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Article

WOA: Wombat Optimization Algorithm for Solving Supply Chain Optimization Problems

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Faculty of Engineering, Liwa College, Abu Dhabi 41009, United Arab Emirates
2
Faculty of Business, Liwa College, Abu Dhabi 41009, United Arab Emirates
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Department of Power and Control Engineering, School of Electrical and Computer Engineering, Shiraz University, Shiraz 71557-13876, Iran
4
Research Laboratory in Economics, Management, and Business Management (LAREGMA), Faculty of Economics and Management, Hassan I University, Settat 26002, Morocco
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(7), 1059; https://doi.org/10.3390/math12071059
Submission received: 19 February 2024 / Revised: 21 March 2024 / Accepted: 24 March 2024 / Published: 1 April 2024
(This article belongs to the Special Issue Innovations in Optimization and Operations Research)

Abstract

:
Supply Chain (SC) Optimization is a key activity in today’s industry with the goal of increasing operational efficiency, reducing costs, and improving customer satisfaction. Traditional optimization methods often struggle to effectively use resources while handling complex and dynamic Supply chain networks. This paper introduces a novel biomimetic metaheuristic algorithm called the Wombat Optimization Algorithm (WOA) for supply chain optimization. This algorithm replicates the natural behaviors observed in wombats living in the wild, particularly focusing on their foraging tactics and evasive maneuvers towards predators. The theory of WOA is described and then mathematically modeled in two phases: (i) exploration based on the simulation of wombat movements during foraging and trying to find food and (ii) exploitation based on simulating wombat movements when diving towards nearby tunnels to defend against its predators. The effectiveness of WOA in addressing optimization challenges is assessed by handling the CEC 2017 test suite across various problem dimensions, including 10, 30, 50, and 100. The findings of the optimization indicate that WOA demonstrates a strong ability to effectively manage exploration and exploitation, and maintains a balance between them throughout the search phase to deliver optimal solutions for optimization problems. A total of twelve well-known metaheuristic algorithms are called upon to test their performance against WOA in the optimization process. The outcomes of the simulations reveal that WOA outperforms the other algorithms, achieving superior results across most benchmark functions and securing the top ranking as the most efficient optimizer. Using a Wilcoxon rank sum test statistical analysis, it has been proven that WOA outperforms other algorithms significantly. WOA is put to the test with twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems to showcase its ability to solve real-world optimization problems. The results of the simulations demonstrate that WOA excels in real-world applications by delivering superior solutions and outperforming its competitors.

1. Introduction

Supply Chain Management (SCM) plays a vital role in the success and competitiveness of modern businesses by ensuring the seamless flow of goods and services from suppliers to the final customer [1]. The complexity of supply chain networks, with multiple overlapping firms, dynamic demand patterns, and resource constraints, poses significant challenges for optimal performance [2]. Supply chain quality is essential for businesses to increase efficiency, reduce costs, and adapt to market demand that changes at every moment [3,4].
Various studies have been presented in the literature in the field of SCM. A novus multi-objective three-stage supply chain network is proposed, which aims to optimize sustainability, resiliency and responsive measures, simultaneously [5]. To uncover optimal strategic and planning choices, a multi-objective mixed integer programming framework is proposed to analyze Supply Chains from environmental, economic, and technological standpoints, while handling multiple periods, products, supply zones, and various feedstocks [6]. A behavioral three-way decision making model with a Fermatean fuzzy Mahalanobis distance is proposed to deal with SCM problems [7]. For supply chain management in hesitant situations, a new model based on an interval type-2 Pythagorean fuzzy set is proposed [8]. A multi-objective, multi-product, and multi-period two-stage sustainable opened- and closed-loop supply chain plan is proposed to maintain the supply among production centers and various hospitals during the COVID-19 pandemic situation [9].
One of the fundamental aspects of SCM is the integration of key processes and functions within and across organizations. Effective SCM requires collaboration not only among different departments within a company but also with external partners such as suppliers, manufacturers, distributors, and retailers. This collaborative approach enables organizations to streamline operations, reduce costs, improve quality, and respond quickly to changing market demands [10]. Furthermore, advancements in technology have revolutionized SCM practices, enabling a real-time visibility and control over supply chain activities. For instance, the use of data analytics, Artificial Intelligence, and the Internet of Things (IoT) has enabled companies to gather and analyze vast amounts of data, leading to better forecasting accuracy, inventory optimization, and decision-making. These technological innovations have transformed SCM into the “Triple-A Supply Chain”—agile, adaptable, and aligned with the overarching business strategy [11]. SCM plays a pivotal role in driving operational efficiency, enhancing customer satisfaction, and creating a competitive advantage. By embracing collaboration, leveraging technology, and addressing emerging challenges, organizations can build resilient, sustainable, and agile supply chains that are well-positioned to thrive in an increasingly complex and interconnected world.
Traditional optimization techniques such as linear programming and mathematical modeling have been widely used for supply chain optimization [12]. However, these methods often struggle to cope with the inherent complexity and uncertainty of real-world supply chains. Consequently, there is an increased interest in developing metaheuristic algorithms that take inspiration from nature to efficiently solve Supply Chain Optimization problems [13]. Among these bio-inspired optimization methods, swarm intelligence-based algorithms have shown promise due to their ability to mimic collective biological systems. One such algorithm inspired by the feeding behavior of the honeybee swarm is Bee Swarm Optimization (BSO). BSO mimics the spatially embedded cooperation and decision-making processes found in honey bee colonies, making it ideally suited for optimization projects characterized by large solution areas and dynamic environments [14].
Optimization problems involve multiple possible solutions, with the goal of finding the most optimal solution from all the options available [15]. These problems are typically represented mathematically through decision variables, constraints, and an objective function. The primary objective in optimization is to assign values to the decision variables in a way that maximizes or minimizes the objective function while satisfying the problem’s constraints [16]. Problem-solving methods for optimizing solutions can fall into deterministic and stochastic categories [17].
Although deterministic approaches excel in solving convex, differentiable, continuous, linear, and low-dimensional optimization problems, they struggle when faced with complex optimization problems, particularly in high dimensions where they may become trapped at local optima [18]. In real-world applications within mathematics, engineering, and other fields, many optimization problems are complex, non-convex, non-derivative, discontinuous, nonlinear, and high-dimensional. The limitations and challenges of deterministic approaches in handling such optimization problems have prompted researchers to develop stochastic approaches [19].
Metaheuristic algorithms are recognized as efficient stochastic methods capable of generating optimal solutions for optimization problems through random search within the problem-solving space. Employing random operators and trial-and-error processes, these algorithms initially create a set of candidate solutions as the algorithm population. Through iteration and algorithm update steps, the positions of population members are altered to enhance these candidates, ultimately leading to the identification of the most optimal solution encountered throughout the algorithm’s execution [20]. The randomness involved in the process of metaheuristic algorithms implies that reaching the global optimum is not guaranteed. However, the results achieved through these algorithms are deemed acceptable as they are situated near the global optimum. Hence, these outcomes are referred to as quasi-optimal solutions. Enhancing these quasi-optimal solutions to be closer to the global optimum stands as a significant goal in the realm of metaheuristic algorithms and optimization. Consequently, this pursuit serves as the primary driving force for researchers in their quest to create numerous innovative metaheuristic algorithms [21].
Effectively managing a random search at both global and local levels is essential for the success of metaheuristic algorithms in optimization processes. Global search involves a deep exploration of the problem-solving space to avoid being trapped in local optima and in order to pinpoint the main optima region. On the other hand, a local search focuses on exploitating the near solutions and potential areas within the problem-solving space to achieve improved outcomes closer to the global optimum. To handle the stochastic search effectively, a metaheuristic algorithm must strike a balance between exploration and exploitation throughout the problem-solving process [22].
The main research question is whether new metaheuristic algorithms are still necessary despite the variety that already exists. The No Free Lunch (NFL) theorem [23] addresses this concern by stating that there is no one universal metaheuristic algorithm that can outperform all others for every optimization problem. This means that the effectiveness of a metaheuristic algorithm for one set of optimization challenges does not guarantee a similar success for a different set. Therefore, the outcome of implementing a metaheuristic algorithm in an optimization scenario cannot be predicted as a definite success or failure. As per the NFL theorem, it is possible for an algorithm to achieve the best solution when dealing with one optimization problem, but it may get trapped in a local optimum when faced with another problem. The NFL theorem promotes ongoing research in metaheuristic algorithms, pushing researchers to develop innovative approaches to solving optimization problems effectively. This theorem has inspired the authors of this paper to introduce a novel metaheuristic algorithm for addressing optimization challenges.
Motivated by the NFL theorem, the aspects of innovation, novelty, and originality of this paper are in introducing a new metaheuristic algorithm called Wombat Optimization Algorithm (WOA) that imitates the behavior of wombats in their habitat.
What is evident from the best knowledge obtained from the literature review, so far, is that no metaheuristic algorithm has been designed based on the simulation of wombats’ natural behaviors. Meanwhile, the activity of foraging and the escaping strategy from predators are intelligent processes among wombats that have a special potential in the design of a new optimizer. In order to address this research gap in the studies of metaheuristic algorithms, in this paper, a new biomimetics metaheuristic algorithm is designed based on the mathematical modeling of wombats’ natural behaviors in the wild.
The key contributions of this paper are as listed:
  • WOA is designed based on simulating wombat’s natural behaviors in the wild.
  • The basic inspiration of WOA is taken from the foraging of the wombat and the strategy of this animal when escaping from its predators.
  • The theory of WOA is expressed and mathematically modeled in two phases: (i) the exploration based on the simulation of wombat movements during foraging and (ii) the exploitation based on simulating wombat movements when it dives towards nearby tunnels to defend against its predators.
  • The capability of WOA in optimization applications has been evaluated in the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100.
  • WOA’s ability to tackle optimization tasks in real-world applications has been evaluated on twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems.
  • Two well-known metaheuristic algorithms are employed to compare with the performance of WOA.
This paper is organized in the following manner: firstly, a literature review is outlined in Section 2. Next, the proposed Wombat Optimization Algorithm (WOA) is introduced and outlined in Section 3. Section 4 covers simulation studies and their results. The efficacy of WOA in addressing real-world applications is explored in Section 5. Lastly, conclusions and recommendations for future research are detailed in Section 6.

