https://doi.org/10.31449/inf.v47i10.5273 Informatica 47 (2023) 71–78 71 
A Modified Emperor Penguin Algorithm for Solving Stagnation in 
Multi-Model Functions 
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: optimization, metaheuristics, emperor penguin algorithm 
Received: October 13, 2023 
Abstract: Metaheuristic algorithms have gained attention in recent years for their ability to solve complex 
problems that cannot be solved using classical mathematical techniques. This paper proposes an 
improvement to the Emperor Penguin Optimizer algorithm, a population-based metaheuristic. The 
original algorithm often gets stuck in local optima for multi-modal functions. To address this issue, this 
paper presents a modification in the relocating procedures that allows the algorithm to utilize information 
gained from the previous positions of each penguin. To demonstrate the effectiveness of the modified 
algorithm, 20 test optimization functions from well-known benchmarks were selected. The implemented 
comparative analysis assesses the proposed algorithm against both the traditional Emperor Penguin 
Optimizer algorithm and one of the most recent algorithm modifications in current research. The findings 
indicate that the proposed algorithm demonstrates significant efficiency, particularly in addressing multi-
modal functions, as evidenced by superior mean results and robustness. 
Povzetek: Predstavljen je izboljšani algoritma "cesarskega pingvina", ki učinkovito rešuje kompleksne 
probleme zlasti v multimodalnih funkcijah.
1 Introduction 
Problem-solving can be approached in various ways, such 
as trial and error, experimental design, or mathematical 
techniques [1]. Operations research is a field that employs 
mathematical formulation to solve complex engineering 
and management problems and gain insight into potential 
solutions [2]. Mathematical approaches used in this field 
can be classified into classical techniques like simplex 
method and dynamic programming, and heuristic and 
metaheuristic methods like the Emperor Penguin 
Optimizer algorithm (EPO) [3]. 
EPO is a population-based metaheuristic algorithm 
proposed by Dhiman & Kumar [4], inspired by the 
behavior of emperor penguins in utilizing crowds to 
survive the Antarctic winter. While the original algorithm 
is efficient in solving unimodal problems, it stagnates with 
complex problems like multi-local minima problems. 
Previous work on EPO in the literature includes the binary 
version by Dhiman et al. [5], multi-objective optimization 
by Kaur et al. [6], photovoltaic system optimization by 
Sameh et al. [7], support vector machine optimization for 
face recognition by Yang and Gao [8], RGB image 
threshold optimization by Jia et al. [9], and energy-
efficient residential building design by Tang et al. [10]. 
Kaur et al. [11] adapted the emperor penguin optimizer 
algorithm to solve multi-objective optimization problems 
using the concept of dynamic archive. Xing [12] proposed 
an EPO algorithm for solving the multilevel threshold for 
color image segmentation. Lu et al. [13] improved the 
EPO algorithm by optimizing its output using the  
 
 
sequential quadratic programming. They used the 
improved algorithm to minimize the market clearing price 
probability function. 
 However, most of these studies focused on applying 
the EPO algorithm to solve real-world problems and did 
not address its stagnation issue in multi-modal problems. 
Table 1 summarizes the previously mentioned literature in 
this paper. 
This paper proposes a modification to the EPO 
algorithm to increase its efficiency. The second section 
discusses the original algorithm's main steps, followed by 
the proposed modification in the third section. The fourth 
section presents comparative results between the original  
and modified EPO, and the fifth section concludes the 
paper. 
2 The original emperor penguin 
optimizer algorithm 
The Emperor Penguin Optimizer (EPO) is a population-
based metaheuristic inspired by the crowd behavior of 
emperor penguins. The algorithm's steps include 
calculating the ambient temperature, distances toward the  
emperor penguins, and effective movers. To calculate the 
temperature around the crowd (𝑇𝐴 ), the temperature of 
each penguin (𝑇 ) is considered, depending on the radius 
(𝑅 ) that surrounds the crowd. The temperature profile 
around the crowd can be calculated using Equation (1), 
where the iteration number (𝐼𝑡𝑟 ) and the maximum 
number of iterations (𝑀𝑎𝑥𝐼𝑡𝑟 ) are taken into account. 
𝑇𝐴 =𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 −𝑀𝑎𝑥𝐼𝑡𝑟 ,∀𝐼𝑡𝑟 <𝑀𝑎𝑥𝐼𝑡𝑟 (1) 

