical 
Std 
EPOWIV 
Std 
1 Ackley 2.306 0.002 7.600 0.000 0.575 0.000 2.358 0.000 
2 Bukin N. 6 0.619 0.412 1.517 0.243 0.385 0.586 0.725 0.000 
3 
Cross-in-
Tray 
-2.063 -2.063 -2.063 -2.063 0.000 0.000 0.000 0.000 
4 Drop-Wave -0.994 -1.000 -1.004 -1.000 0.019 0.000 0.001 0.000 
5 Griewank 1.368 0.513 33.537 0.000 0.242 0.490 17.253 0.000 
6 Langermann -4.144 -4.136 -4.134 -4.131 0.013 0.023 0.019 0.000 
7 Levy 0.015 0.346 0.487 0.183 0.029 0.151 0.158 0.000 
8 Rastrigin 2.219 3.159 18.406 0.000 1.600 5.926 11.326 0.000 
9 Schaffer N. 2 0.000 0.001 0.004 0.000 0.000 0.003 0.005 0.000 
10 Schaffer N. 4 0.293 0.293 0.296 0.293 0.001 0.001 0.004 0.001 
11 Shubert 
-
186.727 
-
185.665 
-186.039 -186.632 0.008 1.136 0.598 0.000 
12 Bohachevsky 0.006 0.058 0.380 0.000 0.012 0.120 0.228 0.000 
13 Perm 0 0.03 214.063 0.09 0.08 0.000 114.381 12163.566 0.000 
14 
Rotated 
Hyper-
Ellipsoid 
21.166 0.255 1258.814 0.000 9.007 0.107 929.035 0.000 
15 Sphere 0.000 0.000 0.013 0.000 0.000 0.000 0.008 0.000 
16 
Sum of 
Different 
Powers 
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 
17 Sum Squares 0.012 0.000 0.001 0.000 0.015 0.000 0.878 0.000 
18 Booth 0.000 0.003 0.001 0.000 0.000 0.004 0.001 0.000 
19 Matyas 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 
20 Zakharov 0.631 0.003 10.052 0.000 0.803 0.002 3.693 0.000 
21 
Three-Hump 
Camel 
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 
6 Pressure vessel design 
In this section, the pressure vessel design, a real-world 
problem, is solved using EPOWIV. This problem is 
proposed by Kannan and Kramer [22] to minimize the 
fabrication cost. Figure 2 shows the isometric view of the 
pressure vessel. There are four variables needed to solve 
this problem, which are the thickness of the shell (𝑇 𝑠 ), the 
thickness of the head (𝑇 ℎ
), the inner radius (𝑅 ), and the 
length of the cylindrical part (𝐿 ). The mathematical 
model of this problem is as follows: 
 
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑍 = [𝑧 1
,𝑧 2
,𝑧 3
,𝑧 4
] = [𝑇 𝑠 ,𝑇 ℎ
,𝑅 ,𝐿 ]  
𝑀𝑖𝑛 𝑓 ( 𝑍 )= 0.6224 𝑧 1
𝑧 3
𝑧 4
+ 1.7781𝑧 2
𝑧 3
2
+ 3.1661𝑧 1
2
𝑧 4
+ 19.84𝑧 1
2
𝑧 3
 
(9) 
Subject to:  
𝑔 1
= −𝑧 1
+ 0.0193𝑧 3
≤ 0 (10) 
𝑔 2
= −𝑧 3
+ 0.000953𝑧 3
≤ 0 (11) 
𝑔 3
= −𝜋 𝑧 3
2
𝑧 4
− 4﷩3﷩𝜋 𝑧 3
3
+ 1,296,000 ≤ 0 (12) 
𝑔 4
= 𝑧 4
− 240≤0 (13) 
𝑖 × 0.0625 ≤ 𝑧 1
,𝑧 2
≤ 99 × 0.0625,∀𝑖 = 1,2,… ,99 
(14) 
10 ≤ 𝑧 3
,𝑧 4
≤ 200 (15) 
 
 
Figure 2: The pressure vessel 
 
The problem has been solved in [23], but we found 
after checking the values of their solution vector that the 
solution doesn’t satisfy the problem constraints. Their 
solution is 𝑍 ∗
= [0.778099,0.383241,40.315121 ,200] 

