https://doi.org/10.31449/inf.v48i7.5711 Informatica 47 (2024) 79–94 79 
A Modified Emperor Penguin Optimizer Algorithm for Solving 
Fixed-Charged Transshipment Problem 
Mohamed Meselhy Eltoukhy
1*
, Mohammad Zakaraia
2
 
1
Department of Information Technology, College of Computing and Information Technology at Khulais, University of 
Jeddah, Jeddah, Saudi Arabia 
2
Faculty of graduate studies for statistical research, Cairo University 
E-mail: mmeltoukhy@uj.edu.sa
1
, dr.mzakaraia@gmail.com
2 
*
Corresponding author 
Keywords: Fixed-Charged transportation problem, transshipment problem, metaheuristics, Emperor penguin optimizer 
algorithm, Taguchi’s orthogonal arrays, design of experiments, Wilcoxon test, Friedman test 
Received: February 8, 2024 
This paper introduces a novel optimization problem termed the Fixed Charged Transshipment Problem 
(FCTP), which incorporates fixed charges for selected routes. A new formulation for this problem is 
presented, aiming to address the combinatorial nature of the challenge. The study further introduces a 
Modified Emperor Penguin Optimizer (EPO) algorithm designed to enhance the solution approach. To 
evaluate the performance of the Modified EPO, a comparative analysis is conducted against the classical 
EPO and Particle Swarm Optimization (PSO) algorithms. 19 problems, including various multi-modal 
test optimization functions, serve as the testing ground. Results demonstrate the efficacy of the Modified 
EPO, establishing its superiority over the classical EPO and PSO. Additionally, a heuristic procedure is 
proposed for solving the combinatorial aspect of the FCTP. This heuristic is hybridized with both the 
Modified EPO and PSO algorithms. 30 FCTP problems are generated using a code available at 
https://github.com/MZakaraia/EPO_Transshipment/. Taguchi's orthogonal arrays are employed to 
optimize parameter levels for both algorithms. The study concludes with the comparison of the Modified 
Hybrid EPO and Hybrid PSO in solving the 30 generated FCTP problems. Remarkably, the Modified 
Hybrid EPO algorithm outperforms the Hybrid PSO, showing its effectiveness in addressing the Fixed 
Charged Transshipment Problem in terms of means and robustness. 
Povzetek: Članek predstavlja spremenjeni algoritem cesarskega pingvina za reševanje problema s 
fiksnimi stroški prenosa, ki kaže premoč nad klasičnimi metodami z robustnostjo in učinkovitostjo rešitev.
1 Introduction  
The fixed charged transshipment problem is a well-known 
optimization problem in the field of logistics and supply 
chain management. It involves determining the optimal 
flow of goods through a network, considering fixed costs 
associated with transshipments between various nodes. 
Solving the FCTP efficiently is crucial for optimizing 
supply chain operations, reducing costs, and enhancing 
overall system performance. In recent years, nature-
inspired optimization algorithms have gained popularity 
as effective tools for solving complex optimization 
problems. One such algorithm is the Emperor Penguin 
Optimizer (EPO), which is inspired by the behavior and 
social interactions of emperor penguins in their natural 
habitat. The EPO algorithm is known for its ability to 
effectively handle continuous and discrete optimization 
problems. This paper presents a modified version of the 
Emperor Penguin Optimizer algorithm tailored 
specifically for addressing the Fixed-Charged 
Transshipment Problem. The main objective of this study 
is to investigate the effectiveness and efficiency of the 
modified EPO algorithm in finding high-quality solutions 
for the FCTP. So, the contribution of this can be 
summarized as follows: 
• Proposing a new formulation for the fixed 
charged transshipment problem by considering 
fixed costs for the routes. 
• Adapting a new modification of the population-
based metaheuristic EPO algorithm for solving 
FTCP. 
• Generating a dataset of 30 problems for FTCP to 
validate the proposed EPO algorithm for solving 
FTCP. 
The remainder of this paper is organized as follows: 
Section 2 provides a literature review of related studies on 
transshipment problems. Section 3 presents the 
mathematical formulation of the Fixed Charge 
Transshipment Problem with a discussion that shows the 
novelty of the proposed formulation. Section 4 describes 
the EPO algorithm in detail, while Section 5 introduces the 
modified EPO and presents comparative results. Section 6 
outlines the adapted EPO for solving FCTP. In this 
Section the computational complexity of the proposed 
EPO is presented. The experimental design is 
implemented to optimize the parameters of the EPO and 
adopted particle swarm optimization algorithm for solving 
the problem. The computational results for 30 generated 
problems are performed to compare between the hybrid 