72 Informatica 47 (2023) 71–78 A. Serag et al. 
   
Table 1: Literature summary
𝑇 = {
0,   𝑖𝑓 𝑅 >1
1,   𝑖𝑓 𝑅 <1
 
The distance between the penguins and their emperor (𝐷 ) 
is to be calculated using some sort of parameters related to 
avoiding collision between penguins (𝐴 ), the position of 
the best penguin (𝑃 𝑏𝑒𝑠𝑡 ), the position of each penguin (𝑃 ), 
the ambient temperature, 𝑇𝐴 , and the social force (𝑆 ) that 
forces the penguins to move towards the direction of best 
solution. The parameter 𝐴 is to be calculated for the 
position 𝑃 𝑖 using the movement parameter (𝑀 ), which is 
set to 2, in equation (2). 
 
𝐴 =𝑀 ×(𝑇𝐴 +|𝑃 𝑏𝑒𝑠𝑡 −𝑃 𝑖 |×𝑟𝑎𝑛𝑑 )−𝑇𝐴 (2) 
 
Equation (3)  is used to calculate the social force. This 
equation is a decreasing function that has three variables, 
𝑓 , 𝑙 , and 𝐼𝑡𝑟 . Both of 𝑓 and 𝑙 are random numbers that each 
as its lower and upper bound. Figure 1 shows an example 
of having 𝑓 belongs to the interval [2,5] and 𝑙 belongs to 
the interval[2,10]. 
 
𝑆 = (√𝑓 .𝑒 −𝐼𝑡𝑟 /𝑙 −𝑒 −𝐼𝑡𝑟 )
2
 
(3) 
 
Figure 1 Social force parameter’s graph 
 
The parameter 𝑆 is used to calculate the distance𝐷 , 
where it increases in the very first iterations to guarantee 
high locality and decrease it gradually until reaching very  
 
 
 
low locality in the higher iterations. So, the distance 𝐷 is 
to be calculated using equation (4) as follows: 
 
𝐷 =|𝑆 .  𝑃 𝑖 −𝑟𝑎𝑛𝑑 𝑃 𝑏𝑒𝑠𝑡 | (4) 
Now, the position of the penguin in the next iteration 
(𝑃 𝑖 +1
) can be calculated using equation  
(5) as follows: 
 
 
𝑃 𝑖 +1
=𝑃 𝑖 −𝐴 .𝐷    
 
(5) 
 
The best position is updated by changing the penguin's 
position in each iteration until the stopping criteria are 
met, and then the best solution is returned. The 
pseudocode for the algorithm can be summarized as 
follows: 
 
1 
𝐼𝑛𝑝𝑢𝑡 𝑡 ℎ𝑒 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑛 𝑠𝑖𝑧𝑒 (𝑁 ),𝑀𝑎𝑥𝐼𝑡𝑟 ,𝑎𝑛𝑑 𝑅 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 
2 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒 𝑡 ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 
3 
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑒𝑎𝑐 ℎ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑠𝑡𝑜𝑟𝑒 𝑡 ℎ𝑒  
𝑏𝑒𝑠𝑡 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝑃 𝑏𝑒𝑠𝑡 ) 
4 𝐼𝑡𝑟 =1 
5 𝑊 ℎ𝑖𝑙𝑒 𝐼𝑡𝑟 ≤𝑀𝑎𝑥𝐼𝑡𝑟 𝑑𝑜 : 
6  𝑖 =1 
7  𝑊 ℎ𝑖𝑙𝑒 𝑖 ≤𝑁 𝑑𝑜 : 
8   𝑇𝐴 =𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 −𝑀𝑎𝑥𝐼𝑡𝑟 
9   𝐴 =𝑀 ×(𝑇𝐴 +|𝑃 𝑏𝑒𝑠𝑡 −𝑃 𝑖 |×𝑟𝑎𝑛𝑑 )−𝑇𝐴 
10   𝑆 = (√𝑓 .𝑒 −𝐼𝑡𝑟 /𝑙 −𝑒 −𝐼𝑡𝑟 )
2
 