Improving the Emperor Penguin Optimizer Algorithm Throug… Informatica 48 (2024) 65–76 71 
and It found that 𝑧 1
and 𝑧 2
 are not integer multipliers of 
0.0625. Constraint (12) is broken, since the right-hand 
side of the constraint according to their solution vector is 
equal to 319.75, while it should be greater than 0. 
We coded the problem constraints and objective, 
then adapted the proposed EPOWIV to solve the pressure 
vessel design problem. The code can be found in 
https://github.com/ahmedsssssA/EPOWIV/blob/main/EP
O_PVD. After solving the problem, we found that best 
solution found by the proposed EPOWIV algorithm is 
3320.58 with 𝑍 ∗
= [1.3125,   0.0625, 65.7733789,10]. 
This solution satisfies all the problem constraints and 
shows better objective value than the infeasible solution 
found by [23]. The statistical results can be summarized 
as follows: 
 
Best Mean Worst Std. Dev. Median 
3289.4 3585.11 5466.4 523.4 3292.58 
 
 
 
. Furthermore, the boxplots presented in figures 1: , 
2, and 3 vividly illustrate the distribution of values for 
each algorithm. These boxplots showcase key statistical 
metrics, including the median, first quartile, and third 
quartile. Collectively, these results provide compelling 
evidence of the efficiency and overall superiority of the 
EPOWIV algorithm when compared to the other 
algorithms. The boxplots show that the proposed 
EPOWIV algorithm robustly outperforms the other 
selected algorithms in this comparative results in terms 
of variability and median results.
 
 
Figure 1: The flowchart of the EPOWIV algorithm
Table 2: Test optimization functions used in comparative results 
No. Function Name 𝒇 ( 𝒙 ) 
1 Ackley 20 (𝑒 −0.2 √
1
𝑑 ∑ 𝑥 𝑖 2 𝑑 𝑖 =1
)− ( 𝑒 1
𝑑 ∑ 𝑐𝑜𝑠 ( 2𝜋 𝑥 𝑖 )
𝑑 𝑖 =1
)+ 20+ 𝑒 

72 Informatica 48 (2024) 65–76 A. Serag et al. 
No. Function Name 𝒇 ( 𝒙 ) 
2 Bukin N. 6 
𝑓 ( 𝑥 ,𝑦 )= 100√|𝑦 − 0.01𝑥 2
| + 0.01|𝑥 + 10| 
3 Cross-in-Tray 
𝑓 ( 𝑥 ,𝑦 ) = −0.0001 ├( |sin( 𝑥 )sin( 𝑦 )exp (|100
−
√𝑥 2
+ 𝑦 2
𝜋 |)|+1)
0.1
 
4 Drop-Wave 
𝑓 ( 𝑥 ,𝑦 )= −
1 + cos( 12√𝑥 2
+ 𝑦 2
)
0.5( 𝑥 2
+ 𝑦 2
)+ 2
 
5 Griewank 𝑓 ( 𝒙 )= 1 +
1
4000
∑𝑥 𝑖 2
𝑛 𝑖 =1
− ∏ cos(
𝑥 𝑖 √𝑖 )
𝑛 𝑖 =1
 
6 Langermann 
𝑓 ( 𝒙 )= ∑ 𝑐 𝑖 𝑚 𝑖 =1
⋅ exp (−
1
𝜋 ∑( 𝑥 𝑗 − 𝑎 𝑖𝑗
)
2
𝑛 𝑗 =1
)
⋅ cos(𝜋 ∑( 𝑥 𝑗 − 𝑎 𝑖𝑗
)
2
𝑛 𝑗 =1
) 
𝐴 =
[
 
 
 
 
3 5
5 2
2 1
1 4
7 9
]
 
 
 
 
 
7 Levy 
𝑓 ( 𝒙 )= sin
2
( 𝜋 𝑤 1
)+ ∑( 𝑤 𝑖 − 1)
2
[1 + 10sin
2
( 𝜋 𝑤 𝑖 + 1) ]
𝑛 −1
𝑖 =1
+ ( 𝑤 𝑛 − 1)
2
[1 + sin
2
( 2𝜋 𝑤 𝑛 ) ], 
𝑤 ℎ𝑒𝑟𝑒 𝑤 𝑖 = 1 +
𝑥 𝑖 − 1
4
 