80 Informatica 48 (2024) 79–94 M.M. Eltoukhy et al. 
EPO and PSO for solving the FCTP. Finally, Section 7 
concludes the paper with a summary of the findings, 
highlighting the advantages and potential applications of 
the proposed algorithm for solving the Fixed Charge 
Transshipment Problem. 
2 Literature review 
This section provides a literature review of prior research 
on the transshipment problem, offering a comprehensive 
overview of existing work and highlighting the motivation 
behind the current study. Herer and Tzur [1] investigated 
the strategy of transshipments in a dynamic, deterministic 
demand environment over a finite planning horizon. Their 
study considered a system of two locations replenished by 
a single supplier, incorporating various costs and deriving 
structural properties of optimal policies, leading to the 
development of an efficient polynomial time algorithm for 
obtaining the optimal strategy and motivating the adoption 
of transshipments in replenishment strategies. Reyes [2] 
used the Shapley value concept from cooperative game 
theory to solve the transshipment problem and 
demonstrated its efficacy through a numerical example. 
Herer et al. [3] examined a supply chain with multiple 
retailers and a supplier, where they established optimal 
replenishment and transshipment policies to minimize 
long-run average costs. Through a sample-path-based 
optimization procedure, they calculated order-up-to 
quantities using a linear programming/network flow 
framework. 
Belgasmi et al. [4] examined a multi-location 
inventory system with centrally coordinated inventory 
choices, allowing lateral transshipments within the same 
echelon to reduce costs and improve service level. They 
proposed a multi-objective model to optimize cost, fill 
rate, and transshipment lead times. They utilized an 
evolutionary multi-objective optimization approach to 
approximate the optimal trade-offs between these 
conflicting objectives. Sharma and Jana [5] developed a 
transshipment planning model for the petroleum refinery 
industry, aiming to minimize costs, maximize production, 
and meet storage and demand requirements. They 
employed a fuzzy goal programming (FGP) model with 
integrated genetic algorithms (GA) to handle imprecision 
and provide flexible solutions. A case example showcased 
the effectiveness of this integrated technique in optimizing 
transshipment operations. Khurana and Arora [6] 
extended the standard transshipment model to include 
inequality constraints.  Their algorithm transformed the 
problem into an equivalent transportation problem to 
obtain the optimal solution. They discussed balanced and 
unbalanced transshipment problems, emphasizing the 
algorithm's applicability in addressing distribution 
problems with mixed constraints and paradoxical 
situations. Özdemir et al.  [7] investigated the coordination 
of stocking locations considering lateral transshipments 
and supply capacity in the transshipment model. They 
formulated the capacitated supply scenario as a network 
flow problem within a stochastic optimization framework. 
They found that system behavior depends on production 
capacity and highlighted the importance of capacity 
flexibility or transshipment flexibility for maintaining 
desired service levels in a production-inventory system.  
Khurana, et al. [8] developed an algorithm to solve a 
transshipment problem with the objective of minimizing 
transportation duration. They transformed the problem 
into an equivalent transportation problem and obtained the 
optimal solution. Their algorithm is easy to understand 
and apply, making it suitable for addressing various 
products distribution problems. In addition, balanced and 
unbalanced time minimization scenarios were discussed 
with numerical examples. Kumar, et al.  [9] addressed the 
challenges of uncertainty in transshipment problems by 
representing parameters as intuitionistic fuzzy numbers. 
Their proposed method is based on ambiguity and 
vagueness indices to derive a fuzzy optimal solution 
without the need for an initial basic feasible solution. The 
technique demonstrated computational efficiency and 
applicability to a wide range of transshipment problems, 
supported by numerical illustrations. Garg et al. [10] 
investigated a fuzzy fractional two-stage transshipment 
problem, using the ratio of costs divided by benefits as the 
objective function. They employed the extension principle 
and Charnes-Cooper transformation method to find the 
fuzzy objective value. The proposed formulation and 
solution method demonstrated superior efficiency 
compared to the existing literature. 
Table 1 shows the literature review summary. To the 
best of our knowledge, the fixed-charge transshipment 
problem, which extends the fixed-charge transportation 
problem by including transshipment nodes, has not yet 
been investigated. Therefore, this paper presents a new 
model for the transshipment problem that incorporates 
both fixed costs and transportation costs. 
 
Table 1: Literature review summary 
Author Problem Approach 
Herer and 
Tzur [1] 
Dynamic 
transshipment 
problem 
Heuristic 
approach 
Reyes [2] Classical 
transshipment 
problem 
Game theory 
approach 
Herer et al. 
[3] 
Multi-location 
transshipment 
problem 
Linear 
programming and 
network follow 
Belgasmi et 
al. [4] 
Multi-objective 
multi-location 
transshipment 
problem 
Strength Pareto 
evolutionary 
algorithm 
Sharma and 
Jana [5] 
Transshipment 
management 
problem 
Fuzzy goal 
programming and 
genetic algorithm 
Khurana 
and Arora 
[6] 
Unbalanced 
transshipment 
problem with 
mixed constraints 
Liner 
Programming 
Özdemir et 
al.  [7] 
Multi-location 
transshipment 
problem with 
Random search, 
Simulation 
applications, and 

A Modified Emperor Penguin Optimizer Algorithm for Solving… Informatica 48 (2024) 79–94 81 
capacitated 
production 
sample average 
approximate 
Khurana, et 
al. [8] 
Time minimizing 
transshipment 
problem 
Heuristic 
approach 
Kumar, et 
al.  [9] 
Fuzzy 
transshipment 
problem 
Heuristic 
approach 
Garg et al. 
[10] 
Fractional two-
stage 
transshipment 
problem under 
uncertainty 
Charnes–Cooper 
transformation 
method, linear 
programming 
 
Given the combinatorial complexity of the problem, 
the classical approaches listed in Table 1, primarily those 
mentioned, are not well-suited to solving the fixed-charge 
transshipment problem. Consequently, this paper adopts 
one of the latest population-based metaheuristics, namely 
the Emperor Penguin Optimizer Algorithm, to address this 
challenge. 
3 Mathematical formulation 
Fixed Charge Transshipment Problem is a combinatorial 
optimization problem in which a set of products or goods 
is transported from source nodes to destination nodes 
through a network of intermediate transshipment nodes. In 
this problem, there is a fixed cost associated with using 
each transshipment node, and a variable cost for 
transporting each unit of product between the nodes. The 
FCTP can be mathematically modeled as follows: 
Notations: 
𝑖 Set of sources 
𝑗 Set of destinations 
𝑘 Set of transshipment nodes 
𝑓 𝑘 Fixed cost associated with transshipment node 
𝑘 
𝑐 𝑖𝑗
 Unit cost of transporting a product from source 
𝑖 to destination 𝑗 via transshipment node 𝑘 
𝑥 𝑖𝑗
 Amount of product transported from source 
𝑖 to destination 𝑗 directly (without 
transshipment) 
𝑠 𝑖 The available quantity produced by source 𝑖 
𝑑 𝑗 The required demand by destination 𝑗 
𝑦 𝑖𝑗𝑘 Amount of product transshipped at node 
𝑘 from source 𝑖 to destination 𝑗 
𝑧 𝑘 A binary variable representing whether 
transshipment node 𝑘 is used or not 
Mathematical model: 
 
min∑𝑓 𝑘 𝑧 𝑘 +∑∑𝑐 𝑖𝑗
(𝑥 𝑖𝑗
+∑𝑦 𝑖𝑗𝑘 𝑘 ∈𝐾 )
𝑗 ∈𝐽 𝑖 ∈𝐼 𝑘 ∈𝐾 
(1) 
 Subject to:  
  
∑(𝑥 𝑖𝑗
+∑𝑦 𝑖𝑗𝑘 𝑘 ∈𝐾 )
𝑗 ∈𝐽 =𝑠 𝑖 ,∀𝑖 ∈𝐼 
(2) 
  ∑(𝑥 𝑖𝑗
+∑𝑦 𝑖𝑗𝑘 𝑘 ∈𝐾 )
𝑖 ∈𝐼 =𝑑 𝑗 ,∀𝑗 ∈𝐽 (3) 
  