2. Literature Review

Metaheuristic algorithms have drawn on influences from various natural sources, including swarm behavior in the animal kingdom, principles of biology and genetics, concepts from the field of physics, studies of human behavior, and evolutionary occurrences. These algorithms can be categorized into five distinct groups based on the design inspiration they exhibit: swarm-based, evolutionary-based, physics-based, human-based, and game-based approaches.
Swarm-based metaheuristic algorithms are inspired by the natural swarming behaviors of various animals in their design, including insects, birds, aquatic creatures, reptiles, and other wildlife species. Among the most famous swarm-based metaheuristic algorithms are: Ant Colony Optimization (ACO) [24], Particle Swarm Optimization (PSO) [25], Firefly Algorithm (FA) [14,26], and Artificial Bee Colony (ABC) [27]. ACO used in its design the skill of ants in finding the optimal communication path between the colony and the food source. PSO is inspired in its design by the search process in the movement of fish and birds with the aim of finding food sources. FA is developed inspired by the optical communication and information exchange between fireflies. The butterfly optimization algorithm (BA) is an advanced metaheuristic for optimization created by Arora that draws inspiration from biology, which is centered on the behavior of butterflies when they are seeking food [28]. ABC is designed based on modeling the hierarchical cooperation and activities of colony bees to obtain food resources. Mantis Search Algorithm (MSA) is developed based on modeling the sexual cannibalism and hunting behavior of praying mantises [29]. Genghis Khan Shark Optimizer (GKSO) is inspired, in its design, by the Genghis Khan shark’s hunting and self-defense strategy in the wild [30]. Gazelle Optimization Algorithm (GOA) uses, in its design, gazelles’ survival ability in their predator-dominated environment [31]. Among natural behaviors in wildlife, foraging, hunting strategy, survival efforts, chasing processes, digging, and migration are very significant; they are employed as sources of inspiration in the design of algorithms such as: Whale Algorithm (WA) [32], Reptile Search Algorithm (RSA) [33], Orca Predation Algorithm (OPA) [34], Grey Wolf Optimizer (GWO) [35], Tunicate Swarm Algorithm (TSA) [36], African Vultures Optimization Algorithm (AVOA) [37], Marine Predator Algorithm (MPA) [38], and White Shark Optimizer (WSO) [39].
Evolutionary metaheuristic algorithms draw upon principles from biology and genetics, incorporating ideas such as natural selection, survival of the fittest, Darwin’s theory of evolution, and other related evolutionary principles in their design. Among the most popular evolutionary-based metaheuristic algorithms are Genetic Algorithm (GA) [40] and Differential Evolution (DE) [41]. GA and DE use, in their design, a simulation of the reproduction process, and apply genetic and biological concepts such as mutation, selection, and crossover. The basic inspiration in the design of Artificial Immune System (AIS) comes from the mechanism of the human body’s defense system against microbes and diseases [42]. Some other evolutionary-based metaheuristic algorithms are: Cultural Algorithm (CA) [43], Genetic programming (GP) [44], and Evolution Strategy (ES) [45].
Physics-based metaheuristic algorithms are use their design of phenomena, transformations, processes, forces, laws, and other concepts of physics. Simulated Annealing (SA) is one of the most widely used physics-based metaheuristic algorithms. The main idea in the design of SA comes from the physical transformations in the metal annealing process, where with the aim of achieving the ideal crystal, the metals are first melted under heat, then slowly cooled and frozen [46]. Gravitational Search Algorithm (GSA) [47] is inspired by modeling the gravitational attraction between objects at different distances. Fick’s Law Optimization (FLA) is designed based on the modeling of Fick’s first law of diffusion, which according to this law, molecules tend to diffuse from higher to lower concentration areas [48]. Rime (RIME) algorithm is designed with inspiration from the physical phenomenon of rime-ice [49]. Some other physics-based metaheuristic algorithms are: Thermal Exchange Optimization (TEO) [50], Water Cycle Algorithm (WCA) [51], Nuclear Reaction Optimization (NRO) [52], Henry Gas Optimization (HGO) [53], Lichtenberg Algorithm (LA) [54], Multi-Verse Optimizer (MVO) [55], Electro-Magnetism Optimization (EMO) [56], Black Hole Algorithm (BHA) [57], and Archimedes Optimization Algorithm (AOA) [58].
Human-based metaheuristic algorithms use, in their design, behaviors, choices, thoughts, decisions, teaching and learning processes, and other human activities in individual and social life. Teaching-Learning Based Optimization (TLBO) [59] can be mentioned among the most widely used human-based metaheuristic algorithms. TLBO uses the modeling of educational communication and knowledge exchange in the classroom environment between the teacher and students and students with each other. Mother Optimization Algorithm (MOA) is proposed with inspiration from Eshrat’s care of her children [60]. Mountaineering Team-Based Optimization (MTBO) is developed with the inspiration of social behavior and human cooperation needed to reach a mountaintop [61]. Deep Sleep Optimizer (DSO) is developed inspired by the sleeping patterns of humans and is based on modeling the fall and rise of homeostatic pressure during the human sleep process [62]. Some other human-based metaheuristic algorithms are: Gaining Sharing Knowledge based Algorithm (GSK) [63], Fireworks Algorithm (FA) [64], War Strategy Optimization (WSO) [65], Coronavirus Herd Immunity Optimizer (CHIO) [66], and Ali Baba and the Forty Thieves (AFT) [67].
Game-based metaheuristic algorithms use, in their design, the rules of individual and team games as well as the behavior of players, coaches, runners, and other influential people in these games. Volleyball Premier League (VPL) [68] and Running City Game Optimizer (RCGO) [69] are examples of game-based metaheuristic algorithms.
Supply chain management (SCM) has evolved from a simple logistical process to a critical strategic tool for businesses worldwide. It involves the coordination of various activities, including procurement, production, inventory management, and distribution, to ensure the smooth flow of goods and services from suppliers to end customers. This holistic approach to managing the flow of materials, information, and finances across the entire supply chain has become indispensable in today’s competitive business environment [70]. With the increasing complexity and dynamic nature of modern supply chains, traditional optimization techniques often fall short in providing optimal solutions within reasonable time frames. This has led to the emergence of metaheuristic algorithms as powerful optimization tools capable of addressing the challenges posed by real-world SCM problems [71]. Unlike traditional optimization methods, which may get stuck in local optima, metaheuristics offer robust and flexible approaches for finding high-quality solutions in a reasonable amount of time [72]. The application of metaheuristic algorithms in SCM is vast and encompasses various areas such as inventory management, facility location, vehicle routing, production scheduling, and supply chain network design. For example, metaheuristic algorithms can be used to optimize inventory replenishment policies, minimize transportation costs, balance production capacities, and design resilient supply chain networks [73].
An overview of the applications of metaheuristic algorithms in dealing with SCM is presented in Table 1.

3. Wombat Optimization Algorithm

In this section, first, the basic inspiration employed in designing the proposed Wombat Optimization Algorithm (WOA) approach is described, then its implementation steps are mathematically modeled.

3.1. Inspiration of WOA

A wombat is a short-legged, muscular quadrupedal marsupials of the family Vombatidae, which is native to Australia [98]. This animal is adaptable and habitat tolerant, and lives in heathland, mountainous, and forested areas of eastern and southern Australia, as well as in an isolated patch in Epping Forest National Park in central Queensland [99]. The appearance characteristics of wombats are as follows: their fur color can vary from gray to black or from a sandy color to brown. Wombats are about 1 m long and weigh between 17 and 39 kg [99,100]. An image of the wombat is shown in Figure 1.
The wombat is an herbivore animal whose main diet is grass and also feeds on roots, bark, herbs, and sedges. Wombats have adapted to survive in habitats with limited nutrition and low food availability. Using the burrows permits wombats to maximize their foraging range and creates a thermally stable environment that helps in the maintenance of a low metabolic rate. Their energy-efficient foraging strategies and low metabolic rates lead to very low energy requirements, allowing them to survive on low-quality food and inhabit areas of scarce food availability. Although wombats can subsist on low-quality diets, their survival still relies on the availability of a habitat that provides adequate nutrition, as nutritional stress increases the risk of illness and death for animals [101].
Wombats have great skill and ability in digging. They dig extensive burrow systems with their powerful claws and rodent-like front teeth. The backward-facing pouch is a special distinctive adaptation in wombats. This pouch has the advantage of the wombat not collecting soil on its young in the pouch while digging. In addition to the effect of an increased foraging range, these excavated tunnels are a suitable defensive position for wombats when they are attacked by their predators. Wombats are slow animals and move slowly; however, when attacked and threatened, they run away at a speed of 40 km/h. Tasmanian and Dingos are the main predators of wombats. When a wombat is attacked by a predator, it dives into one of the nearby tunnels and uses its rump to stop the pursuing predator. The main defense of wombats is their toughened rear hide, with most of the posterior made of cartilage. This feature, as well as not having a substantial tail, makes the stalking predator who enters the tunnel unable to bite the wombat and hunt it [99].
Among the wombat’s natural behaviors and lifestyles in the wild, two stand out the most: (i) their extensive foraging activity in the wild and (ii) their escape strategy from predators by diving into nearby tunnels. The mathematical modeling of these intelligent processes in the wombat lifestyle was employed in order to design a new metaheuristic algorithm called Wombat Optimization Algorithm (WOA), which is presented below.

3.2. Algorithm Initialization

The proposed WOA approach is a population-based metaheuristic algorithm, where wombats form the population members of the algorithm. In order to visualize and create a mentality, the habitat of wombats in the wild corresponds to the problem-solving space, and the position of each wombat in this habitat corresponds to the position of a candidate solution in the problem-solving space. Each wombat determines values for decision variables based on its position in the problem-solving space. Therefore, each wombat as a WOA member corresponds to a candidate solution to the problem that can be mathematically modeled using a vector. The community of wombats in the form of these vectors together makes the WOA population, which can be mathematically modeled using a matrix according to Equation (1). At the beginning of the implementation of the algorithm, the position of each wombat in the problem-solving space is randomly initialized using Equation (2).
X = X 1 X i X N N × m = x 1,1 x 1 , d x 1 , m x i , 1 x i , d x i , m x N , 1 x N , d x N , m N × m
x i , d = l b d + r · ( u b d l b d )
Here, X is the WOA population matrix, X i is the i th wombat (i.e., candidate solution), x i , d is its d th dimension in the search space (i.e., decision variable), N is the number of wombat population, m is the number of decision variables, r is a random number in interval 0 ,   1 , and l b d and u b d are the lower bound and upper bound of the d th decision variable, respectively.
As stated, the position of each wombat represents a candidate solution to the problem. Therefore, the objective function of the problem corresponding to each wombat can be evaluated. The set of evaluated values for the objective function of the problem can be represented using a vector using Equation (3).
F = F 1 F i F N N × 1 = F ( X 1 ) F ( X i ) F ( X N ) N × 1
Here, F is the vector of the evaluated objective function and F i is the evaluated objective function based on the i th wombat.
The evaluated values for the objective function are considered as a suitable criterion for measuring the quality of candidate solutions and population members. Thus, the best evaluated value for the objective function corresponds to the best WOA member, and the worst value obtained for the objective function corresponds to the worst WOA member. Since the position of the wombats in the problem-solving space is updated in each iteration, the best member of the population must also be updated in each iteration.

3.3. Mathematical Modelling of WOA

The proposed approach of WOA is able to provide suitable solutions for optimization problems in an iteration-based process based on the searching power of its members in the problem-solving space. The design of WOA was inspired by the natural behavior of wombats in nature. In the WOA design, the position of the wombats in the problem-solving space is updated in each iteration in two phases: (i) an exploration based on the simulation of the foraging process of the wombats and (ii) an exploitation based on the simulation of the strategy of the wombats escaping from their predators towards the tunnels. The full description and mathematical model of each of these phases of updating the position of wombats in the problem-solving space is presented below.