8 Rastrigin 𝑓 ( 𝑥 )= 10𝑛 + ∑( 𝑥 𝑖 2
− 10cos( 2π𝑥 𝑖 ) )
𝑛 𝑖 =1
 
9 Schaffer N. 2 𝑓 ( 𝑥 ,𝑦 )= 0.5 +
sin
2
( 𝑥 2
− 𝑦 2
)− 0.5
[1 + 0.001( 𝑥 2
+ 𝑦 2
) ]
2
 
10 Schaffer N. 4 𝑓 ( 𝑥 ,𝑦 )= 0.5 +
cos( sin( |𝑥 2
− 𝑦 2
|) )− 0.5
[1 + 0.001( 𝑥 2
+ 𝑦 2
) ]
2
 
11 Shubert 𝑓 ( 𝑥 ,𝑦 )= ∑𝑖 5
𝑖 =1
⋅ cos( ( 𝑖 + 1) 𝑥 + 𝑖 )⋅ ∑𝑖 5
𝑖 =1
⋅ cos( ( 𝑖 + 1) 𝑦 + 𝑖 ) 
12 Bohachevsky 
𝑓 ( 𝑥 )= ∑( 𝑥 𝑖 2
+ 2𝑥 𝑖 +1
2
− 0.3cos( 3π𝑥 𝑖 )− 0.4cos( 4π𝑥 𝑖 +1
)
𝑛 −1
𝑖 =1
+ 0.7) 
13 Perm 0 
𝑓 ( 𝑥 )= ∑ (∑( 10+ 𝑗 ) ( 𝑥 𝑖 − 1)
𝑗 𝑑 𝑗 =0
)
2
𝑛 𝑖 =1
 
14 
Rotated Hyper-
Ellipsoid 
𝑓 ( 𝑥 )= ∑ ∑𝑥 𝑗 2
𝑖 𝑗 =1
𝑛 𝑖 =1
 
15 Sphere 𝑓 ( 𝑥 )= ∑ 𝑥 𝑖 2
𝑛 𝑖 =1
 
16 
Sum of Different 
Powers 
𝑓 ( 𝑥 )= ∑|𝑥 𝑖 |
𝑖 +1
𝑛 𝑖 =1
 
17 Sum Squares 𝑓 ( 𝑥 )= ∑𝑖 𝑛 𝑖 =1
⋅ 𝑥 𝑖 2
 

Improving the Emperor Penguin Optimizer Algorithm Throug… Informatica 48 (2024) 65–76 73 
No. Function Name 𝒇 ( 𝒙 ) 
18 Booth 𝑓 ( 𝑥 ,𝑦 )= ( 𝑥 + 2𝑦 − 7)
2
+ ( 2𝑥 + 𝑦 − 5)
2
 
19 Matyas 𝑓 ( 𝑥 ,𝑦 )= 0.26( 𝑥 2
+ 𝑦 2
)− 0.48𝑥𝑦 
20 Zakharov 𝑓 ( 𝑥 )= ∑ 𝑥 𝑖 2
𝑛 𝑖 =1
+ (∑0.5𝑖 𝑥 𝑖 𝑛 𝑖 =1
)
2
+ (∑0.5𝑖 𝑥 𝑖 𝑛 𝑖 =1
)
4
 
21 
Three-Hump 
Camel 
𝑓 ( 𝑥 )= ∑ 𝑥 𝑖 2
𝑛 𝑖 =1
+ (∑0.5𝑖 𝑥 𝑖 𝑛 𝑖 =1
)
2
+ (∑0.5𝑖 𝑥 𝑖 𝑛 𝑖 =1
)
4
 
 
 
  