11   𝐷 =|𝑆 .  𝑃 𝑖 −𝑟𝑎𝑛𝑑 𝑃 𝑏𝑒𝑠𝑡 | 
12   𝑃 𝑖 +1
=𝑃 𝑖 −𝐴 .𝐷 
13   𝑖𝑓 𝑓 (𝑃 𝑖 +1
)≤𝑓 (𝑃 𝑏𝑒𝑠𝑡 ) 𝑡 ℎ𝑒𝑛 : 
14    𝑃 𝑏𝑒𝑠𝑡 =𝑃 𝑖 +1
 
15   𝑖 =𝑖 +1 
16  𝐼𝑡𝑟 =𝐼𝑡𝑟 +1 
17 𝑅𝑒𝑡𝑢𝑟𝑛 𝐺𝑏𝑒𝑠𝑡 
3 The proposed modified EPO 
algorithm 
In this section, a new modification to the EPO algorithm 
is proposed, which involves utilizing information gained 
from the current position of each penguin before 
relocating it. This modification constructs an information 
vector using a threshold selected from the interval [0,1]. 
The purpose of this proposed mutation process is to 
achieve increased locality when the lower threshold is 
reached. This is done by creating a new solution with more 
components from the current solution. Conversely, when 
Author Modification Application 
Dhiman and Kumar [4] First Proposed Test Optimization Problems 
Jia et al. [9] disruptive polynomial mutation 
Levy flight 
thermal exchange operator 
Satellite image segmentation 
Xing [12] Gaussian mutation 
Levy flight 
multilevel threshold for color image segmentation 
Kaur et al. [6]  Muti-objective optimization 
Yang and Gao [8]  Face recognition 
Kaur et al. [11]  Multi-objective optimization 
Lu et al. [13] Sequential quadratic programming Market clearing price 
Dhiman et al. [5] Binary emperor penguin optimizer  
Sameh et al. [7]  Photovoltaic control system 
Tang et al. [10]  Energy consumption of the residential buildings 

A Modified Emperor Penguin Algorithm for Solving Stagnation… Informatica 47 (2023) 71–78 73 
the higher threshold is reached, the process aims to 
enhance diversity by incorporating more components 
from the relocated solution. The current position, 𝑃 𝑖 , and 
the position generated using equation  
(5) are used to create the information vector, 𝑃 𝐼𝑉
. The new 
procedure selects the components of 𝑃 𝐼𝑉
 using the 
threshold. The first step of the new procedure is to 
generate a uniform random number that will be compared 
to the selected threshold. If the generated uniform random 
number is greater than the threshold, then component 𝑗 is 
selected from 𝑃 𝑖 +1
; otherwise, it is selected from 𝑃 𝑖 . The 
following steps show the procedure for creating the 
𝑃 𝐼𝑉
 position: 
 
1 𝑗 =1  
2 𝑊 ℎ𝑖𝑙𝑒 𝑗 ≤dim(𝑃 ) 𝑑𝑜 : 
3  𝑖𝑓 𝑟𝑎𝑛𝑑 >𝑡 ℎ𝑟𝑒𝑠 ℎ𝑜𝑙𝑑 : 
4   𝑃 𝐼𝑉
[𝑗 ]=𝑃 𝑖 +1
[𝑗 ] 
5  𝑒𝑙𝑠𝑒 : 
6   𝑃 𝐼𝑉
[𝑗 ]=𝑃 𝑖 [𝑗 ] 
7  𝑗 =𝑗 +1 
 