3.3.1. Phase 1: Foraging Process (Exploration Phase)

In the first phase of WOA, the position of wombats in the problem-solving space is updated based on the simulation of this animal’s foraging strategy. The wombat is a herbivorous animal that has a high searching power for finding forage in a wide range of its habitat. Modeling the position change of the wombat while moving towards forage leads to the creation of extensive changes in the position of WOA members in the problem-solving space and, as a result, increasing the exploration power of the algorithm in order to manage the global search. In the WOA design, for each wombat, the position of other population members that have a better value for the objective function is considered as the forage position. The set of forage positions for each wombat is identified using Equation (4).
C F P i = X k : F k < F i   a n d   k i ,   w h e r e   i = 1,2 ,   ,   N   a n d   k 1,2 ,   ,   N
Here, C F P i is the set of candidate forage positions for the i th wombat, X k is the population member with a better objective function value than the i th wombat, and F k is its objective function value.
In the design of WOA, it is assumed that the wombat chooses one of these fodder positions completely randomly and moves towards it. Based on the modeling of the wombat’s movement towards the selected forage in this foraging process, a new position for each WOA member is calculated using Equation (5). This new position, if it leads to the improvement of the objective function value, replaces the previous position of the corresponding member using Equation (6).
x i , j P 1 = x i , j + r i , j   · ( S F P i , j I i , j · x i , j ) ,  
X i = X i P 1 , F i P 1 F i , X i , e l s e ,
Here, S F P i is the selected forage position for the i th wombat, S A F P i , j is its j th dimension, X i P 1 is the new position calculated for the i th wombat based on the foraging phase of the proposed WOA, x i , j P 1 is its j th dimension, F i P 1 is its objective function value, r i , j are random numbers from the interval 0 ,   1 , and I i , j are numbers which are randomly selected as 1 or 2.

3.3.2. Phase 2: Escape Strategy (Exploitation Phase)

In the second phase of WOA, the position of wombats in the problem-solving space is updated based on the simulation of the escape strategy of this animal against the attacks of its predators. The wombat, with its high digging ability, makes many tunnels in its habitat. When the wombat is in danger and attacked by a predator, it escapes by diving towards one of the tunnels located near it and tries to save itself. Modeling the wombat’s position change while escaping from the predator towards the tunnel leads to small changes in the position of the WOA members in the problem-solving space, and, as a result, increases the exploitation power of the algorithm in order to manage the local search.
In the WOA design based on the modeling of the wombat’s position change and dive towards the nearby tunnel, a new position for each member of the WOA is calculated using Equation (7). This new position, if it leads to the improvement of the value of the objective function, replaces the previous position of the corresponding member using Equation (8).
x i , j P 2 = x i , j + 1 2   r i , j ·   u b j l b j t  
X i = X i P 2 , F i P 2 F i X i , e l s e
Here, X i P 2 is the new position calculated for the i th wombat based on the escape phase of the proposed WOA, x i , j P 2 is its j th dimension, F i P 2 is its objective function value, r i , j are random numbers from the interval 0 ,   1 , and t is the iteration counter.

3.4. Repetition Process, Pseudocode, and Flowchart of WOA

The first iteration of WOA is completed after updating the position of all wombats in the problem-solving space based on foraging and escape phases. After that, with the updated values, the algorithm enters the next iteration; the process of updating the position of wombats in the problem-solving space continues until the last iteration of the algorithm based on Equations (4)–(8). In each iteration, the position of the best wombat is identified and stored as the best candidate solution. After the full implementation of the algorithm, the best candidate solution obtained during the iterations of the algorithm is presented as the WOA solution for the given problem.
Different criteria can be considered as the stopping condition of the algorithm. Among these criteria, we can mention: (i) the maximum number of algorithm iterations—in this case, the algorithm stops after passing the specified number of iterations; (ii) the maximum number of objective function evaluations (FEs)—in this case, after the number of evaluations of the objective function reaches the maximum number of FEs (MFEs), which is specified at the beginning of the implementation, the algorithm stops; (iii) the error determined between the successive solutions obtained—in this case, when the difference between the solutions obtained during several iterations of the algorithm is very small, based on the comparison of this difference with a “specified value”, the algorithm stops.
The implementation steps of WOA are shown as a flowchart in Figure 2, and the WOA pseudocode is presented in Algorithm 1.
Algorithm 1. Pseudocode of WOA
Start WOA.
1.Input problem information: variables, objective function, and constraints.
2.Set WOA population size (N) and iterations (T).
3.Generate the initial population matrix at random using Equation (2). x i , d l b d + r · ( u b d l b d )
4.Evaluate the objective function.
5. For t = 1 to T
6. For i = 1 to N
7. Phase 1: foraging process (exploration phase)
8. Determine the candidate foraging positions set for the ith wombat using Equation (4). C F P i X k i : F k i < F i   a n d   k i i
9. Select the target foraging position for the ith wombat at random.
10. Calculate new position of ith wombat using Equation (5). x i , d P 1 x i , d + r · S F P i , d I · x i , d
11. Update ith wombat using Equation (6). X i X i P 1 , F i P 1 < F i X i , e l s e
12. Phase 2: escape strategy (exploitation phase)
13. Calculate new position of ith wombat using Equation (7). x i , d P 2 x i , d + ( 1 2 r ) · u b d l b d t
14. Update ith wombat using Equation (8). X i X i P 2 , F i P 2 < F i X i , e l s e
15. end
16. Save the best candidate solution so far.
17. end
18.Output the best quasi-optimal solution obtained with the WOA.
End WOA.

3.5. Computational Complexity of WOA

In this subsection, the computational complexity of the proposed WOA approach is evaluated. The preparation and initialization steps of WOA have a computational complexity equal to O(Nm), where N is the number of wombats and m is the number of decision variables of the problem. In each iteration, the position of the wombats in the problem-solving space is updated in two phases: foraging and escape. Therefore, the process of updating the position of wombats has a computational complexity equal to O(2NmT), where T is the maximum number of iterations of the algorithm. According to this, the total computational complexity of the proposed WOA approach is equal to O(Nm(1 + 2T)).

4. Simulation Studies and Results

In this section, the ability of WOA to tackle optimization problems is evaluated on the CEC 2017 test suite.

4.1. Performance Comparison

Twelve well-known metaheuristic algorithms consisting of GA [40], PSO [25], GSA [47], TLBO [59], MVO [55], GWO [35], WA [32], MPA [38], TSA [36], RSA [33], AVOA [37], and WSO [39] are employed to compete with the performance of WOA. The simulation results are presented using six statistical indicators: mean, best, worst, standard deviation (std), median, and rank.
Considering that in order to optimize each of the benchmark functions, metaheuristic algorithms are used in several independent implementations, the statistical indicators are described as follows:
  • Mean: represents the average values obtained for the objective function from independent executions.
  • Best: indicates the best value obtained for the objective function among the values obtained from independent executions.
  • Worst: represents the worst value obtained for the objective function among the values obtained from independent executions.
  • std: represents the standard deviation between the values obtained for the objective function from independent runs.
  • Median: represents the median index between the values obtained for the objective function from independent executions.
  • Rank: indicates the rank of each metaheuristic algorithm in competition with other metaheuristic algorithms in dealing with the corresponding benchmark function. The evaluated values for the mean index have been applied as a ranking criterion for metaheuristic algorithms in handling each of the benchmark functions.
It should be mentioned that in order to provide a fair comparison, in the simulation studies, the original versions of competing algorithms published by their main researchers have been used. Also, regarding GA and PSO, the standard versions published by Professor Seyed Ali Mirjalili have been used. Also, the complete information and details about the experimental test suites and their optimal values are available in their respective references introduced in each subsection.

4.2. Evaluation CEC 2017 Test Suite

In this subsection, the ability of WOA and competitor algorithms to tackle the CEC 2017 test suite is challenged. The CEC 2017 test suite consists of thirty benchmark functions, as follows: (i) three unimodal functions of C17-F1 to C17-F3, (ii) seven multimodal functions of C17-F4 to C17-F10, (iii) ten hybrid functions of C17-F11 to C17-F20, and (iv) ten composition functions of C17-F21 to C17-F30. Among the functions of this test suite, C17-F2 is not considered in simulation studies due to its unstable behavior (as with all similar papers). Complete information, details, and descriptions of the CEC 2017 test suite are available at [102]. Based on the mentioned reference, the CEC 2017 test suite is designed to evaluate the performance of metaheuristic algorithms in handling optimization problems. Also, in order to provide a scalability analysis, it is recommended that this evaluation be conducted for the problem size equal to 10, 30, 50, and 100. The WOA approach along with competitor algorithms are employed in handling the CEC 2017 test suite in fifty-one independent runs, where each run consists of 10,000 m function evaluations (FEs), where m is the number of problem dimensions.
The implementation results of the WOA and competitor algorithms, in order to tackle the CEC 2017 test suite, are reported in Table 2, Table 3, Table 4 and Table 5. The boxplot diagrams obtained from the implementation of metaheuristic algorithms in the CEC 2017 test suite are plotted in Figure 3, Figure 4, Figure 5 and Figure 6. What is evident based on the analysis of simulation results and the performance of metaheuristic algorithms, in handling the CEC 2017 test suite for the problem dimension equal to 10 (m = 10), is that WOA has been the first best optimizer for the functions C17-F1, C17-F3 to C17-F21, C17-F23, C17-F24, and C17-F26 to C17-F30. For the problem dimension equal to 30 (m = 30), the proposed WOA approach is the first best optimizer for functions C17-F1, C17-F3 to C17-F22, C17-F24, C17-F25, and C17-F27 to C17-F30. For the problem dimension equal to 50 (m = 50), the proposed WOA approach is the first best optimizer for functions C17-F1, C17-F3 to C17-F25, and C17-F27 to C17-F30. For the problem dimension equal to 100 (m = 100), the proposed WOA approach is the first best optimizer for functions C17-F1 and C17-F3 to C17-F30.
As mentioned, the CEC 2017 test suite has functions of unimodal, multimodal, hybrid, and composition types. Each of these types of functions have been chosen with a special motivation to measure the quality of metaheuristic algorithms.
Unimodal functions C17-F1 and C17-F3 are types of functions that do not have local optima and only have one main optimal solution. These types of functions are suitable in order to evaluate the exploitation ability of metaheuristic algorithms in local search management with the aim of achieving solutions closer to the global optimum. The findings obtained from the optimization results of unimodal functions show that WOA has a high ability in the exploitation and local search by providing better results for these functions and obtaining the rank of the first best optimization.
Multimodal functions C17-F4 to C17-F10 are types of functions that have several local optimal solutions in addition to the main optimum. These types of functions challenge the exploration ability of metaheuristic algorithms in providing a global search and escaping from local optima. The findings obtained from the optimization of multimodal functions show that WOA, by achieving better results in most of the multimodal benchmark functions C17-F4 to C17-F10 and obtaining the rank of the first best optimizer, has a successful performance in its exploration to manage the global search in the problem-solving space.
Hybrid functions C17-F11 to C17-F20 and composition functions C17-F21 to C17-F30 are complex optimization problems for which it is very challenging to find a suitable solution. These types of optimization problems are very suitable for evaluating the ability of metaheuristic algorithms in balancing the exploration and exploitation during the search process. The analysis of the simulation results shows that WOA achieved better results in most of the benchmark functions C17-F11 to C17-F30 compared to the competing algorithms and was identified as the first best optimizer for these functions. The findings obtained from the optimization results of hybrid and composition functions show that WOA has a high ability to balance the exploration and exploitation during the search process in the problem-solving space in order to identify the main optimal area, escape from local optima, and converge towards the global optima.
The optimization findings are that WOA has achieved suitable solutions for the CEC 2017 test suite with its high capability in managing exploration and exploitation, and balancing them during the search process. The simulation findings are that WOA provided a superior performance by providing better results for most of the benchmark functions and obtained the rank of the first best optimizer in order to tackle the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100.