∑∑𝑦 𝑖𝑗𝑘 𝑗 ∈𝐽 𝑖 ∈𝐼 ≤𝑄 𝑘 𝑧 𝑘 ,∀𝑘 ∈𝐾 
(4) 
  𝑥 𝑖𝑗
≥0,∀𝑖 ∈𝐼 ,𝑗 ∈𝐽 (5) 
  𝑦 𝑖𝑗𝑘 ≥0,∀𝑖 ∈𝐼 ,𝑗 ∈𝐽 ,𝑘 ∈𝐾 (6) 
  𝑧 𝑘 ∈{0,1},∀𝑘 ∈𝐾 (7) 
 
The fixed-charged transshipment problem, as shown 
in the mathematical model, is a combinatorial 
optimization problem known for its NP-hardness, making 
it very difficult to solve using classical approaches. 
Therefore, metaheuristics are the preferred choice for 
tackling such problems. These approaches can be broadly 
classified into two categories: single-based metaheuristics 
(e.g., simulated annealing, Tabu search, and variable 
neighborhood algorithms) and population-based 
metaheuristics (e.g., particle swarm optimization, gray 
wolf, genetic algorithm, and the proposed emperor 
penguin optimizer algorithm (EPO) presented in this 
paper) [11]–[14]. 
In this paper a new form of the transshipment problem 
is presented, which considers fixed charges associated 
with selected routes. To fill this gap, we propose a 
hybridized EPO algorithm that incorporates a heuristic 
procedure based on priority vectors to achieve solution 
improvement. Before presenting the hybridized algorithm, 
we propose a modified EPO algorithm and compare its 
performance with both the classical EPO and a particle 
swarm optimization algorithm.  
4 Emperor penguin optimizer 
algorithm 
The emperor penguin optimizer algorithm is a population-
based metaheuristic that was first proposed by Dhiman 
and Kumar [15]. It mimics the emperor penguins’ 
huddling behavior. The steps of the algorithm include 
generating the boundaries of the huddle, computing the 
temperature of the huddle, the distances between specific 
penguins, and finding the emperor penguins by obtaining 
the effective mover. The proposed algorithm by Dhiman 
and Kumar [15] consists of four phases: 
• Generating positions based on huddle 
boundaries. 
• Calculating the temperature around the huddle. 
• Determining the distances between emperor 
penguins. 
• Relocating procedure. 
The next subsections show these phases followed by 
the full pseudo code of the basic POA algorithm. 
4.1 Generating positions based on huddle 
boundaries 
In this phase, the positions of the penguins are to be 
generated using the huddle boundaries, where they are 
restricted by the lower bound (𝐿𝐵 ) and the upper bound 
(𝑈𝐵 ). So, each penguin position is to be generated using 
equation (8) for all 𝑛 penguins in the huddle. 

82 Informatica 48 (2024) 79–94 M.M. Eltoukhy et al. 
𝑃𝑜𝑠
𝑖 =𝐿𝐵 +𝑟𝑎𝑛𝑑 (0,1)(𝑈𝐵 −𝐿𝐵 ),∀𝑖 ={1,…,𝑛 } 
(8) 
4.2 Calculating temperature profile 
around the huddle 
In this phase, the temperature profile of the huddle is to be 
calculated using the radius of the huddle (𝑅 ). If the radius 
of the huddle is less than 1, then the temperature (𝑇 ) equals 
to 1, and 𝑇 equals 0 if 𝑅 greater than or equals 1. The 
radius of the huddle in the algorithm is to be generated 
randomly in each iteration from the interval [0,1]. So, the 
new temperature profile around the huddle (𝑇 ′
) can be 
calculated according to equation (9). 
 
𝑇 ′
=𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 −𝑀𝑎𝑥𝐼𝑡𝑟 
(9 ) 
𝑇 ={
1,𝑅 <0
0,𝑅 ≤1
 
 
4.3 Determining the distance between the 
emperor penguins 
The distance between emperor penguins and the best 
penguin can be calculated using two vectors that prevent 
collision 𝐴⃗
 and 𝐶⃗
, the position of the penguin 𝑖 in the 
current iteration (𝑃 𝐼𝑡𝑟 (𝑖 ) ), a social force 𝑆 , and the position 
of the current optimal emperor penguin (𝑃 𝑜𝑝𝑡 ). The 
proposed equation by Dhiman and Kumar [15] to calculate 
the distance (𝐷 ) is presented in equation (10). The 
calculations of the two collision vectors 𝐴⃗
 and 𝐶⃗
 are 
shown in equations (11) and (12), respectively. The 
𝑃 𝑔𝑟𝑖𝑑 variable found in equation (13) is the absolute value 
of the difference between the position of the best emperor 
penguin and the current penguin 𝑖 , where the equation of 
the 𝑃 𝑔𝑟𝑖𝑑 is equation (13). The social force function can be 
calculated using equation (14). 
 
𝐷 =|𝑆 (𝐴⃗
)𝑃 𝑜𝑝𝑡 −𝐶⃗
𝑃 𝐼𝑡𝑟 (𝑖 )| (10) 
𝐴⃗
=(2×𝑇 ′
+ 𝑃 𝑔𝑟𝑖𝑑 ×𝑟𝑎𝑛𝑑 (0,1))−𝑇 ′
 (11) 
𝐶⃗
=𝑟𝑎𝑛𝑑 (0,1) (12) 
𝑃 𝑔𝑟𝑖𝑑 =|𝑃 𝑜𝑝𝑡 −𝑃 𝐼𝑡𝑟 (𝑖 )| (13) 
𝑆 =(
√
𝑟𝑎𝑛𝑑 (2,3)𝑒 −
𝐼𝑡𝑟 𝑟𝑎𝑛𝑑 (1.5,2)
−𝑒 −𝐼𝑡𝑟 )
2
 
(14) 
4.4 Relocating procedure 
In this phase, the position of each emperor penguin is to 
be modified using the calculated distance (𝐷 ) as found in 
equation (15). 
 