After creating the information vector, 𝑃 𝐼𝑉
, it will be 
evaluated using the objective function, and if its 
evaluation is better than both 𝑃 𝑖 and 𝑃 𝑖 +1
, then 𝑃 𝐼𝑉
 will 
replace 𝑃 𝑖 to be used in the next iteration of the algorithm. 
The full steps of the Modified Emperor Penguin 
Algorithm Optimizer (MEPO) can be shown as follows: 
 
1 𝐼𝑛𝑝𝑢𝑡 𝑡 ℎ𝑒 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑛 𝑠𝑖𝑧𝑒 (𝑁 ),𝑀𝑎𝑥𝐼𝑡𝑟 ,𝑎𝑛𝑑 𝑅 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 
2 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒 𝑡 ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 
3 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑒𝑎𝑐 ℎ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑠𝑡𝑜𝑟𝑒 𝑡 ℎ𝑒 𝑏𝑒𝑠𝑡 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝑃 𝑏𝑒𝑠𝑡 ) 
4 𝐼𝑡𝑟 =1 
5 𝑊 ℎ𝑖𝑙𝑒 𝐼𝑡𝑟 ≤𝑀𝑎𝑥𝐼𝑡𝑟 𝑑𝑜 : 
6  𝑖 =1 
7  𝑊 ℎ𝑖𝑙𝑒 𝑖 ≤𝑁 𝑑𝑜 : 
8   𝑇𝐴 =𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 −𝑀𝑎𝑥𝐼𝑡𝑟 
9   𝐴 =𝑀 ×(𝑇𝐴 +|𝑃 𝑏𝑒𝑠𝑡 −𝑃 𝑖 |×𝑟𝑎𝑛𝑑 )−𝑇𝐴 
10   𝑆 = (√𝑓 .𝑒 −𝐼𝑡𝑟 /𝑙 −𝑒 −𝐼𝑡𝑟 )
2
 
11   𝐷 =|𝑆 .  𝑃 𝑖 −𝑟𝑎𝑛𝑑 𝑃 𝑏𝑒𝑠𝑡 | 
12   𝑃 𝑖 +1
=𝑃 𝑖 −𝐴 .𝐷 
13   𝑗 =1 
14   𝑊 ℎ𝑖𝑙𝑒 𝑗 ≤dim(𝑃 ) 𝑑𝑜 : 
15    𝑖𝑓 𝑟𝑎𝑛𝑑 >𝑡 ℎ𝑟𝑒𝑠 ℎ𝑜𝑙𝑑 : 
16     𝑃 𝐼𝑉
[𝑗 ]=𝑃 𝑖 +1
[𝑗 ] 
17    𝑒𝑙𝑠𝑒 : 
18     𝑃 𝐼𝑉
[𝑗 ]=𝑃 𝑖 [𝑗 ] 
19    𝑗 =𝑗 +1 
20   𝑖𝑓 𝑓 (𝑃 𝑖 +1
)≤𝑓 (𝑃 𝑏𝑒𝑠𝑡 ) 𝑡 ℎ𝑒𝑛 : 
21    𝑃 𝑏𝑒𝑠𝑡 =𝑃 𝑖 +1
 
22   𝑖 =𝑖 +1 
23  𝐼𝑡𝑟 =𝐼𝑡𝑟 +1 
24 𝑅𝑒𝑡𝑢𝑟𝑛 𝐺𝑏𝑒𝑠𝑡 
 
 
 
 
 