4.3. Statistical Analysis

Comparing the performance of metaheuristic algorithms using statistical indicators showed that WOA provided a superior performance against competitor algorithms in handling the CEC 2017 test suite. In this subsection, using statistical analysis, it has been checked whether this superiority of WOA is statistically significant or not. For this purpose, the Wilcoxon rank sum test [103] is employed, which is a non-parametric test effective in determining the significant difference between the means of two data samples. In this test, the presence or absence of a significant difference between the performance of two metaheuristic algorithms is determined using a criterion called the p-value.
The results of implementing the Wilcoxon rank sum test on the performance of WOA against the performance of each of the competitor algorithms are reported in Table 6. Based on the obtained results, in cases where the p-value is less than 0.05, WOA has a statistically significant superiority in its competition with the corresponding algorithm. The findings obtained from the statistical analysis are that WOA has a significant statistical superiority over all twelve competitor algorithms in order to tackle the CEC 2017 test suite for all four problem dimensions equal to 10, 30, 50, and 100.

5. WOA for Real-World Applications

One of the most special applications of metaheuristic algorithms is their efficiency in handling real world applications. In this section, the ability of WOA to tackle optimization tasks in real world applications is evaluated. With this aim, twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems have been selected.

5.1. Evaluation of CEC 2011 Test Suite

In this subsection, the capability of WOA and competitor algorithms to tackle the CEC 2011 test suite is challenged. The CEC 2011 test suite consists of twenty-two constrained optimization problems from real-world applications. The complete information, details, and description of the CEC 2011 test suite are available at [104]. The proposed WOA approach and each of the competitor algorithms are implemented in the CEC-2011 functions in twenty-five independent implementations, where each implementation contains 150,000 FEs.
The optimization results of the CEC 2011 test suite using WOA and competitor algorithms are reported in Table 7. The boxplot diagrams obtained from the implementation of metaheuristic algorithms on this test suite are plotted in Figure 7. The optimization results confirm that WOA has a high ability to manage exploration and exploitation, and establishing a balance between them has achieved suitable solutions for CEC 2011 test suite optimization problems. Simulation results show that WOA has provided a superior performance compared to competitor algorithms in order to tackle the CEC 2011 test suite, by achieving better results for all twenty-two optimization problems, from C11-F1 to C11-F22. In addition, the results obtained from the statistical analysis (reported in the last row of Table 7) confirm that WOA has a significant statistical superiority against all twelve competitor algorithms. The simulation findings are that WOA has an effective and promising performance for solving optimization tasks in real-world applications.

5.2. Pressure Vessel Design Problem

Pressure vessel design is a design challenge in engineering according to the schematic shown in Figure 8, whose main goal is to minimize the construction cost. The mathematical model of this design is as follows [105]:
C o n s i d e r :   X = x 1 ,   x 2 ,   x 3 ,   x 4 = T s ,   T h ,   R ,   L . M i n i m i z e :   f x = 0.6224 x 1 x 3 x 4 + 1.778 x 2 x 3 2 + 3.1661 x 1 2 x 4 + 19.84 x 1 2 x 3 .
Subject to:
g 1 x = x 1 + 0.0193 x 3     0 ,       g 2 x = x 2 + 0.00954 x 3   0 , g 3 x = π x 3 2 x 4 4 3 π x 3 3 + 1,296,000   0 ,       g 4 x = x 4 240     0 .
With
0 x 1 , x 2 100   a n d     10 x 3 , x 4 200 .
The results of tackling the pressure vessel design by employing WOA and competitor algorithms are presented in Table 8 and Table 9. The convergence curve of WOA, which shows the process of achieving a suitable solution for the pressure vessel design, is drawn in Figure 9. Based on the optimization results, WOA has provided the optimal design with the value of the objective function equal to (5882.8955) and the values of the design variables equal to (0.7780271, 0.3845792, and 40.312284, 200). From the simulation results, WOA has a superior performance compared to competitor algorithms for solving the pressure vessel design; this is because it provides better results for statistical indicators and design variables.

5.3. Speed Reducer Design Problem

The speed reducer design is a design challenge in engineering according to the schematic shown in Figure 10, whose main goal is to minimize the weight of the speed reducer. The mathematical model of this design is as follows [106,107]:
Consider :   X = x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 = b , m , p , l 1 , l 2 , d 1 , d 2 . Minimize :   f x = 0.7854 x 1 x 2 2 3.3333 x 3 2 + 14.9334 x 3 43.0934 1.508 x 1 x 6 2 + x 7 2 + 7.4777 x 6 3 + x 7 3 + 0.7854 ( x 4 x 6 2 + x 5 x 7 2 ) .
Subject to:
g 1 x = 27 x 1 x 2 2 x 3 1 0 ,         g 2 x = 397.5 x 1 x 2 2 x 3 1 0 , g 3 x = 1.93 x 4 3 x 2 x 3 x 6 4 1 0 ,         g 4 x = 1.93 x 5 3 x 2 x 3 x 7 4 1 0 , g 5 x = 1 110 x 6 3 745 x 4 x 2 x 3 2 + 16.9 × 10 6 1 0 , g 6 ( x ) = 1 85 x 7 3 745 x 5 x 2 x 3 2 + 157.5 × 10 6 1 0 , g 7 x = x 2 x 3 40 1 0 ,     g 8 x = 5 x 2 x 1 1 0 , g 9 x = x 1 12 x 2 1 0 ,     g 10 x = 1.5 x 6 + 1.9 x 4 1 0 , g 11 x = 1.1 x 7 + 1.9 x 5 1 0 .
With
2.6 x 1 3.6 , 0.7 x 2 0.8 , 17 x 3 28 , 7.3 x 4 8.3 , 7.8 x 5 8.3 , 2.9 x 6 3.9 ,   a n d   5 x 7 5.5 .
The results of employing WOA and competitor algorithms to tackle the speed reducer design are published in Table 10 and Table 11. The convergence curve of WOA while achieving the optimal solution for the speed reducer design is plotted in Figure 11. Based on the optimization results, WOA has provided the optimal design with the value of the objective function equal to (2996.3482) and the values of the design variables equal to (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, and 5.2866832). From the analysis of simulation results, it can be concluded that WOA has provided a superior performance compared to competitor algorithms in order to tackle the speed reducer design.

5.4. Welded Beam Design

Welded beam design is a design challenge in engineering according to the schematic shown in Figure 12, whose main goal is to minimize the fabrication cost. The mathematical model of this design is as follows [32]:
C o n s i d e r :   X = x 1 ,   x 2 ,   x 3 ,   x 4 = h ,   l ,   t ,   b . M i n i m i z e :   f ( x ) = 1.10471 x 1 2 x 2 + 0.04811 x 3 x 4   ( 14.0 + x 2 ) .
Subject to:
g 1 x = τ x 13,600     0 ,         g 2 x = σ x 30,000     0 , g 3 x = x 1 x 4   0 ,         g 4 ( x ) = 0.10471 x 1 2 + 0.04811 x 3 x 4   ( 14 + x 2 ) 5.0     0 , g 5 x = 0.125 x 1   0 ,         g 6 x = δ   x 0.25     0 , g 7 x = 6000 p c   x   0 .
where
τ x = τ 2 + 2 τ τ x 2 2 R + τ 2   ,         τ = 6000 2 x 1 x 2 ,         τ = M R J , M = 6000 14 + x 2 2 ,         R = x 2 2 4 + x 1 + x 3 2 2 , J = 2 x 1 x 2 2 x 2 2 12 + x 1 + x 3 2 2   ,   σ x = 504,000 x 4 x 3 2 ,         δ   x = 65,856,000 30 · 1 0 6 x 4 x 3 3   ,         p c   x = 4.013 30 · 1 0 6 x 3 2 x 4 6 36 196 1 x 3 28 30 · 1 0 6 4 ( 12 · 1 0 6 ) .
With
0.1 x 1 ,   x 4 2   a n d   0.1 x 2 ,   x 3 10 .
The implementation results of the WOA and competitor algorithms to tackle a welded beam design are reported in Table 12 and Table 13. The convergence curve of WOA, which shows the process of achieving the solution for a welded beam design, is drawn in Figure 13. Based on the optimization results, WOA has provided the optimal design with the value of the objective function equal to (1.7246798) and the values of the design variables equal to (0.2057296, 3.4704887, 9.0366239, and 0.2057296). Comparing the performance of metaheuristic algorithms shows that WOA has provided a superior performance by achieving better results for statistical indicators and design variables in competition with competitor algorithms in order to tackle the welded beam design.

5.5. Tension/Compression Spring Design

Tension/compression spring design is a design challenge in engineering according to the schematic shown in Figure 14, whose main goal is to minimize the construction cost. The mathematical model of this design is as follows [32]:
C o n s i d e r :   X = x 1 ,   x 2 ,   x 3   = d ,   D ,   P . M i n i m i z e :   f x = x 3 + 2 x 2 x 1 2 .
Subject to:
g 1 x = 1 x 2 3 x 3 71,785 x 1 4     0 ,   g 2 x = 4 x 2 2 x 1 x 2 12,566 ( x 2 x 1 3 ) + 1 5108 x 1 2 1   0 , g 3 x = 1 140.45 x 1 x 2 2 x 3   0 , g 4 x = x 1 + x 2 1.5 1     0 .
With
0.05 x 1 2 ,   0.25 x 2 1.3 a n d 2   x 3 15
The results of dealing with a tension/compression spring design by employing WOA and competitor algorithms are published in Table 14 and Table 15. The convergence curve of WOA during reaching the appropriate solution for the tension/compression spring design is drawn in Figure 15. Based on the optimization results, WOA provided the optimal design with the value of the objective function equal to (0.0126019) and the values of the design variables equal to (0.0516891, 0.3567177, and 11.288966). The analysis of simulation results and the performance of metaheuristic algorithms shows that WOA provided a superior performance compared to competitor algorithms in order to tackle the tension/compression spring design, by achieving better results for statistical indicators and design variables. The WOA convergence curve shown in Figure 15 shows the process of achieving the optimal solution for the tension/compression spring objective function during successive iterations of the algorithm. As it turns out, WOA identified the region containing the original optimum in the initial iterations with high power in global search and exploration. Then WOA, relying on its high ability in local search and exploitation, tries, until the last iterations of the algorithm, to obtain better values for the objective function and converge to the global optimum. The convergence curve shows that WOA has a high power in exploring, exploiting, and balancing them during algorithm iterations.

5.6. Application and Advantages of WOA for Supply Chain Management

While introducing the applications of the Wombat Optimization Algorithm (WOA) in supply chain management (SCM), its unique capabilities are distinguished by its optimal risk management, multi-objective information, agile operations, decisions to be collaborative, and sustainable development efforts, and offer advantages over traditional quality systems.
  • Risk management and resilience: WOA can help improve the supply chain resilience by identifying and mitigating potential risks such as supply chain disruptions, natural disasters, and demand fluctuations on the snow. By incorporating risks into the optimization process, WOA helps companies create robust supply chain models that can better adapt to unexpected disruptions compared to traditional models.
  • Multi-Objective Optimization: WOA can handle multi-objective optimization problems, where conflicting objectives such as cost minimization, lead time minimization, and service level maximization need to be balanced by the exploration–exploitation equilibrium of WOA on its Pareto front in contrast to the feasible search for trade-offs, giving decision makers the best solutions to choose from; traditional systems may struggle to meet many objective optimization problems and manage them effectively, as they often require complex changes or goal accumulations.
  • Dynamic and real-time optimization: WOA can be optimized in dynamic and real-time optimization scenarios where supply chain conditions change over time, such as demand fluctuations, disruptions, or capacity constraints. By constantly updating solutions based on the latest information, WOA enables companies to make the right decisions in a timely manner to optimize supply chains. Traditional systems may require a periodic reassessment or manual intervention to accommodate dynamic situations, resulting in suboptimal solutions or increased response times.
  • Collaboration and coordination optimization: WOA can optimize collaborative and coordinated decision-making among multiple entities within the supply chain, such as suppliers, manufacturers, distributors, and retailers. By optimizing decisions across the entire supply chain network, WOA helps companies achieve synergies and efficiencies that may not be achievable through localized optimizations. Traditional schemes often focus on optimizing individual components of the supply chain in isolation, leading to a suboptimal overall performance due to the lack of coordination and collaboration.
  • Sustainability and green logistics: WOA can incorporate sustainability criteria such as carbon emissions, energy consumption, and the environmental impact into the optimization process, enabling companies to design more sustainable and environmentally friendly supply chain strategies. By optimizing supply chain operations with sustainability objectives in mind, WOA helps companies reduce their ecological footprint and achieve corporate social responsibility goals. Traditional schemes may overlook sustainability considerations or treat them as constraints rather than optimization objectives, resulting in less environmentally sustainable supply chain designs.
In summary, the Wombat Optimization Algorithm offers several advantages over traditional optimization schemes in various areas of supply chain management, including risk management, multi-objective optimization, dynamic optimization, collaboration optimization, and sustainability. Its nature-inspired approach and flexibility make it well-suited for addressing the complex and dynamic challenges faced by modern supply chains.