𝑃 𝐼𝑡𝑟 +1
=𝑃 𝐼𝑡𝑟 +𝐴 ⃗
𝐷 
(15 ) 
Now the pseudo code of the basic algorithm 
developed by Dhiman and Kumar [15] can be summarized 
as follows: 
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 𝑜𝑝𝑡 ) 
𝐼𝑡𝑟 =1 
𝑊 ℎ𝑖𝑙𝑒 𝐼𝑡𝑟 ≤𝑀𝑎𝑥𝐼𝑡𝑟 𝑑𝑜 : 
 𝑖 =1 
 𝑊 ℎ𝑖𝑙𝑒 𝑖 ≤𝑁 𝑑𝑜 : 
  𝑇 ′
=𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 −𝑀𝑎𝑥𝐼𝑡𝑟 
  𝐴 =𝑀 ×(𝑇 ′
+|𝑃 𝑜𝑝𝑡 −𝑃 𝐼𝑡𝑟 (𝑖 )|×𝑟𝑎𝑛𝑑 (0,1))−𝑇 ′
 
  
𝑆 =(
√
𝑟𝑎𝑛𝑑 (2,3)𝑒 −
𝐼𝑡𝑟 𝑟𝑎𝑛𝑑 (1.5,2)
−𝑒 −𝐼𝑡𝑟 )
2
 
  𝐷 =|𝑆 .  𝑃 𝐼𝑡𝑟 (𝑖 )−𝑟𝑎𝑛𝑑 𝑃 𝑜𝑝𝑡 | 
  𝑃 𝐼𝑡𝑟 +1
(𝑖 )=𝑃 𝐼𝑡𝑟 (𝑖 )−𝐴⃗
𝐷 
  𝑖𝑓 𝑓 (𝑃 𝐼𝑡𝑟 +1
(𝑖 ))≤𝑓 (𝑃 𝑜𝑝𝑡 ) 𝑡 ℎ𝑒𝑛 : 
   𝑃 𝑜𝑝𝑡 =𝑃 𝐼𝑡𝑟 +1
(𝑖 ) 
  𝑖 =𝑖 +1 
 𝐼𝑡𝑟 =𝐼𝑡𝑟 +1 
𝑅𝑒𝑡𝑢𝑟𝑛 𝑃 𝑜𝑝𝑡 
5 Modified penguin optimizer and 
comparative results 
This section presents a modification of the EPO algorithm 
to adapt it for solving FCTP. The new modification of the 
algorithm considers adding an information vector 
(𝑃 𝐼𝑉
(𝑖 ) )[16]. Such information vector will be created 
during the relocating procedure of the algorithm. The 
creation of this vector is done using the positions of two 
emperor penguins. The first position is associated with the 
position of penguin (𝑖 ) at iteration (𝐼𝑡𝑟 ) in the population 
(𝑃 𝐼𝑡𝑟 (𝑖 ) ), while the second position is associated with the 
relocated position of that penguin (𝑃 𝐼𝑡𝑟 +1
(𝑖 ) ). The 
threshold herein is used as a predetermined number from 
the interval [0,1]. It is used to determine whether to select 
a component from the 𝑃 𝐼𝑡𝑟 (𝑖 ) or from 𝑃 𝐼𝑡𝑟 +1
(𝑖 ) . So, the 
steps to create the information vector can be summarized 
as follows: 
 
𝑗 =1  
𝑊 ℎ𝑖𝑙𝑒 𝑗 ≤dim(𝑃 𝐼𝑡𝑟 (𝑖 )) 𝑑𝑜 : 
 𝑖𝑓 𝑟𝑎𝑛𝑑 >𝑡 ℎ𝑟𝑒𝑠 ℎ𝑜𝑙𝑑 : 
  𝑃 𝐼𝑉
(𝑖 )[𝑗 ]=𝑃 𝐼𝑡𝑟 +1
(𝑖 )[𝑗 ] 
 𝑒𝑙𝑠𝑒 : 
  𝑃 𝐼𝑉
(𝑖 )[𝑗 ]=𝑃 𝐼𝑡𝑟 (𝑖 )[𝑗 ] 
 𝑗 =𝑗 +1 
The created 𝑃 𝐼𝑉
(𝑖 ) replaces the 𝑃 𝐼𝑡𝑟 (𝑖 ) if its fitness 
value is better.   Now, the modified version of the emperor 
penguin optimizer can be summarized as follows: 
𝐼𝑛𝑝𝑢𝑡 𝑡 ℎ𝑒 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑛 𝑠𝑖𝑧𝑒 (𝑁 ),𝑀𝑎𝑥𝐼𝑡𝑟 ,𝑎𝑛𝑑 𝑅 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒 𝑡 ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 

A Modified Emperor Penguin Optimizer Algorithm for Solving… Informatica 48 (2024) 79–94 83 
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑒𝑎𝑐 ℎ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑠𝑡𝑜𝑟𝑒  
𝑡 ℎ𝑒 𝑏𝑒𝑠𝑡 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝑃 𝑜𝑝𝑡 ) 
𝐼𝑡𝑟 =1 
𝑊 ℎ𝑖𝑙𝑒 𝐼𝑡𝑟 ≤𝑀𝑎𝑥𝐼𝑡𝑟 𝑑𝑜 : 
 𝑖 =1 
 𝑊 ℎ𝑖𝑙𝑒 𝑖 ≤𝑁 𝑑𝑜 : 
  