 
The time complexity of the algorithm is the same as the 
classical EPO presented by Dhiman and Kumar [4], which 
is 𝑜 (𝑘 ×𝑛 ×𝑀𝑎𝑥𝐼𝑡𝑟 ×𝑑 ×𝑁 ) . The next section shows 
the comparative results which implemented to compare 
the MEPO with the classical EPO and another 
modification of the EPO algorithm presented by Xing 
[12]. All of the are coded using python programming and 
they code is available on https://github.com/ 
ahmedsssssA/EPOIV. 
4 Comparative results 
Both algorithms, EPO and MEPO, were implemented 
using Python programming language on a PC equipped 
with a Core i5 processor clocked at 3.40GHz and 4 
gigabytes of RAM. Another algorithm is selected for the 
comparative results from the previous work listed in Table 
1. The selected algorithm is the EPO that utilizes the Levy 
flight and gaussian mutation (EPOLG) presented by Xing 
[12].To evaluate the performance of the algorithms, a set 
of benchmark problems available at https://www.sfu 
.ca/~ssurjano/optimization.html are selected as shown in 
Table 2. The parameters of the algorithms are set as high 
as possible to demonstrate their effectiveness in finding 
the global minimum. Specifically, the population size (𝑁 ) 
is set to 100, the maximum number of iterations (𝑀𝑎𝑥𝐼𝑡𝑟 ) 
is set to 100, and 𝑅 is randomly selected for each solution 
in each iteration from the interval [0, 1]. 
The selected optimization test functions are chosen to 
cover functions with different shapes, including those with 
many local minima, bowl-shaped, plate-shaped, steep 
ridges/drops, and valley-shaped. Figure 2 and Figure 3 
present the comparison between EPO, EPOLG, and 
MEPO. The figures show that the proposed MEOP 
algorithm has a higher efficiency in terms of convergence 
and solving the stagnation problem of the multi-modal 
functions. The functions Ackley, Bent Ciger, Easom, 
Michalewicz, and Schwefel show stagnation with both 
EPO and EPOLG algorithms, and this stagnation is solved 
with the MEPO algorithm. Moreover, Table 3 presents the 
results of the implementation for each algorithm. The 
optimized results are shown in bold, and all of them are 
associated with the MEPO algorithm. MEPO shows 
significant improvement in terms of mean results and the 
absolute relative standard deviation (RSD), which reflects 
the robustness of the MEPO algorithm’s results. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

74 Informatica 47 (2023) 71–78 A. Serag et al. 
 
 
 
Table 2 Selected test optimization problems 
No. 
Function 
Name 
𝑓 (𝑥 ) Global Minimum 
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 ,𝑥 𝑖 ∈[−33,33] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(0,…,0) 
2 Bent Cigar x
1
2
+10
6
∑𝑥 𝑖 2
𝑛 𝑖 =2
,𝑥 𝑖 ∈[−100,100] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(0,…,0) 
3 Bohachevsky 
𝑥 1
2
+2𝑥 2
2
−0.3cos(31𝜋 𝑥 1
)−0.4cos(31𝜋 𝑥 2
)+0.7,
𝑥 𝑖 ∈[−100,100] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(0,0) 
4 Booth (𝑥 1
+2𝑥 2
−7)
2
+(2𝑥 1
+𝑥 2
−5)
2
, 𝑥 𝑖 ∈[−10,10] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(1,3) 
5 Bukin 100√|𝑥 2
−0.01𝑥 1
2
|+0.01|𝑥 1
+10|, 𝑥 𝑖 ∈[−15,3] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(−10,0) 
6 Cross-in-Tray 
−0.0001
(
 
 
|
|
𝑠𝑖𝑛 (𝑥 1
) 𝑠𝑖𝑛 (𝑥 2
)𝑒 | |100 − 
√𝑥 1
2
+𝑥 2
2
𝜋 | |
|
|
+1
)
 