6. Conclusions and Future Works

In this paper, a new biomimetics metaheuristic algorithm named Wombat Optimization Algorithm (WOA) is used for supply chain optimization, and imitates wombat behaviors in nature. WOA’s basic inspiration was taken from the wombat’s foraging process and the animal’s escape strategy when faced with its predators. The theory of WOA was expressed and then mathematically modeled in two phases: (i) exploration based on the simulation of wombat movement during foraging and trying to find food and (ii) exploitation based on simulating wombat movements when diving towards nearby tunnels to defend against its predators. WOA’s ability to solve optimization problems was tested in the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100. The optimization results showed that WOA achieves suitable solutions for optimization problems with its high capability for managing exploration and exploitation, and balancing them during the search process. The quality of the results obtained from WOA was compared with the performance of twelve well-known metaheuristic algorithms. The simulation results showed that WOA has provided a superior performance in competition with the compared algorithms, by providing better results in most of the benchmark functions and achieving the rank of the first best optimizer. WOA’s ability to tackle optimization tasks in real-world applications was challenged in twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems. The results of this implementation showed that WOA, while providing better results compared to competitor algorithms, has an effective performance when addressing optimization issues in real world applications.
The introduction of WOA presents several research proposals for further work. The development of binary and multi-objective versions of WOA is one of the most special research potentials of this study for future works. Another research potential of WOA is its application to deal with supply chain applications. Employing WOA to tackle optimization problems in various sciences and optimization tasks in real-world applications is another suggestion of this paper for further work in the future.