𝑇𝐴 =𝑇 −
𝑀𝑎𝑥𝐼𝑡𝑟 𝐼𝑡𝑟 −𝑀𝑎𝑥𝐼𝑡𝑟 
  𝐴 =𝑀 ×(𝑇𝐴 +|𝑃 𝑜𝑝𝑡 −𝑃 𝑖 |×𝑟𝑎𝑛𝑑 )−𝑇𝐴 
  
𝑆 = (√𝑓 .𝑒 −𝐼𝑡𝑟 /𝑙 −𝑒 −𝐼𝑡𝑟 )
2
 
  𝐷 =|𝑆 .  𝑃 𝐼𝑡𝑟 (𝑖 )−𝑟𝑎𝑛𝑑 𝑃 𝑜𝑝𝑡 | 
  𝑃 𝐼𝑡𝑟 +1
(𝑖 )=𝑃 𝐼𝑡𝑟 (𝑖 )−𝐴 .𝐷 
  𝑗 =1 
  𝑊 ℎ𝑖𝑙𝑒 𝑗 ≤dim(𝑃 𝐼𝑡𝑟 (𝑖 )) 𝑑𝑜 : 
   𝑖𝑓 𝑟𝑎𝑛𝑑 >𝑡 ℎ𝑟𝑒𝑠 ℎ𝑜𝑙𝑑 : 
    𝑃 𝐼𝑉
(𝑖 )[𝑗 ]=𝑃 𝐼𝑡𝑟 +1
(𝑖 )[𝑗 ] 
   𝑒𝑙𝑠𝑒 : 
    𝑃 𝐼𝑉
(𝑖 )[𝑗 ]=𝑃 𝐼𝑡𝑟 (𝑖 )[𝑗 ] 
   𝑗 =𝑗 +1 
  𝑖𝑓 𝑓 (𝑃 𝐼𝑡𝑟 +1
(𝑖 ))≤𝑓 (𝑃 𝑜𝑝𝑡 ) 𝑡 ℎ𝑒𝑛 : 
   𝑃 𝑜𝑝𝑡 =𝑃 𝐼𝑡𝑟 +1
(𝑖 ) 
  𝑖 =𝑖 +1 
 𝐼𝑡𝑟 =𝐼𝑡𝑟 +1 
𝑅𝑒𝑡𝑢𝑟𝑛 𝑃 𝑜𝑝𝑡 
The information vector proves efficiency in the 
proposed modification of the EPO algorithm that prevents 
stagnation in local optima, especially in the multi-model 
test optimization functions. EPO mainly uses a vector-
based methodology to deal with positions and modify 
them using its relocating procedure and the new proposed 
information vector creation process. In order to test the 
performance of the algorithm, 19 test optimization 
problems are selected from https://www.sfu.ca/~s 
surjano/optimization.html to be solved using the EPO 
algorithm. 
Figure 1 and Figure 2 show the 3D plots of the 19 
optimization functions. To evaluate the effectiveness of 
the modified EPO, a comparison is done with the classical 
EPO and particle swarm optimization (PSO) algorithms. 
These algorithms are implemented in Python, and the 
comparisons are conducted on a PC featuring a core-i5 
3.40 GHz CPU and 4 GB of memory. Table 3 shows the 
comparative results, where the highlighted values in the 
table prove the effectiveness of the modified EPO in terms 
of objective values and robustness. Furthermore, the 
Friedman test [17] is applied to prove that the null 
hypothesis is rejected, since the p-value for means is 
0.00093, and for standard deviations is 0.00064. The 
comparative results show that the modified EPO 
outperforms the other algorithms in 15 problems in terms 
of means and standard deviations. 
 
Table 2: 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 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) 
3 Booth (𝑥 1
+2𝑥 2
−7)
2
+(2𝑥 1
+𝑥 2
−5)
2
, 𝑥 𝑖 ∈[−10,10] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(1,3) 
4 Bukin 
100√|𝑥 2
−0.01𝑥 1
2
|+0.01|𝑥 1
+10|, 𝑥 𝑖 ∈[−15,3] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(−10,0) 
5 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) 
6 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) 

84 Informatica 48 (2024) 79–94 M.M. Eltoukhy et al. 
No. Function Name 𝑓 (𝑥 ) Global Minimum 
7 Discus 10
6
𝑥 1
2
+∑𝑥 𝑖 2
𝐷 𝑖 =2
,𝑥 𝑖 ∈[0,100] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(0,…,0) 
8 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 ) 
9 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) 
10 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) 
11 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) 
12 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) 
13 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] 
14 Rastrigin 10𝑑 + ∑[𝑥 2
−10𝑐𝑜𝑠 (2𝜋 𝑥 𝑖 )]
𝑑 𝑖 =1
, 𝑥 𝑖 ∈[−5,5] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=[0,…,0] 
15 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) 
16 Schwefel 418.9829𝑑 −∑𝑥 𝑖 𝑠𝑖𝑛 (√|𝑥 𝑖 |)
𝑑 𝑖 =1
, 𝑥 𝑖 ∈[−500,500] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=(420.9687,…,420.9687) 

A Modified Emperor Penguin Optimizer Algorithm for Solving… Informatica 48 (2024) 79–94 85 
No. Function Name 𝑓 (𝑥 ) Global Minimum 
17 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) 
18 Sphere ∑𝑥 𝑖 2
𝑛 𝑖 =1
,𝑥 𝑖 ∈[−5,5] 
𝑓 (𝑥 ∗
)=0,𝑥 ∗
=[0,…,0] 
19 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 comparative results of the modified EPO with the classical EPO and PSO in 19 test optimization 
problems 
Functions EPO mean EPO Classical mean PSO mean EPO std EPO Classical std PSO std 
Ackley 3.30 7.71 9.64 0.62 1.14 2.43 
Bohachevsky 0.22 0.23 0.01 0.20 0.21 0.02 
Booth 0.00 0.00 0.00 0.00 0.00 0.00 
Bukin 0.89 1.62 0.29 0.44 0.57 0.22 
Cross-in-Tray -2.06 -2.06 -2.06 0.00 0.00 0.00 
Drop Wave -1.00 -0.97 -0.99 0.00 0.03 0.03 
Discus 101.99 313.97 424.81 22.33 75.34 38.10 
Easom -0.98 -0.59 -1.00 0.00 0.30 0.00 
Eggholder -942.86 -933.63 -915.82 10.38 17.30 35.44 
Griewank 1.48 35.68 6.07 0.24 15.66 1.96 
Holder Table -19.21 -19.18 -19.21 0.00 0.01 0.01 
Michalewicz -8.29 -5.92 -5.12 0.22 0.52 0.65 
Modified 
Schwefel 
73.83 1574.72 1135.02 24.10 64.48 71.17 
Rastrigin 6.59 25.61 40.20 2.68 7.23 9.12 
Rosenbrock 11.81 49.92 2811.51 1.75 27.63 2926.26 
Schwefel 263.11 1953.86 2114.75 142.06 109.59 338.20 
six-hump -1.03 -1.03 -1.03 0.00 0.00 0.00 
Sphere 0.01 0.01 1.70 0.00 0.00 0.79 
Zakharov 2.58 11.62 50.02 1.27 6.37 24.62 
 