 
0.1
, 𝑥 𝑖 ∈[−15,15] 
𝑓 (𝑥 ∗
)
=−2.06261,𝑥 ∗
=(1.3491,−1.3491) 
7 Drop Wave 
−
1+cos(12√𝑥 1
2
+𝑥 2
2
)
0.5(𝑥 1
2
+𝑥 2
2
)+2
,𝑥 𝑖 ∈[−5.12,5.12] 
𝑓 (𝑥 ∗
)=−1.5,𝑥 ∗
=(0,0) 
8 Discus 10
6
𝑥 1
2
+∑𝑥 𝑖 2
𝐷 𝑖 =2
,𝑥 𝑖 ∈[0,100] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(0,…,0) 
9 Easom −cos(𝑥 1
)cos(𝑥 2
) 𝑒 (−(𝑥 1
−𝜋 )
2
−(𝑥 2
−𝜋 )
2
)
, 𝑥 𝑖 ∈[−100,100] 
𝑓 (𝑥 ∗
)=−1,𝑥 ∗
=(𝜋 ,𝜋 ) 
10 Eggholder 
−(𝑥 2
+47)sin(√|𝑥 2
+
𝑥 1
2
+47|)−𝑥 1
sin(√|𝑥 1
−(𝑥 2
+47)|),𝑥 ∈[−500,500] 
𝑓 (𝑥 ∗
)
=−959.6407,𝑥 ∗
=(512,404.2319) 
11 Griewank ∑
𝑥 𝑖 2
4000
−∏cos(
𝑥 𝑖 √𝑖 )+1
𝑑 𝑖 =1
𝑑 𝑖 =1
,𝑥 ∈[−600,600] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(0,…,0) 
12 Holder Table −
|
|
sin(𝑥 1
)cos(𝑥 2
)𝑒 (
 
| |1−
√𝑥 1
2
+𝑥 2
2
𝜋 | |
)
 
|
|
,𝑥 𝑖 ∈[−10,10] 
𝑓 (𝑥 ∗
)
=−19.2085,𝑥 ∗
=(8.05502,−9.66459) 
13 Michalewicz −∑sin(𝑥 𝑖 )sin
20
(
𝑖 𝑥 𝑖 2
𝜋 )
𝑑 𝑖 =1
,𝑥 ∈[0,𝜋 ] 
𝑓 (𝑥 ∗
)=−1.8013,𝑥 ∗
=(2.20,1.57) 
14 
Modified 
Schwefel 
418.9829×𝐷 −∑𝑔 (𝑧 𝑖 )
𝐷 𝑖 =1
,𝑧 𝑖 =𝑥 𝑖 +4.209687462275036e+002,𝑥 𝑖 ∈[−500,500] 
𝑔 (𝑧 𝑖 )
=
{
 
 
 
 
𝑧 𝑖 sin(|𝑧 |
1
2), 𝑖𝑓 |𝑧 𝑖 |≤500
(500−𝑚𝑜𝑑 (𝑧 𝑖 ,500))sin(√500−|𝑚𝑜𝑑 (𝑧 𝑖 ,500)|)−
(𝑧 𝑖 −500)
2
1000𝐷 ,𝑖𝑓 𝑧 𝑖 >500
(𝑚𝑜𝑑 (|𝑧 𝑖 |,500)−500)sin(√|𝑚𝑜𝑑 (|𝑧 𝑖 |,500)−500|)−
(𝑧 𝑖 +500)
1000𝐷 ,𝑖𝑓 𝑧 𝑖 <−500
  
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=[0,…,0] 
15 Rastrigin 10𝑑 + ∑[𝑥 2
−10cos(2𝜋 𝑥 𝑖 )]
𝑑 𝑖 =1
, 𝑥 𝑖 ∈[−5,5] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=[0,…,0] 
16 Rosenbrock ∑[100(𝑥 𝑖 +1
−𝑥 𝑖 2
)
2
+(𝑥 𝑖 −1)
2
]
𝑑 −1
𝑖 =1
, 𝑥 𝑖 ∈[−5,10] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(1,…,1) 