Author Contributions

Conceptualization, Z.B. and K.K.; Methodology, Z.B., M.D. and Y.G.; Software, M.D. and Y.G.; Validation, K.K.; Investigation, K.K.; Writing—original draft, K.K. and M.D. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Wombat taken from: free media Wikimedia Commons.
Figure 1. Wombat taken from: free media Wikimedia Commons.
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Figure 2. Flowchart of WOA.
Figure 2. Flowchart of WOA.
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Figure 3. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2017 test suite (dimension = 10).
Figure 3. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2017 test suite (dimension = 10).
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Figure 4. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2017 test suite (dimension = 30).
Figure 4. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2017 test suite (dimension = 30).
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Figure 5. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2017 test suite (dimension = 50).
Figure 5. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2017 test suite (dimension = 50).
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Figure 6. Boxplot diagrams of WOA and competitor algorithms performances on CEC 2017 test suite (dimension = 100).
Figure 6. Boxplot diagrams of WOA and competitor algorithms performances on CEC 2017 test suite (dimension = 100).
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Figure 7. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2011 test suite.
Figure 7. Boxplot diagrams of WOA and competitor algorithms’ performances on CEC 2011 test suite.
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Figure 8. Schematic of pressure vessel design.
Figure 8. Schematic of pressure vessel design.
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Figure 9. WOA’s performance convergence curve on pressure vessel design.
Figure 9. WOA’s performance convergence curve on pressure vessel design.
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Figure 10. Schematic of speed reducer design.
Figure 10. Schematic of speed reducer design.
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Figure 11. WOA’s performance convergence curve on speed reducer design.
Figure 11. WOA’s performance convergence curve on speed reducer design.
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Figure 12. Schematic of welded beam design.
Figure 12. Schematic of welded beam design.
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Figure 13. WOA’s performance convergence curve on welded beam design.
Figure 13. WOA’s performance convergence curve on welded beam design.
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Figure 14. Schematic of tension/compression spring design.
Figure 14. Schematic of tension/compression spring design.
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Figure 15. WOA’s performance convergence curve on tension/compression spring.
Figure 15. WOA’s performance convergence curve on tension/compression spring.
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Table 1. Applications of metaheuristic algorithms on SCM problem.
Table 1. Applications of metaheuristic algorithms on SCM problem.
ReferenceDescriptionYear
1[14]This paper conducts a comprehensive comparison study of the Firefly algorithm’s performance using various test functions, emphasizing its application in the lot size optimization within supply chain management. Demonstrating a superior performance over deterministic methods, the Firefly algorithm efficiently addresses the complexities arising from cost minimization and service level maximization conflicts in the supply chain evolution.2018
2[74]This paper introduces a closed-loop supply chain network configuration model addressing research gaps, and employs an innovative metaheuristic algorithm called improved PSO (IPSO) for location-allocation decisions and a gradient descent search method for pricing–inventory decisions. IPSO, integrating mutation and replicator dynamics, demonstrates a superior performance compared to traditional PSO, simulated annealing (SA), and genetic algorithm (GA) methods; this is confirmed through numerical evaluations across various problem scales.2018
3[75]This paper addresses a distribution–allocation problem in a two-stage supply chain, formulating it as an integer–programming model to minimize total supply chain operation costs. Employing an Ant Colony Optimization (ACO), the study demonstrates computational efficiency in obtaining solutions within a reasonable time frame, with an average gap of approximately 10% from optimal solutions.2018
4[76]This paper presents an enhanced artificial bee colony (ABC) optimization algorithm tailored for supply chain network (SCN) management, addressing the challenge of finding multi-objective Pareto optimal solutions (POS) efficiently. By extending the application field of SCN based on complex networks and integrating a naive Bayes classifier to accelerate the search speed, the proposed approach demonstrates its capability in optimizing a three-echelon SCN, achieving a global multi-objective POS while improving the solution-finding speed.2019
5[77]This paper introduces a bi-level optimization model for rice supply chain management, aiming to minimize the total cost while considering the perspectives of two decision-makers. Utilizing meta-heuristic algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), along with hybrid and modified versions, this study demonstrates the effectiveness of the proposed model in optimizing the rice supply chain, with the modified algorithm (GPA) showing promising results.2019
6[78]This paper addresses the evolving role of inventory management in the context of supply chain management, emphasizing the need for new strategies to enhance the supply chain integration and agility. By leveraging the system theory and integration theory, this paper proposes an optimized inventory management model utilizing ant colony algorithm and fuzzy model that aim to improve the supply chain efficiency and market responsiveness.2019
7[79]This paper presents a hybrid algorithm combining a genetic algorithm and particle swarm optimization to optimize supply chain scheduling in a mass customization mode, leveraging the genetic algorithm’s global search capability and the particle swarm optimization’s fast convergence speed.2019
8[80]This paper introduces an enhanced African Buffalo Optimization (ABO) algorithm for petroleum supply chain distribution, leveraging swarm intelligence to optimize product scheduling and distribution costs. By applying standard ABO and its improved variants, such as chaotic ABO and chaotic-Levy ABO, it demonstrates a superior performance compared to existing exact algorithms, offering efficient solutions for complex real-world supply chain networks.2020
9[81]This paper addresses the challenges of perishable product management in supply chains, proposing a holistic model integrated with the improved bacteria-forging algorithm (IBFA) to optimize production, inventory, and distribution processes. Through two case studies, the IBFA demonstrates effectiveness in optimizing perishable supply chain networks, offering valuable insights for decision-makers in managing time-sensitive products efficiently.2020
10[82]This paper presents a risk-based optimization framework for supply chain management, addressing strategic, tactical, and operational decisions to mitigate internal and external risks. Utilizing a new genetic algorithm integrated with an artificial neural network, it effectively minimizes supply–demand mismatches and reduces inventory levels, enhancing the profitability compared to traditional techniques and regular genetic algorithms.2020
11[83]This paper introduces a novel multi-level serial closed-loop supply chain model, incorporating batch deliveries, a quality-dependent return rate, a random defective rate, a rework process, and learning effects, particularly focusing on the impact of learning on inventory control. It employs metaheuristic algorithms such as a genetic algorithm, invasive weed optimization algorithm, and moth flame optimization algorithm to address the complexity of the proposed model, demonstrating the significant influence of the learning effect on the manufacturing/remanufacturing time and system costs in closed-loop supply chain problems.2021
12[84]This paper introduces a novel approach for a dual-channel, multi-product, multi-period, multi-echelon closed-loop supply chain network design (SCND) under uncertainty, specifically tailored for the tire industry. It utilizes a fuzzy approach to handle uncertain parameters and proposes two hybrid meta-heuristic algorithms, integrating red deer and whale optimization algorithms with a genetic algorithm and simulated annealing, respectively; this demonstrates their effectiveness in delivering high-quality solutions within a reasonable computational time.2021
13[85]This paper presents a location-inventory optimization model for supply chain configuration, addressing stochastic customer demand and replenishment lead time. It employs a two-phase approach integrating the queuing theory and stochastic optimization to determine optimal distribution center locations and inventory policies, with a hybrid genetic algorithm designed to handle the NP-hard complexity of the problem; this offers a computationally tractable solution for supply chain optimization.2021
14[86]This paper aims to optimize economic and environmental dimensions in a sustainable supply chain network through a mixed-integer linear programming (MILP) model, integrating sustainable supplier selection and performance optimization. Utilizing multi-objective genetic and particle swarm algorithms, it achieves a balance between cost minimization, time efficiency, and sustainability indexes, offering robust solutions for supply chain managers seeking to enhance their sustainability performance.2021
15[87]This paper introduces a sustainable Closed-Loop Supply Chain Network (CLSCN) design for the olive industry, integrating economic, environmental, and social factors through a multi-objective optimization framework. It proposes novel hybrid optimization algorithms, including the Virus Colony Search (VCS) algorithm with simulated annealing (SA) and Electromagnetism-like Algorithm (EMA) with Genetic Algorithm (GA), demonstrating a superior efficiency in addressing the complex challenges of large-scale networks, which offer valuable insights for supply chain managers in the olive industry.2022
16[88]This paper explores the utilization of the Particle Swarm Optimization (PSO) algorithm for the supply chain network design, and aims to optimize network configurations and improve operational efficiency. By leveraging PSO, the study offers insights into enhancing supply chain network design processes through efficient optimization techniques.2022
17[89]This paper introduces a hybrid MDE_Restart and modified differential evolution (MDE) tailored for designing closed-loop supply chain networks, considering quantity discounts and fixed-charge transportation. By incorporating these algorithms, the algorithms efficiently optimize supply chain network configurations, addressing cost-saving strategies and logistical complexities.2022
18[90]This paper focuses on designing a novel supply chain network by considering transportation delays and employing meta-heuristic techniques. It explores the application of meta-heuristics to optimize the supply chain network design, taking into account transportation delays for enhanced efficiency and performance.2022
19[91]This paper introduces a novel metaheuristic approach tailored for a multi-objective supply chain network design, hybridizing a simulated annealing, tabu search, and variable neighborhood algorithms, along with linear programming. By combining these techniques, the approach aims to leverage the strengths of each algorithm, enhancing the solution quality and efficiency in supply chain network optimization.2023
20[92]This paper presents a hybrid metaheuristic approach combining a greedy randomized adaptive search procedure (GRASP) and genetic algorithm (GA) integrated with a learning component to address a real-world supply chain scheduling problem effectively. By combining metaheuristic techniques with learning mechanisms, it offers a robust solution framework tailored to enhance scheduling efficiency in complex supply chain environments.2023
21[93]This paper applies an improved multi-objective particle swarm optimization algorithm to address disruptions in the two-stage vehicle routing problem with time windows. By leveraging enhanced optimization techniques, it effectively balances multiple objectives, ensuring efficient routing solutions despite disruptions, and thus enhances the overall supply chain performance.2023
22[94]This paper proposes the integration of the Grey Wolf Optimizer and Whale Optimization Algorithm to address stochastic inventory management challenges in a two-level supply chain for reusable products. By combining both algorithms, it enhances inventory control strategies, optimizing stock levels and minimizing costs in dynamic supply chain environments.2023