86 Informatica 48 (2024) 79–94 M.M. Eltoukhy et al. 
 
Figure 1: The plots of the first 10 optimization functions
 

A Modified Emperor Penguin Optimizer Algorithm for Solving… Informatica 48 (2024) 79–94 87 
 
Figure 2: The plots of the 11 to 19 optimization functions 
6 Modified EPO for solving FCTP 
To adapt EPO for solving FCTP, a priority rule that uses a 
weighted vector is developed in this paper. The length of 
the weighted vector equals the ordered product which 
consists of the set of all ordered pairs of supplies and 
demands. The transshipment problem involves transient 
nodes that can be considered for both supplies and 
demands simultaneously. Hence, the number of supply 
nodes (𝑆𝑁 ) is equal to the sum of the number of supplies 
and the number of transient nodes, while the number of 
demand nodes (𝐷𝑁 ) is equal to the sum of the number of  
 
demands and the number of transient nodes. So, the 
number of ordered pairs in our case herein can be 
calculated using equation (16): 
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑑𝑒𝑟𝑒𝑑 𝑝𝑎𝑖𝑟𝑠 =𝑆𝑁 ×𝐷𝑁 (16) 
 
Each ordered pair consists of two components. The 
first component is the supply node number, and the second 
component is the demand node number. By arranging 
these ordered pairs and solving the problem according to 
this arrangement, a heuristic solution can be obtained. As 
aforementioned, a proposed weighted vector is used to 

88 Informatica 48 (2024) 79–94 M.M. Eltoukhy et al. 
generate a priority rule for these ordered pairs, where each 
ordered pair has an associated weight, and the highest 
weighted ordered pair will be assigned first. Table 4 shows 
an example of the ordered product for a problem where the 
number of supplies equals three and the number of 
demands equals four. 
 
Table 4: An example of an ordered pair for a problem of 
three supplies and four demands 
Supplies Demands Ordered Pairs 
1 1 (1,1) 
1 2 (1,2) 
1 3 (1,3) 
1 4 (1,4) 
2 1 (2,1) 
2 2 (2,2) 
2 3 (2,3) 
2 4 (2,4) 
3 1 (3,1) 
3 2 (3,2) 
3 3 (3,3) 
3 4 (3,4) 
 
In the heuristic procedure, the arrangement of ordered 
pairs is to be arranged according to the weighted vector, 
which can be initially generated randomly for each 
ordered pair. For illustration, Table 5 shows an example 
of arranging these ordered pairs according to a weighted 
vector. 
 
Table 5: An example of using the weighted vector to 
rearrange the problem’s ordered pairs. 
Ordered Pairs Weights 
(3,3) 0.99 
(2,3) 0.90 
(1,1) 0.83 
(1,3) 0.71 
(3,2) 0.44 
(1,2) 0.43 
(3,1) 0.43 
(2,1) 0.37 
(3,4) 0.10 
(2,2) 0.05 
(2,4) 0.05 
(1,4) 0.00 
 
The proposed heuristic procedure of the algorithm can 
be implemented using the arranged ordered pairs, where it 
considers assigning the quantities of the problem 
according to the arrangement found by the weighted 
vector. The heuristic procedure now can be illustrated 
using the following pseudo code: 
 
𝑂𝑃
𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 = 𝑡 ℎ𝑒 𝑎𝑟𝑟𝑎𝑔𝑛𝑒𝑑 𝑜𝑟𝑑𝑟𝑒𝑑 𝑝𝑎𝑖𝑟𝑠 𝑤𝑖𝑡 ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡  
𝑡𝑜 𝑎 𝑤𝑒𝑖𝑔 ℎ𝑡𝑒𝑑 𝑣𝑒𝑐𝑡𝑜𝑟  
𝑆𝑒𝑡 𝑆 ={𝑠 𝑖 |𝑖 ∈𝐼 } 𝑎𝑛𝑑 𝐷 ={𝑑 𝑗 |𝑗 ∈𝐽 } 
𝐶𝑟𝑒𝑎𝑡𝑒 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑡 ℎ𝑎𝑡 𝑖𝑠 𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑖𝑧𝑒𝑑 𝑏𝑦 𝑧𝑒𝑟𝑜𝑠 𝑤𝑖𝑡 ℎ  
𝑆𝑁 𝑟𝑜𝑤𝑠 𝑎𝑛𝑑 𝐷𝑁 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 
𝐼𝑛𝑑𝑒𝑥 =0 
𝑊 ℎ𝑖𝑙𝑒 𝑂𝑃
𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 𝑖𝑠 𝑛𝑜𝑡 𝑒𝑚𝑝𝑡𝑦 𝑑𝑜 : 
 (𝑎 ,𝑏 )=𝑂𝑃
𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 [𝐼𝑛𝑑 𝑒 𝑥 ]  
 𝐼𝑓 𝑠 𝑎 =𝑑 𝑏 𝑡 ℎ𝑒𝑛 : 
  𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝑎 ,𝑏 )=𝑠 𝑎 
  𝑠 𝑎 =0 
  𝑑 𝑏 =0 
  
𝑂𝑃
𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 ={(𝑖 ,𝑗 )|𝑖 ∈𝐼 𝑎𝑛𝑑 𝑖 ≠𝑎 ,𝑗 ∈𝐽 𝑎𝑛𝑑 𝑗 ≠𝑏 } 
 𝐸𝑛𝑑 𝑖𝑓 
 𝐼𝑓 𝑠 𝑎 <𝑑 𝑏 𝑡 ℎ𝑒𝑛 : 
  𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝑎 ,𝑏 )=𝑠 𝑎 
  𝑑 𝑏 =𝑑 𝑏 −𝑠 𝑎 
  𝑠 𝑎 =0 
  𝑂𝑃
𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 ={(𝑖 ,𝑗 )|𝑖 ∈𝐼 𝑎𝑛𝑑 𝑖 ≠𝑎 ,𝑗 ∈𝐽 } 
 𝐸𝑛𝑑 𝑖𝑓 
 𝐼𝑓 𝑠 𝑎 >𝑑 𝑏 𝑡 ℎ𝑒𝑛 : 
  𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝑎 ,𝑏 )=𝑑 𝑏 
  𝑠 𝑎 =𝑠 𝑎 −𝑑 𝑏 
  𝑑 𝑏 =0 
  𝑂𝑃
𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 ={(𝑖 ,𝑗 )|𝑖 ∈𝐼 ,𝑗 ∈𝐽 𝑎𝑛𝑑 𝑗 ≠𝑏 } 
 𝐸𝑛𝑑 𝑖𝑓 
𝐸𝑛𝑑 𝑤 ℎ𝑖𝑙𝑒 
𝑅𝑒𝑡𝑢𝑟𝑛 𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 𝑎𝑛𝑑 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥  
 