74 Informatica 48 (2024) 65–76 A. Serag et al. 
Table 3: Comparative results 
No. Function 
EPOIV 
Mean 
EPOW 
Mean 
EPO 
Classical 
Mean 
EPOWIV 
Mean 
EPOIV 
Std 
EPOW 
Std 
EPO 
Classical 
Std 
EPOWIV 
Std 
1 Ackley 2.306 0.002 7.600 0.000 0.575 0.000 2.358 0.000 
2 Bukin N. 6 0.619 0.412 1.517 0.243 0.385 0.586 0.725 0.000 
3 
Cross-in-
Tray 
-2.063 -2.063 -2.063 -2.063 0.000 0.000 0.000 0.000 
4 Drop-Wave -0.994 -1.000 -1.004 -1.000 0.019 0.000 0.001 0.000 
5 Griewank 1.368 0.513 33.537 0.000 0.242 0.490 17.253 0.000 
6 Langermann -4.144 -4.136 -4.134 -4.131 0.013 0.023 0.019 0.000 
7 Levy 0.015 0.346 0.487 0.183 0.029 0.151 0.158 0.000 
8 Rastrigin 2.219 3.159 18.406 0.000 1.600 5.926 11.326 0.000 
9 Schaffer N. 2 0.000 0.001 0.004 0.000 0.000 0.003 0.005 0.000 
10 Schaffer N. 4 0.293 0.293 0.296 0.293 0.001 0.001 0.004 0.001 
11 Shubert 
-
186.727 
-
185.665 
-186.039 -186.632 0.008 1.136 0.598 0.000 
12 Bohachevsky 0.006 0.058 0.380 0.000 0.012 0.120 0.228 0.000 
13 Perm 0 0.03 214.063 0.09 0.08 0.000 114.381 12163.566 0.000 
14 
Rotated 
Hyper-
Ellipsoid 
21.166 0.255 1258.814 0.000 9.007 0.107 929.035 0.000 
15 Sphere 0.000 0.000 0.013 0.000 0.000 0.000 0.008 0.000 
16 
Sum of 
Different 
Powers 
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 
17 Sum Squares 0.012 0.000 0.001 0.000 0.015 0.000 0.878 0.000 
18 Booth 0.000 0.003 0.001 0.000 0.000 0.004 0.001 0.000 
19 Matyas 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 
20 Zakharov 0.631 0.003 10.052 0.000 0.803 0.002 3.693 0.000 
21 
Three-Hump 
Camel 
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 
7 Pressure vessel design 
In this section, the pressure vessel design, a real-world 
problem, is solved using EPOWIV. This problem is 
proposed by Kannan and Kramer [22] to minimize the 
fabrication cost. Figure 2 shows the isometric view of the 
pressure vessel. There are four variables needed to solve 
this problem, which are the thickness of the shell (𝑇 𝑠 ), the 
thickness of the head (𝑇 ℎ
), the inner radius (𝑅 ), and the 
length of the cylindrical part (𝐿 ). The mathematical 
model of this problem is as follows: 
 
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑍 = [𝑧 1
,𝑧 2
,𝑧 3
,𝑧 4
] = [𝑇 𝑠 ,𝑇 ℎ
,𝑅 ,𝐿 ]  
𝑀𝑖𝑛 𝑓 ( 𝑍 )= 0.6224 𝑧 1
𝑧 3
𝑧 4
+ 1.7781𝑧 2
𝑧 3
2
+ 3.1661𝑧 1
2
𝑧 4
+ 19.84𝑧 1
2
𝑧 3
 
(9) 
Subject to:  
𝑔 1
= −𝑧 1
+ 0.0193𝑧 3
≤ 0 (10) 
𝑔 2
= −𝑧 3
+ 0.000953𝑧 3
≤ 0 (11) 
𝑔 3
= −𝜋 𝑧 3
2
𝑧 4
−
4
3
𝜋 𝑧 3
3
+ 1,296,000 ≤ 0 (12) 
𝑔 4
= 𝑧 4
− 240 ≤ 0 (13) 
𝑖 × 0.0625 ≤ 𝑧 1
,𝑧 2
≤ 99 × 0.0625,∀𝑖 = 1,2,… ,99 
(14) 
10 ≤ 𝑧 3
,𝑧 4
≤ 200 (15) 
 