A Modified Emperor Penguin Algorithm for Solving Stagnation… Informatica 47 (2023) 71–78 75 
No. 
Function 
Name 
𝑓 (𝑥 ) Global Minimum 
17 Schwefel 418.9829𝑑 −∑𝑥 𝑖 sin(√|𝑥 𝑖 |)
𝑑 𝑖 =1
, 𝑥 𝑖 ∈[−500,500] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(420.9687,…,420.9687) 
18 six-hump (4−2.1𝑥 1
2
+
𝑥 1
4
3
)𝑥 1
2
+𝑥 1
𝑥 2
+(−4+4𝑥 2
2
)𝑥 2
2
, 𝑥 𝑖 ∈[−3,3] 
𝑓 (𝑥 ∗
)=−1.0316,𝑥 ∗
=(0.0898,−0.7126) 
19 Sphere ∑𝑥 𝑖 2
𝑛 𝑖 =1
,𝑥 𝑖 ∈[−5,5] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=[0,…,0] 
20 Zakharov 
∑𝑥 𝑖 2
𝑑 𝑖 =1
+(∑0.5𝑖𝑥 𝑖 𝑑 𝑖 =1
)
2
+(∑0.5𝑖 𝑥 𝑖 𝑑 𝑖 =1
)
4
, 𝑥 𝑖 ∈[−5,10] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(0,…,0) 
Table 3 The results of algorithms 
Function 
EPO 
Means 
EPOLG 
Means 
MEPO 
Means 
EPO 
RSD 
EPOLG 
RSD 
MEPO 
RSD 
Ackley 7.235 3.711 0.277 0.255 0.228 0.233 
Bent Cigar 14.572 16.693 0.367 10.314 14.028 0.907 
Bohachevsky 0.497 0.528 0.024 0.649 0.555 0.142 
Booth 0.003 0.023 0.000 1.554 1.178 1.131 
Bukin 1.335 1.731 0.455 0.655 0.424 0.290 
Cross in tray -2.063 -2.062 -2.063 0.000 0.000 0.000 
Drop Wave -0.989 -0.998 -0.998 0.024 0.033 0.024 
Discus 20.622 17.352 0.030 12.782 10.151 0.309 
Easom -0.765 -0.449 -1.000 0.350 0.627 0.001 
Eggholder -938.908 -939.129 -950.677 0.012 0.011 0.010 
Griewank 26.045 10.557 1.259 0.533 0.918 0.124 
Holder Table -19.197 -19.179 -19.208 0.001 0.001 0.000 
Michalewicz -6.149 -4.682 -8.462 0.101 0.076 0.016 
Modified 
Schwefel 
1311.743 1199.826 1.475 0.182 0.360 0.129 
Rastrigin 20.361 9.611 0.023 0.462 0.872 0.437 
Rosenbrock 7.525 11.891 4.198 1.240 1.325 0.325 
Schwefel 1920.748 1909.122 16.103 0.080 0.084 0.043 
Six hump -1.032 -1.031 -1.032 0.000 0.000 0.000 
Sphere 0.019 0.023 0.002 1.209 0.683 0.262 
Zakharov 10.105 0.632 0.506 0.609 0.664 0.074 
5 Conclusion 
This paper presents a new improvement to the EPO 
algorithm presented by Dhiman and Kumar [4]. After 
coding the algorithm using Python programming and 
implementing it on a set of test optimization problems 
with various shapes such as many local minima, bowl-
shaped, plate-shaped, steep ridges/drops, and valley-
shaped, it was found that the EPO algorithm stagnated in 
some functions without showing any improvement. The 
proposed modification provides more accurate results than 
the original algorithm in most of the considered test 
functions, making the proposed algorithm more reliable 
than the original one. In addition, the algorithm is  
 
compared with another modification of the algorithm, 
EPOLG, that proposed by Xing [12] that proposed a 
modification in the relocating procedure of the algorithm 
by having levy flight and gaussian mutations. Although of 
his new modifications, the algorithm still stagnates on the 
multi-modal functions. The proposed MEPO in this paper 
shows a high efficiency on solving the stagnation of the 
multi-modal functions and shows a high robustness in 
comparison with the other algorithms. 
Future research may include one of the following 
points: 

76 Informatica 47 (2023) 71–78 A. Serag et al. 
• Developing further improvements to 
increase the efficiency of the algorithm and 
apply it to CEC2017 problems. 
• Applying the algorithm to solve real-world 
problems, such as transshipment, 
transportation, and job shop scheduling 
problems. 
• Optimizing the parameters of the algorithm 
using design of experiments. 
• Hybridizing the algorithm with other 
metaheuristic algorithms to improve its 
efficiency
 
 
 