The proposed modified Penguin Algorithm can now 
solve the problem by utilizing the previously mentioned 
heuristic procedure and a weighted vector. These 
weighted vectors represent the positions of the penguins. 
By modifying the weighted vector, new solutions can be 
obtained using the heuristic procedure. The heuristic 
procedure can now serve as the optimization function that 
needs to be optimized, with the weighted vectors 
representing the positions of the penguins. 
6.1 Computational complexity 
The algorithm initializes with a number of priority vectors 
equal to 𝑁 solutions with 𝑑 dimensions. So, the 
initialization process requires 𝑂 (𝑁 ×𝑑 ) . The heuristic 
procedure step count is less than 𝑑 , since not all of the 
ordered pairs are selected in the heuristic procedure. Thus, 
the heuristic procedure requires 𝑂 (𝑁 ×𝑑 ×𝑀𝑎𝑥𝐼𝑡𝑟 ) , 
hence it will repeat until the maximum number of 
iterations is reached. The step count of the rest of the 
functions required for calculating new positions in the 
EPO equals 𝑁 with 𝑘 formulas. So, it this requires 
𝑂 (𝑁 ×𝑘 ) . The total time complexity required for the 
algorithm is 𝑂 (𝑁 ×𝑑 ×𝑀𝑎𝑥𝐼𝑡𝑟 ×𝑘 ) and the space 
complexity is 𝑂 (𝑁 ×𝑑 ) , since the algorithm only works 

A Modified Emperor Penguin Optimizer Algorithm for Solving… Informatica 48 (2024) 79–94 89 
with population that initialized by 𝑁 solutions with 𝑑 
dimensions. 
6.2 Experimental design 
In order to obtain the optimal settings of the algorithm, an 
experimental design is done on both the hybrid EPO and 
PSO algorithms for solving the fixed charged 
transshipment problem. The code of the hybrid algorithms 
and the other codes related to the problem are coded using 
python and can be found in https://github.com 
/MZakaraia/EPO_Transshipment/. The selected problems 
for experimental design are generated using the generate 
problem’s function found in previously mentioned GitHub 
repository. The modified EPO algorithm has 3 parameters, 
which are 𝑀𝑎𝑥𝐼𝑡𝑟 , the population size (𝑃𝑜𝑝𝑆𝑖𝑧𝑒 ), and 
𝑅𝑎𝑑𝑖𝑢𝑠 . For each parameter, 4 levels are chosen as shown 
in Table 6. 
The full factorial design requires 4
3
×5=320 trails 
for 5 replicates. This number of experiments can be 
reduced using Taguchi’s orthogonal arrays. In order to 
select the convenient orthogonal array, the degrees of 
freedom should be calculated. So, the degree of freedom 
for such experiment is 1 of the overall mean and 3 for each 
parameter, which means the total degrees of freedom 
herein is 10. The most convenient orthogonal array for this 
experiment is 𝐿 16
(4
3
) . The 16 runs of the experimental 
design are implemented each 5 times to calculate the 
signal to noise ratio (𝑆𝑁𝑅 ) using equation (17) after 
normalizing the outputs. Figure 3shows the optimized 
parameter levels for each parameter, which is 20 
iterations, 20 penguins, and the radius should be equal 2. 
 
Table 6: Parameter levels for the modified EPO 
algorithm 
𝑀𝑎𝑥𝐼𝑡𝑟 𝑃𝑜𝑝𝑆𝑖𝑧𝑒 𝑅𝑎𝑑𝑖𝑢𝑠 
20 20 0 
50 50 -1 
70 70 1 
100 100 2 
 
  
 
Figure 3: The main effect plots of signal to ratio of EPO 
 
𝑆𝑁 𝑅 =10log(
𝜇 2
𝜎 2
) 
(17) 
 
For the hybridized PSO algorithm, there are 5 
parameters, which are 𝑀𝑎𝑥𝐼𝑡𝑟 , 𝑃𝑜𝑝𝑆𝑖𝑧𝑒 , Inertia weight, 
personal weight, and the global weight. The proposed 
levels for each parameter are found in Table 7. 
 
Table 7: Parameter levels for the PSO algorithm 
𝑀𝑎𝑥𝐼𝑡𝑟 𝑃𝑜𝑝𝑆𝑖𝑧𝑒 
Inertia 
weight 
personal 
weight 
global 
weight 
20 20 0.1 0.1 0.1 
50 50 0.3 0.3 0.3 
70 70 0.5 0.5 0.5 
100 100 0.6 0.6 0.6 
 
 
 
 
The required number of trails for the full factorial 
design for the hybrid PSO according to the levels in Table 
7 is 4
5
×5=5120 trails. The Taguchi’s orthogonal 
arrays again can be used to reduce this number using the 
𝐿 16
(4
5
) , where the number of trails is 80 trails for 5 
replicates. Figure 4 shows the optimized parameter levels 
for the hybrid PSO algorithm, which are 20 iterations, 20 
particles, 0.1 for the inertia weight, 0.6 for personal 
weight, and 0.6 for global weight. The optimized 
parameter levels are to be used in the computational 
results section to show the comparative results between 
the EPO and PSO algorithms. 
 
 
Figure 4: The main effect plots of signal to ratio of PSO 

90 Informatica 48 (2024) 79–94 M.M. Eltoukhy et al. 
6.3 Computational results 
This section presents the implementation of the modified 
EPO algorithm for solving 30 generated problems, which 
can be found at https://github.com/MZakaraia/EPO_T 
transshipment. The problem sizes cover three different 
forms of transshipment problem sizes: 3×3×2, 4×4×
3, and 5×5×4. All these problems are generated using 
the generate problems function in the Transshipment.py 
file. This function allows for generating more fixed-
charge transshipment problems with different sizes. 
Therefore, the included problems are considered 
benchmarks for future comparisons. 
 