23[95]This paper introduces a multi-objective dragonfly algorithm tailored for optimizing sustainable supply chains, particularly under resource-sharing conditions. By employing the dragonfly algorithm, it efficiently balances multiple objectives, enhancing sustainability practices in supply chain management through an optimized resource allocation.2024
24[96]This paper presents a hybrid meta-heuristic approach aimed at designing a bi-objective cosmetic tourism supply chain, demonstrating its applicability through a case study. By leveraging meta-heuristic methods, it offers an optimized framework to balance the cost-efficiency and service quality within the cosmetic tourism sector.2024
25[97]This paper proposes a hybrid whale optimization algorithm tailored for optimizing limited capacity vehicle routing in supply chain management. By integrating whale optimization techniques, it enhances routing efficiency, and addresses constraints and complexities inherent in supply chain logistics.2024
Table 2. Optimization results of CEC 2017 test suite (dimension = 10).
Table 2. Optimization results of CEC 2017 test suite (dimension = 10).
WOAWSOAVOARSAMPATSAWAMVOGWOTLBOGSAPSOGA
C17-F1mean524.9613.68 × 10911,334,9276.88 × 10935,084,1571.18 × 10915,674,08711,337,40370,711,2701.1 × 10811,332,84211,334,45719,310,478
best100.13373.12 × 1095502.35.98 × 10911,140.132.51 × 1083,927,3339510.76822,309.9844,355,6733666.0263831.3827,525,902
worst1643.9274.61 × 10941,171,8138.19 × 1091.27 × 1082.56 × 10944,332,88941,178,9242.57 × 1082.39 × 10841,171,76941,172,25050,095,279
std815.43527.07 × 10821,743,6831.09 × 10967,320,9401.11 × 10920,897,37621,746,6501.36 × 10895,173,73921,744,86821,743,91022,413,972
median177.89173.5 × 1092,081,1976.68 × 1096,430,7109.61 × 1087,218,0632,080,58812,960,03779,143,9972,077,9682,080,8739,810,366
rank11241381165910237
C17-F3mean300.01335557.276656.8946946.6861400.9747992.8351617.886655.65642519.006942.49177357.19655.619610,395.98
best300.00593581.433457.75484316.696788.39823686.656697.5888457.76331284.363572.99284624.991457.75483742.932
worst300.02697096.0241020.3438967.9832521.69510,914.742497.2711017.7014777.811370.4329845.6271017.61716,177.44
std0.0105421671.14281.92852309.198870.31353337.817985.4242281.15081754.111380.35482325.601281.11517133.237
median300.01035775.823574.73917251.0331146.9028684.9741638.342573.58062006.926913.27087479.071573.553210,831.78
rank19410612738511213
C17-F4mean400.0001751.2581405.36191042.661406.6922520.9873419.1006404.4076410.0676408.3385405.2283415.8432412.0756
best400586.876401.9679703.3198402.434453.5618406.6237402.2056404.8878407.2968403.3221400.8574409.9993
worst400.0001900.4412407.46741375.483411.3215597.1338450.3308406.3686422.7592409.3592407.1639450.4723415.491
std5.78 × 10−5152.68852.677108312.05054.77109176.6653722.766221.9219519.2800251.2497112.26933925.334622.583055
median400.0001758.8575406.00611045.92406.5066516.6267409.7239404.528406.3117408.349405.2136406.0215411.406
rank11241351110276398
C17-F5mean501.2465546.0639534.1456553.7107512.9544547.9623532.0536520.3087513.0506527.3509540.8192523.167523.243
best500.9952534.7172522.2683542.2803508.392534.761519.3133512.3122508.6237522.2608536.0831510.4081519.8081
worst501.9918552.3599548.102565.0885518.0915568.2777554.9786528.5494519.6668531.4124547.4567540.5667528.8225
std0.540738.79043414.0531613.956465.54997216.7901417.812177.2284815.2939444.5211725.22948215.445054.31056
median500.9995548.5892533.1059553.737512.6671544.4053526.9612520.1866511.9559527.8652539.8686520.8466522.1708
rank11191321284371056
C17-F6mean600622.427612.2172628.1889601.2043617.3465616.2108601.8572601.1587605.0758612.1393605.4632607.3961
best600619.185611.5093625.8807600.7171610.5267605.3716600.5972600.6819603.4816602.2599601.1933604.9474
worst600625.5096613.8456630.9364602.4192627.8731631.1437603.7271601.5874607.7086624.9582613.9345610.6874
std1.07 × 10−52.9167981.197432.5475560.8837378.07112611.755411.5300680.4647162.08841811.511536.3145012.755938
median600622.5067611.7571627.9693600.8405615.4931614.164601.5523601.1827604.5565610.6696603.3624606.9748
rank11291331110425867
C17-F7mean711.1269773.8425752.7079779.2108724.7594795.6833750.3349729.0217725.6989743.4845719.6236730.2951733.1163
best710.6728763.7127739.358771.616720.5128769.7413741.4602719.2818718.4864741.0113717.5532725.009725.6534
worst711.7996781.9515770.3003788.1838729.2209824.3351770.0493743.626739.1095749.3562722.4981738.5156737.3097
std0.5573498.2028615.955278.4309843.99231925.7754214.483411.2397210.095144.2788012.6436076.3366845.80094
median711.0176774.8529750.5867778.5216724.6519794.3284744.915726.5895722.5999741.7853719.2215728.8279734.751
rank11110123139548267
C17-F8mean801.493836.5212825.3888840.8137812.7774837.1173828.9696812.2052814.9513829.8901817.6965819.6812815.5957
best800.9951832.909816.7434833.08808.9252825.9835817.4319809.8082810.071825.6459812.9463815.4581813.4852
worst801.9913839.8716836.8082844.1263814.9398849.0655837.2411814.2366818.9738834.1117821.7669824.7868819.6825
std0.6256434.1062459.0898775.6407053.02147911.216219.4110051.9818524.1219495.2486664.3066345.3483243.024761
median801.4927836.6521824.0018843.0243813.6223836.7101830.6028812.3881815.3803829.9015818.0364819.24814.6075
rank11181331292410675
C17-F9mean900.00011258.6331098.5631289.325905.24671230.1611226.46902.2423909.8499909.7766901.6947904.5936905.1876
best9001154.645936.81891222.699900.33061084.9351020.774900.1655900.4985906.7986900.1068900.7778903.7061
worst900.00021353.1041424.4621385.924913.46771425.5381416.941904.3529926.9905914.1982904.3501908.9463906.639
std7.79 × 10−594.16625245.83975.61816.435353160.7559181.1962.3948713.432423.4170262.0786483.6889391.633171
median9001263.3921016.4851274.339903.59421205.0861234.063902.2255905.9554909.0547901.161904.3251905.2027
rank11291361110387245
C17-F10mean1006.1852048.6381692.9332234.8951516.1811865.4681860.3241694.9421657.4591959.8362031.6521806.7181650.865
best1000.2911839.7371453.3772112.6311390.9111703.3871464.1241437.1551490.8111654.8711835.7941505.4141440.627
worst1012.6732158.6722116.9272501.241591.2332041.8452239.5082057.3391862.2532179.1572126.1462105.821942.696
std7.243172154.8632323.4461195.0429102.7074193.9076389.1079310.5663166.8985244.947145.7272269.4419230.3456
median1005.8882098.0731600.7132162.8561541.2911858.3211868.8311642.6361638.3852002.6582082.3331807.821610.069
rank11251329864101173
C17-F11mean11002683.3821141.5133058.5051127.0074055.8881143.1721127.3191146.0881143.1411135.2181138.1491975.904
best11001839.0281127.5831349.5351113.1813951.5281127.721110.8571118.8751132.6831128.4791126.3711129.13
worst1100.0013501.6861173.0674740.7091158.6964106.4021156.5271137.3241205.7431153.1391150.9541162.9164405.576
std0.000283815.328223.127821661.68223.3904876.7483514.4753712.7492344.1657710.8784511.553618.173651763.102
median11002696.4061132.7013071.8891118.0754082.8111144.221130.5471129.8681143.3711130.721131.6551184.456
rank11161221383974510
C17-F12mean1359.8252.4 × 108929,553.94.78 × 108568,255.9888,2671,779,172881,104.41,142,8933,608,347875,189.1189,031.5593,640.7
best1327.31353,915,999484,966.21.06 × 10819,884.71371,868.6313,281.2202,852.737,2191,113,386565,303.213,728.52362,439
worst1448.9314.19 × 1081,359,2968.36 × 108889,303.61,107,6302,934,3342,197,5361,788,9046,350,1761,176,097296,681.6920,789.6
std64.771412.01 × 108496,292.94.02 × 108416,840.9377,092.81,261,767976,176840,175.82,935,492295,957.3134,450.5296,135.4
median1331.5292.43 × 10893,6976.94.85 × 108681,917.61,036,7851,934,537562,014.41,372,7243,484,914879,677.9222,858545,667.2
rank11281337106911524
C17-F13mean1305.37411,655,63714,180.7923,302,9845439.06210,388.916892.5236316.1448736.13813,092.158581.796243.41638,661.03
best1303.143972,780.53993.2551,935,7303723.5417241.3214322.3752751.9795612.1611,906.264996.233423.7657889.229
worst1308.55138,684,34722,836.0477,357,4286652.15914,879.4911,845.7810,488.9811,89914,978.2611,762.0113,427.3124,179.4
std2.4752519,652,73810,768.3639,303,4531520.153548.8943725.0734326.3232821.5551439.7793025.5135229.26962,071.96
median1304.9013,482,71114,946.946,959,3895690.2739717.4195700.9666011.8118716.69512,742.048784.464061.29811,287.73
rank11210132854796311
C17-F14mean1400.7533207.71996.7964252.91941.3512922.7961655.6361691.512217.0311704.3584401.2072657.5759419.632
best14002606.8911642.9463737.1371435.1151476.2681476.8981432.2841460.5461495.4093613.6731433.4132990.105
worst1401.0133876.4342385.6185141.4422906.8594249.9151986.2452289.4944304.6122036.7726062.1215278.61117,992.18
std0.545908565.9558341.6936704.1598751.23841521.565251.4321442.81591516.819263.55961232.2471962.0276855.379
median1400.9993173.7381979.3114066.5111711.7152982.5021579.71522.1321551.4821642.6263964.5161959.1388348.119
rank11061159237412813
C17-F15mean1500.3618206.4934878.42410,697.313981.3276034.935502.8342329.8735228.6962443.42617,486.457388.2244370.894
best1500.0423488.992446.2873143.543227.0912862.2922592.8652075.4443462.4482166.2828842.9643174.4072648.946
worst1500.5313,039.779793.321,824.414899.5289552.29510704.092624.6816095.7012655.13725,606.1411,618.16725.968
std0.2502094560.7423631.98934.354755.06543082.4633882.324246.72071315.114222.60998586.3243809.9922135.416
median1500.4378148.6043637.0548910.653899.3455862.5674357.1912309.6845678.3182476.14217,748.347380.1934054.331
rank11161249827313105
C17-F16mean1600.7611900.5161769.4261909.7451684.3991930.6921865.0151773.9531714.4281679.4261948.2411846.8071764.529
best1600.3571858.3531665.831785.9551641.8311791.5171738.351699.3411624.2461662.431848.9731777.2561717.698
worst1601.1211979.7981834.8152094.5931715.0632054.9791951.4991825.6991790.1091726.1512090.3181959.9361790.358
std0.34360758.8526879.01002142.765934.28364130.2233108.790858.2245874.3545533.96555117.498388.8149835.40944
median1600.7831881.9561788.5311879.2161690.3511938.1361885.1061785.3851721.6781664.5611926.8371825.0191775.03
rank11061131297421385
C17-F17mean1700.11794.4961746.1711791.9031735.8261780.9231807.8661808.4641758.081751.1961811.1731747.111749.567
best1700.0211778.5271731.061789.1611721.9661766.7551757.1781760.9891724.3121739.8391739.631738.7381743.096
worst1700.3321810.9881771.5691793.6851775.0991793.031852.7521894.2331840.7091768.621909.5891761.1951763.32
std0.16875614.4735720.151182.19632528.5073711.8813343.628368.3066760.1477314.1916291.7080110.9158610.10979
median1700.0231794.2341741.0261792.3831723.121781.9531810.7661789.3161733.6491748.1621797.7361744.2541745.926
rank11039281112761345
C17-F18mean1805.4721,937,81211,592.433,859,32811,046.4411,729.4619,340.817,743.3517,038.923,534.3510,141.5818,372.2112,240.5
best1800.031104,413.28000.928195,598.34158.1019774.9367238.4688762.7295623.32917,578.487200.726277.8137047.84
worst1820.5365,610,34915,888.7811,198,38616512.913,065.4528,477.6428,146.9827,452.7327,845.9912,745.9130,442.8717,841.83
std10.93482,776,0833713.9515,548,9946115.0741540.45310,707.219103.72112,055.935135.6763058.82814,952.744993.52
median1800.6591,018,24311,2402,021,66411,757.3812,038.7320,823.5517,031.8517,539.7724,356.4710,309.8518,384.0712,036.17
rank11241335108711296
C17-F19mean1900.489264,028.96348.121478,141.95596.93886,747.5425,361.633104.1135451.7654986.74229,163.3418,689.695992.339
best1900.04418,200.912222.25531,876.912317.7112226.2126057.3742041.3552139.1682329.75510,083.212650.0344067.144
worst1901.65553,293.811,531.751,024,0059413.121172,704.745,687.834337.45411913.979200.89740,551.0855,068.569728.734
std0.843567254,409.14310.376486,854.93935.834105,192.317,629.551270.0074791.5923211.50615,355.6626,810.952785.874
median1900.132242,310.55819.24428,342.75328.46186,029.6524,850.653018.8223876.9584208.15733,009.538520.0815086.739
rank11271351192431086
C17-F20mean2000.3132175.2722145.2232180.7252092.2822170.1382169.6072124.2442144.7862078.532201.4822144.1522063.786
best2000.3132138.4842060.8592140.0172072.7172099.6042106.1962060.4492112.0752064.7492153.6052126.7052052.918
worst2000.3142214.2922222.7722228.0792122.7772240.642218.2952190.9242206.1252090.2572274.3332159.3832073.555
std0.00028333.8192980.3237744.7342223.3447363.820660.6422858.2418345.6956212.2100463.2143215.684429.418732
median2000.3132174.1572148.632177.4022086.8172170.1542176.972122.8022130.4712079.5572188.9962145.2612064.335
rank11181241095731362
C17-F21mean2200.0012280.4772227.832263.9342257.2132303.2422292.8732254.4682295.22285.9872332.4582298.9272284.957
best2200.0012249.0312221.5392234.9162254.7232232.6282230.7062217.6812291.5462221.2122319.8182292.6652237.284
worst2200.0012299.2352245.1122279.7512259.752335.2552321.9952291.1232299.3852311.3812343.9622304.2552307.614
std2.8 × 10−525.2315712.5473421.86152.31533552.2269645.5728745.218343.51026847.2557910.939356.31934534.98016
median2200.0012286.8222222.3342270.5352257.1892322.5432309.3952254.5342294.9342305.6772333.0252299.3952297.466
rank16254129310813117
C17-F22mean2300.0732571.2082307.7052718.9422305.0092582.0692317.7462291.982307.4462314.8812301.6212310.612313.767
best23002490.1792305.9842576.9922300.9452401.372314.8792252.492301.1632310.0862300.3052300.7372310.489
worst2300.292652.7862309.0222824.2412309.3712724.3362321.6132305.3962318.2132324.2442303.0272332.7242316.405
std0.15788178.27431.374266113.21763.86487156.75793.57358928.652288.2227717.2226821.24674816.196322.736905
median23002570.9352307.9082737.2672304.8612601.2842317.2462305.0162305.2042312.5982301.5772304.492314.087
rank21161341210159378
C17-F23mean2600.922665.4812633.2282672.9432614.3622688.4412637.7392618.3932613.9662633.552734.8612634.7332642.783
best2600.0032642.3142624.6022652.4752611.9162628.432626.2882609.9542609.1422627.0412691.1532629.0552630.113