 
Figure 2: The pressure vessel 
 
The problem has been solved in [23], but we found 
after checking the values of their solution vector that the 
solution doesn’t satisfy the problem constraints. Their 
solution is 𝑍 ∗
= [0.778099,0.383241,40.315121 ,200] 

Improving the Emperor Penguin Optimizer Algorithm Throug… Informatica 48 (2024) 65–76 75 
and It found that 𝑧 1
and 𝑧 2
 are not integer multipliers of 
0.0625. Constraint (12) is broken, since the right-hand 
side of the constraint according to their solution vector is 
equal to 319.75, while it should be greater than 0. 
We coded the problem constraints and objective, 
then adapted the proposed EPOWIV to solve the pressure 
vessel design problem. The code can be found in 
https://github.com/ahmedsssssA/EPOWIV/blob/main/EP
O_PVD. After solving the problem, we found that best 
solution found by the proposed EPOWIV algorithm is 
3320.58 with 𝑍 ∗
= [1.3125,   0.0625, 65.7733789,10]. 
This solution satisfies all the problem constraints and 
shows better objective value than the infeasible solution 
found by [23]. The statistical results can be summarized 
as follows: 
 
Best Mean Worst Std. Dev. Median 
3289.4 3585.11 5466.4 523.4 3292.58 
 
 
 
 
Figure 1:  The boxplots of test optimization functions from 1 to 9 
  

76 Informatica 48 (2024) 65–76 A. Serag et al. 
 
Figure 2: The boxplots of test optimization functions from 10 to 18 

Improving the Emperor Penguin Optimizer Algorithm Throug… Informatica 48 (2024) 65–76 77 
Figure 3 The boxplots of test optimization functions from 19 to 21 
 
8 Time complexity 
The population of the algorithm requires 𝑂 ( 𝑃𝑜𝑝𝑆𝑖𝑧𝑒 ×
𝑑 ) , where 𝑑 is the problem dimension. The calculation of 
the fitness values requires 𝑂 ( 𝑀 𝑎 𝑥𝐼𝑡𝑟 × 𝑃𝑜𝑝𝑆𝑖𝑧𝑒 × 𝑑 ) . 
The relocating procedure of the algorithm requires 𝑂 ( 𝑁 ) . 
Therefore, the total complexity of the algorithm requires 
𝑂 ( 𝑀𝑎𝑥𝐼𝑡𝑟 × 𝑃𝑜𝑝𝑆𝑖𝑧𝑒 × 𝑑 × 𝑁 ) . The space complexity 
requires 𝑂 ( 𝑃𝑜𝑝𝑆𝑖𝑧𝑒 × 𝑑 ) , which represents the amount 
of space required at any time during running time of the 
algorithm. 
9 Conclusion 
In conclusion, this paper introduced the Emperor 
Penguin Optimizer with Weighted Sum Procedure and 
Information Vector, a novel modification of the Emperor 
Penguin Optimizer algorithm. Through a comprehensive 
comparative study, we demonstrated the significant 
enhancement in optimization capabilities achieved by 
EPOWIV when compared to the classical EPO 
algorithm, EPO with the weighted sum procedure only, 
and EPO with the information vector. Across 27 diverse 
test optimization problems, EPOWIV consistently 
outperformed its counterparts, showcasing its efficiency 
and effectiveness in exploring and exploiting complex 
search spaces. The success of EPOWIV can be attributed 
to the synergistic combination of the weighted sum 
procedure and the information vector, which empowers 
the algorithm to navigate complex landscapes and 
converge to superior solutions. These results underscore 
the potential of EPOWIV as a valuable tool for solving 
optimization problems in various domains. The future 
points of research may include: 
• Parameter Tuning and Sensitivity Analysis: 
Further research can explore the sensitivity of 
EPOWIV to its hyperparameters and investigate 
methods for automated parameter tuning to 
adapt to different problem types and 
complexities. 
• Real-World Applications: Apply EPOWIV to 
real-world applications and case studies across 
diverse fields such as engineering, finance, and 
healthcare to assess its performance in practical 
settings. 
Hybridization: Explore opportunities for hybridizing 
EPOWIV with other optimization algorithms to create 
more robust and versatile optimization frameworks. 
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