 
 
Both the EPO and PSO algorithms were implemented 
to solve the 30 problems using the optimized parameter 
levels found through the experimental design. The 
convergence curves for the problems are shown in Figure 
5, 6, and 7. Table 8 presents the comparative results. The 
Wilcoxon test [18] was performed on selected metrics 
(mean, standard deviation, maximum, minimum) between 
the EPO and PSO results. The p-value for each metric 
indicates rejection of the null hypothesis since the p-value 
for means is 1.89×10
−9
 and for standard deviations is 
0.00046. Therefore, the comparative results in Table 8 
conclude that the proposed EPO algorithm outperforms 
the PSO algorithm in terms of mean results and robustness 
for solving fixed-charge transshipment problems. 
 
Table 8: The comparative results of the 30 fixed charged transshipment problems between the modified EPO and PSO 
Problem 
EPO 
Mean 
PSO 
Mean 
EPO Std PSO Std 
EPO 
Max 
PSO 
Max 
EPO 
Min 
PSO 
Min 
3X3X2_0 41797 42206.3 92.29084462 508.4504007 42031 43136 41719 41719 
3X3X2_1 40680.5 41229.8 267.1895395 567.3982376 41237 42530 40547 40547 
3X3X2_2 33693.5 34348.6 88.86534758 572.6377913 33960 35163 33663 33663 
3X3X2_3 29758.4 30133 144.9497844 424.4848643 30117 30874 29688 29688 
3X3X2_4 42124.7 42336.3 153.5689096 240.9514681 42445 42713 42030 42030 
3X3X2_5 37551 37904.9 184.3805847 383.2465134 37908 38456 37432 37432 
3X3X2_6 38157 38436.7 23.37947818 247.5019394 38198 38968 38142 38180 
3X3X2_7 33565 34001.8 0 411.3178333 33565 34920 33565 33565 
3X3X2_8 30994 31619.5 0 645.4463959 30994 32779 30994 30994 
3X3X2_9 31027.1 31353.8 102.3 298.0412052 31334 31873 30993 30993 
4X4X3_0 47553.3 49048.4 632.032602 1151.657953 49443 50903 47324 47324 
4X4X3_1 58361.5 59400.7 128.9234269 351.1797403 58663 60087 58217 58880 
4X4X3_2 60169.8 60771.1 813.7788152 594.8965372 61489 61697 59351 59667 
4X4X3_3 71064.5 72923.7 1106.743895 620.5789313 73668 73803 70269 71847 
4X4X3_4 65386 67792.3 762 1303.083961 67672 71176 65132 65740 
4X4X3_5 64932.1 65240.9 945.1474435 811.3807306 66407 67011 63926 64370 
4X4X3_6 56582.9 57742.8 95.05519449 869.7954702 56853 59901 56538 56634 
4X4X3_7 64673.8 65989.4 242.1808415 599.2033378 65260 66880 64432 65080 
4X4X3_8 64801.4 66226.3 598.4196187 632.1170857 66091 67494 64249 65245 
4X4X3_9 64173.9 65578 452.4350672 450.8871256 65236 66499 63846 64718 
5X5X4_0 99127 101002 525.5029971 1238.38217 99858 103173 98290 98948 
5X5X4_1 103558.6 105635.4 971.9633944 1316.379824 105953 107749 102571 104127 
5X5X4_2 118153.9 120260 500.3739502 1535.788136 119108 123992 117548 118137 
5X5X4_3 100174.3 103129.5 222.7321486 1607.177977 100741 107405 99905 100916 
5X5X4_4 105533.9 107772.6 563.3280483 1053.34811 106665 110543 104657 106575 
5X5X4_5 101425.3 103906.4 165.7950844 1003.615285 101794 105193 101214 102351 
5X5X4_6 97936.6 100878.4 1137.296901 2000.878567 100376 104848 96986 97753 
5X5X4_7 106153.2 107832.2 1274.031224 757.3083652 108707 108989 105099 106449 
5X5X4_8 86597.7 87972.9 1604.486089 1036.310229 89020 89773 84759 86412 
5X5X4_9 92603.3 94816.2 307.2627703 1282.450217 93358 97327 92113 92515 

A Modified Emperor Penguin Optimizer Algorithm for Solving… Informatica 48 (2024) 79–94 91 
 
Figure 5: The convergence curve of the 3×3×2 problems 
 
 
Figure 6: The convergence curve of the 4×4×3 problems 
 

92 Informatica 48 (2024) 79–94 M.M. Eltoukhy et al. 
 
Figure 7: The convergence curve of the 5×5×4 problems
 
7 Conclusion 
In conclusion, this paper introduced a modified 
Emperor Penguin algorithm tailored for solving FCTP. 
The algorithm demonstrated its effectiveness in finding 
high-quality solutions by utilizing new benchmarks 
specifically designed for the problem. The computational 
results presented in this study provide valuable insights 
into the algorithm's performance. The mean results 
showcased the algorithm's ability to achieve competitive 
solutions for the FCTP, while the standard deviation and 
Relative Standard Deviation offered measures of its 
robustness. The findings of this research contribute to the 
field of logistics and supply chain management by 
offering an optimized algorithmic approach for addressing 
the FCTP. The modified Emperor Penguin algorithm, with 
its robustness and improved solution quality, holds great 
potential for enhancing supply chain operations and 
optimizing transshipment processes.  
Future research directions may involve further fine-
tuning of the modified algorithm and expanding the 
benchmark suite to encompass a wider range of real-world 
scenarios. Additionally, investigating the algorithm's 
performance on larger-scale instances and exploring its 
applicability to other related optimization problems would 
be beneficial. The future research also may include 
extending the formulations of the transshipment problem 
to cover: 
 
 
• The solid transshipment problem by including 
constraints related to the type of transportation 
and products. 
• The capacitated fixed charged transshipment by 
considering capacity constraints related to each 
transshipment node.  
In conclusion, this paper's findings highlight the 
promising capabilities of the modified Emperor Penguin 
algorithm for tackling the Fixed Charged Transshipment 
Problem, providing a valuable tool for optimizing supply 
chain operations and fostering efficiency in logistics 
management.  
 
Acknowledgement 
This work was funded by the University of Jeddah, 
Jeddah, Saudi Arabia, under grant No. (UJ-02-068-DR). 
The authors, therefore, acknowledge with thanks the 
University of Jeddah technical and financial support. 
 
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