Figure 2: The results of the first 10 functions

A Modified Emperor Penguin Algorithm for Solving Stagnation… Informatica 47 (2023) 71–78 77 
 
 
 
 
 
Figure 3: The results of the second 10 functions 
  

78 Informatica 47 (2023) 71–78 A. Serag et al. 
References 
[1] M. A. Kader, K. Z. Zamli, and B. S. Ahmed, “A 
systematic review on emperor penguin 
optimizer,” Neural Comput. Appl., vol. 33, no. 23, 
pp. 15933–15953, 2021, doi: 10.1007/s00521-
021-06442-4. 
[2] P. R. Murthy, Operations research (linear 
programming). bohem press, 2005. 
[3] L. M. Abualigah, A. T. Khader, and E. S. 
Hanandeh, “Hybrid clustering analysis using 
improved krill herd algorithm,” Appl. Intell., vol. 
48, no. 11, pp. 4047–4071, 2018, doi: 
10.1007/s10489-018-1190-6. 
[4] G. Dhiman and V. Kumar, “Emperor penguin 
optimizer: A bio-inspired algorithm for 
engineering problems,” Knowledge-Based Syst., 
vol. 159, pp. 20–50, 2018, doi: 
10.1016/j.knosys.2018.06.001. 
[5] G. Dhiman et al., “BEPO: A novel binary emperor 
penguin optimizer for automatic feature 
selection,” Knowledge-Based Syst., vol. 211, p. 
106560, 2021, doi: 
10.1016/j.knosys.2020.106560. 
[6] H. Kaur, A. Rai, S. S. Bhatia, and G. Dhiman, 
“MOEPO: A novel Multi-objective Emperor 
Penguin Optimizer for global optimization: 
Special application in ranking of cloud service 
providers,” Eng. Appl. Artif. Intell., vol. 96, p. 
104008, 2020, doi: 
10.1016/j.engappai.2020.104008. 
[7] M. A. Sameh, M. I. Marei, M. A. Badr, and M. A. 
Attia, “An optimized pv control system based on 
the emperor penguin optimizer,” Energies, vol. 
14, no. 3, p. 751, 2021, doi: 10.3390/en14030751. 
[8] J. Yang and H. Gao, “Cultural Emperor Penguin 
Optimizer and Its Application for Face 
Recognition,” Math. Probl. Eng., vol. 2020, 2020, 
doi: 10.1155/2020/9579538. 
[9] H. Jia, K. Sun, W. Song, X. Peng, C. Lang, and Y. 
Li, “Multi-Strategy Emperor Penguin Optimizer 
for RGB Histogram-Based Color Satellite Image 
Segmentation Using Masi Entropy,” IEEE Access, 
vol. 7, pp. 134448–134474, 2019, doi: 
10.1109/ACCESS.2019.2942064. 
[10] F. Tang, J. Li, and N. Zafetti, “Optimization of 
residential building envelopes using an improved 
Emperor Penguin Optimizer,” Eng. Comput., vol. 
38, no. 2, pp. 1395–1407, 2022, doi: 
10.1007/s00366-020-01112-w. 
[11] H. Kaur, A. Rai, S. S. Bhatia, and G. Dhiman, 
“MOEPO: A novel Multi-objective Emperor 
Penguin Optimizer for global optimization: 
Special application in ranking of cloud service 
providers,” Eng. Appl. Artif. Intell., vol. 96, 2020, 
doi: 10.1016/j.engappai.2020.104008. 
[12] Z. Xing, “An improved emperor penguin 
optimization based multilevel thresholding for 
color image segmentation,” Knowledge-Based 
Syst., vol. 194, 2020, doi: 
10.1016/j.knosys.2020.105570. 
[13] X. Lu, Y. Yang, P. Wang, Y. Fan, F. Yu, and N. 
Zafetti, “A new converged Emperor Penguin 
Optimizer for biding strategy in a day-ahead 
deregulated market clearing price: A case study in 
China,” Energy, vol. 227, 2021, doi: 
10.1016/j.energy.2021.120386.