worst2602.872679.8612646.1592701.3722617.0682718.0742650.6532627.1222619.3972639.0762828.2852642.3012647.923
std1.43688619.0301210.6953224.515512.65774844.2232614.339568.1286525.6716746.12932370.405676.2389099.268331
median2600.4032669.8752631.0752668.9612614.2322703.632637.0082618.2482613.6622634.0422710.0032633.7882646.549
rank11051131284261379
C17-F24mean2630.4882736.6422723.6222779.4792630.6532656.2022718.8732666.4032710.8412715.632709.952722.2352693.422
best2516.6782703.6852702.5112761.62616.9122561.422696.7922538.6422690.1292703.1642545.0132715.9622569.749
worst2732.3182784.692737.0332822.6562637.4772755.0032741.2052719.3842722.252724.2152813.8782737.5982754.381
std126.786940.660217.3204731.5132910.15127114.467919.799793.1465416.0472810.58233125.603611.1828491.04191
median2636.4772729.0962727.4722766.8292634.1122654.1922718.7482703.7932715.4932717.5712740.4562717.6892724.778
rank11211132394786105
C17-F25mean2932.6393066.7392914.8793161.2962917.8693064.1892910.8442920.7212931.9772928.4722920.8372921.5582941.165
best2898.0483017.4022904.6373116.492913.6552911.4452812.6022905.1522918.7942915.3222906.2492904.3472931.162
worst2945.7933173.632940.7413212.3792923.3223419.1662945.9082937.1312938.6052941.3192935.2732938.3182950.055
std25.1284979.2024818.811843.355034.454837260.21771.3475518.784159.72365615.9949416.5388119.64328.745852
median2943.3593037.9622907.073158.1582917.252963.0722942.4332920.32935.2542928.6242920.9122921.7842941.722
rank91221331114875610
C17-F26mean2900.0013382.1762990.3463517.293011.9313425.3173128.2232936.2553184.0763144.2483588.7832938.9092934.265
best29003252.2312846.1123388.7822892.0963063.1833005.5022897.5782944.4272917.4052846.1122909.2252766.749
worst2900.0053502.9923201.7213719.1053294.5333840.0033368.5133027.5133710.9573689.6334010.8452971.583052.759
std0.002495125.409190.2343170.3872205.9921381.348178.626366.51778385.449397.2118558.118536.64296135.6199
median29003386.742956.7753480.6382930.5473399.0423069.4382909.9643040.4592984.9763749.0872937.4162958.777
rank11051261173981342
C17-F27mean3089.5183176.163115.123190.5333104.7293155.5183165.9663095.8633112.4793111.7873187.0753126.0283142.268
best3089.5183139.6273096.1853115.9833092.253101.0183163.9333090.5913093.7393095.7693175.5983095.5433110.642
worst3089.5193229.1813165.8943318.7133133.923186.7813169.9183103.993163.0823146.6873197.6723167.573180.087
std0.00025841.3160736.8716196.4177221.3330442.617642.9410286.23161136.77626.1151410.365333.2610431.2902
median3089.5183167.9163099.2023163.7193096.3733167.1363165.0063094.4363096.5473102.3473187.5153120.4993139.172
rank11161339102541278
C17-F28mean3100.0013472.4473230.5973598.7363218.6913467.9553264.9613232.3733304.3683290.913376.0883277.7453237.54
best3100.0013448.593140.1433544.7493167.0153338.9773180.8913121.733185.8323223.6423362.8533192.4133152.068
worst3100.0023495.8743318.4463645.8463243.6133611.5653337.2893336.9433357.8953337.1063390.3853327.3613420.354
std0.00046721.1414186.6624946.2050738.58309152.425581.7028124.746786.6770654.305112.3349967.19946134.348
median3100.0023472.6633231.93602.1753232.0673460.6383270.8323235.4093336.8733301.4463375.5573295.6033188.87
rank11231321164981075
C17-F29mean3132.2423295.93258.5813320.1213203.3113225.8033302.2633203.0273245.453209.7853300.223246.0423226.48
best3130.0773269.6283196.2363259.6543166.0713172.4583229.7743150.1433186.5913182.2273223.2063167.3813187.635
worst3134.8423320.1223310.0543371.1493244.883261.333396.2693264.2413336.4043238.6223500.7643296.543248.091
std2.70173722.7163264.1047564.0607737.7531341.0560775.1988351.0048978.4298726.51012145.714763.8565431.07757
median3132.0233296.9253264.0173324.8413201.1463234.7123291.5043198.8633229.4033209.1463238.4543260.1243235.097
rank11091335122741186
C17-F30mean3423.7071,659,881332,889.12,617,672413,992.4548,981.2804,162.1338,396.8766,098.9174,698.1662,631.2395,413.91,165,787
best3394.8341,300,276112,282.8734,672.715,907.74273,256.2142,94910,169.8827,840.6724,936.49582,053.210,531.89532,136.6
worst3449.4442,183,767675,809.43,928,496611,011.8883,172.62,706,390955,766.71,099,071228,475.1733,085.5715,543.42,356,340
std31.91655413,046.5262,887.11,473,594294,132.7283,788.31,380,451458,510.5550,547.6108,731.468,368.52385,165.3931,541.8
median3425.2751,577,740271,732.12,903,759514,525519,748183,654.5193,825.3968,742222,690.3667,693427,790.1887,336.2
rank11231367104928511
Sum rank38319178351107287240117188191240184199
Mean rank1.310345116.13793112.103453.6896559.8965528.2758624.0344836.4827596.5862078.2758626.3448286.862069
Total rank1114122109367958
Table 3. Optimization results of CEC 2017 test suite (dimension = 30).
Table 3. Optimization results of CEC 2017 test suite (dimension = 30).
WOAWSOAVOARSAMPATSAWAMVOGWOTLBOGSAPSOGA
C17-F1mean100.3211.77 × 101010,685.552.76 × 101026,598.071.21 × 10101.14 × 109370,179.61.12 × 1094.15 × 1097,073,0879.45 × 1081.2 × 108
best100.14711.52 × 10104334.5552.47 × 101012,240.797.57 × 1099.03 × 108284,867.61.85 × 1082.62 × 1097494.3226473.38689,553,528
worst100.48862.21 × 101014,216.893.4 × 101040,439.211.64 × 10101.42 × 109471,460.93.38 × 1096.19 × 10924,666,8073.78 × 1091.65 × 108
std0.1796423.5 × 1094844.6794.69 × 10914,830.934.5 × 1092.89 × 108100,748.31.65 × 1091.62 × 10912,897,3612.05 × 10935,735,136
median100.32411.67 × 101012,095.382.6 × 101026,856.131.21 × 10101.13 × 109362,195.14.63 × 1083.9 × 1091,809,0232,161,4901.12 × 108
rank11221331194810576
C17-F3mean300.009765,817.2730,416.0649,852.41097.13432,095.35156,312.31549.65728,37523,677.9964,815.8521,801.7112,811.6
best300.006660,055.3516,758.5638,605.67847.385830,389.58129,3451309.70824,870.120,293.1955,814.1115,738.9885,487.74
worst300.012772,311.2739,191.4854,205.291350.31833,835.91179,603.52032.34631,610.3525,577.0471,261.1627,862.37156,686.7
std0.0029776574.40710,469.648190.228245.78461825.46322,754.38358.5273004.5852601.6867609.2786045.36336,800.58
median300.009665,451.2332,857.153,299.321095.41632,077.96158,150.41428.28828,509.7724,420.8766,094.0621,802.72104,535.9
rank11179281336510412
C17-F4mean458.5624497.82507.40496769.478492.92313218.49737.6698495.3664545.8814771.9046561.2869581.0463707.4596
best458.56182595.808489.47934395.695482.5108864.6459692.8323486.716515.4298631.5341545.044507.0297669.8195
worst458.56226028.379518.55639395.177514.54395241.204788.9501511.4194563.61191048.057575.9539705.3547728.9632
std0.0001941548.39614.715532259.03516.00972010.66245.9563411.9323422.85959203.847713.7695897.1460128.81583
median458.56194683.547510.79196643.519487.31893384.056734.4484491.665552.242704.0139562.0749555.9004715.5279
rank11241321193510678
C17-F5mean502.4884759.2186678.7108785.6342582.8792724.9132744.6379607.2809608.9598709.0273676.9627616.1536663.1888
best500.9962743.9431652.0313766.2516560.4361703.8315717.82599.627589.0987686.5317665.416597.8302637.5721
worst503.9807768.9604725.745802.0721605.8082740.3558761.3818623.7057629.428734.2369692.8394647.6527707.261
std1.39779412.2792335.6868420.4392420.6934918.3989620.3977612.1383821.0836522.5955212.5408423.7359133.09432
median502.4883761.9856668.5334787.1066582.6362727.7327749.6749602.8954608.6563707.6703674.7977609.5657653.961
rank11281321011349756
C17-F6mean600654.6673632.4048656.7921603.2431652.7484652.2338617.4266609.0318630.1329638.9866632.5584621.3114
best600653.9089630.8969652.8186601.9816642.2756645.0398609.885604.0154624.9456638.0784623.9718616.4128
worst600.0001656.0022634.622661.0946604.636658.9347655.7002625.5434613.3883637.7483640.1017639.599624.5343
std1.52 × 10−50.9981031.8215074.1639431.255218.3598275.4029378.2053284.2406615.9806290.9095547.6041793.935982
median600654.3791632.0501656.6276603.1775654.8916654.0976617.139609.3618628.919638.8832633.3314622.1492
rank11271321110436985
C17-F7mean733.47941149.7181047.8681177.137846.37551099.9941155.758851.3168872.59221000.192930.0361867.6688927.5223
best732.81981112.149987.15011159.234819.4609996.65061125.986806.6683819.8287954.6034902.5773844.6827892.1004
worst734.52191179.251148.1941210.139899.9331197.9181201.539895.7276898.42051048.75972.079880.6517981.6722
std0.82100234.0432380.5556824.7752739.6051799.3895635.2141243.2253539.0530955.2744432.8437417.7899341.52581
median733.28791153.7371028.0631169.588833.0541102.7031147.754851.4358886.0599998.7069922.744872.6705918.1583
rank11191321012358746
C17-F8mean803.33091021.091930.21961046.227890.55741003.64984.4257892.4033891.5049979.1958938.1944912.7277954.795
best801.20341011.594912.44271032.472883.8529976.3113944.1651871.6459886.7197966.3352924.3735904.7129941.7634
worst804.15851034.679944.46311064.184898.6831073.6811015.226910.3595896.62791004.115954.112923.136968.5517
std1.54628411.6946416.2674316.484386.66304151.086132.8041919.185774.43978718.4179914.711088.38397613.55689
median803.98081019.045931.98631044.126889.8469982.2843989.1557893.8039891.336973.1666937.146911.5309954.4324
rank11261321110439758
C17-F9mean900.00227729.3493598.7547501.1261083.688081.8997771.0834022.9871751.64236.9393098.0172740.351223.889
best900.00046605.9512725.8037351.551929.60875012.3265997.4023226.3781415.7433109.1492785.481804.641093.835
worst900.00418766.8434100.4257639.3741235.59210819.739225.7365938.4772231.5696265.5753681.6723942.4381392.341
std0.001864984.4253663.0424128.2036153.92092607.1271782.9451396.078423.13811550.553449.7933994.7306150.2506
median900.00227772.3013784.3957506.7891084.7618247.7687930.5963463.5461679.5443786.5152962.4582607.1611204.69
rank11171021312849653
C17-F10mean2293.2876161.4734970.3926622.9483984.5895717.1345674.2924429.2044522.9736636.1884562.9634692.3835436.048
best1851.7875913.344392.4666187.5343628.0264920.6495125.3054124.0724226.6846274.2254281.894528.7135127.049
worst2525.0416425.8395438.0087098.434431.0836171.8766716.8894608.6994889.4356841.1124892.3244922.595962.139
std326.89228.4198488.6168410.0172403.212596.2858796.7545244.2174302.233282.4372272.9881194.1375422.2879
median2398.166153.3565025.5486602.9143939.6245888.0065427.4874492.0234487.8876714.7074538.824659.1145327.501
rank11171221093413568
C17-F11mean1102.9875439.9261229.0886315.6851169.6033839.9685650.4651267.1031860.9161721.1512335.0931223.3756563.117
best1100.9964528.5021194.6035196.9551122.1642846.3814178.431222.4951316.981463.1391877.741202.5072653.878
worst1105.9786181.071257.0577071.2741203.0825608.6558180.9551305.6583315.6912216.0722798.6271247.86311,992.73
std2.342682788.369228.52432933.663938.062131355.9851902.0446.139611056.634366.1953470.454920.287184354.439
median1102.4885525.0661232.3466497.2561176.5833452.4175121.2371270.1291405.4961602.6972332.0031221.5665802.928
rank11041229115768313
C17-F12mean1744.7934.74 × 10914,077,9247.36 × 10921,567.553.42 × 1091.67 × 1087,582,67535,462,6642.04 × 1081.34 × 1081,736,4965,192,897
best1722.033.92 × 1091,986,4366.56 × 10915,405.161.76 × 109427417143,524,2303,450,3451.3 × 10825,975,685194,511.43,598,379
worst1765.1026.02 × 10934,372,9379.27 × 10927,519.434.47 × 1093.34 × 10818,338,24474,357,4273.54 × 1084.29 × 1083,445,0886,796,896
std21.925689.78 × 10815,515,0921.4 × 1095617.7991.28 × 1091.46 × 1087,815,44833,643,2651.1 × 1082.14 × 1081,525,4521,578,420
median1746.0214.51 × 1099,976,1626.81 × 10921,672.813.72 × 1091.46 × 1084,234,11332,021,4411.66 × 10841,173,5491,653,1925,188,156
rank11261321195710834
C17-F13mean1315.7983.85 × 109101,532.17.12 × 1091887.5039.87 × 108610,702.261,970.73509,838.459,480,60025,222.2322,449.958,037,399
best1314.591.88 × 10956,657.53.73 × 1091613.80913308283288,394.325,154.762,267.141,306,52120,486.399619.2512,181,265
worst1318.655.4 × 109160,172.58.74 × 1092423.7373.43 × 109902,630.9123,812.21,580,77187,708,50336,797.0950,129.1817,287,717
std2.1054221.59 × 10946,800.182.49 × 109398.41841.78 × 109348,460.450,365.05787,069.221,850,0508488.76820,311.387,045,508
median1314.9764.07 × 10994,649.28 × 1091756.2332.54 × 108625,891.749,457.99198,157.754,453,68921,802.7115,025.686,340,306
rank11261321185710439
C17-F14mean1423.0171,277,259183,187.81,480,0891440.332791,751.81,498,44414,169.61359,607.794,702.5771,02013,109.591,352,841
best1422.014787,819.426,021.81744,142.81436.987566,674.524,657.713833.5123,619.6855,233.67500,470.52607.809224,311.9
worst1423.9931,616,727423,495.82,203,7741445.0921,118,3444,576,70923,795.94770,200.4108,892.31,163,89723,553.042,280,465
std0.879535422,235.6190,861.1764,154.84.048836275,535.42,274,9569358.04412,483.528,647.14339,781.19957.2071,032,057
median1423.0311,352,245141,616.71,486,2201439.623740,994.6696,204.114,524.49322,305.3107,342709,85613,138.761,453,294
rank11061229134758311
C17-F15mean1503.132.05 × 10825,743.454.02 × 1081618.3169,678,7593,396,91729,333.4710,656,3803,457,07511,347.813746.102643,946.8
best1502.4631.77 × 1087875.2733.47 × 1081580.9893,813,143156,980.817,207.1366,683.28785,3678201.2441804.924118,632.2
worst1504.2672.27 × 10841,482.54.44 × 1081635.0522,514,05411,028,24848,172.539,897,7456,507,17515,205.026514.5161,442,109
std0.9312326,757,11615,445.9351,771,10827.296239,394,6475,612,36714,605.7321,226,0822,553,2973185.3242268.093659,103.4
median1502.8952.08 × 10826,8084.09 × 1081628.6126,193,9201,201,21925,977.131,330,5453,267,87910,992.53332.485507,522.8
rank11251321086119437
C17-F16mean1663.4743568.4392681.4114011.1152025.8582863.8933518.5662403.5612373.82993.5283129.432634.2812647.031
best1614.7283307.6572463.1933449.3881732.3022484.5543053.1242208.1942178.5592769.2712909.4992432.8652320.343
worst1744.123799.7853079.2444528.8462279.9023082.4574174.8152533.4932538.9633234.5393233.0522868.6642965.881
std67.44314219.693298.9312563.965268.2093284.3054518.5014157.2443187.9758224.2454162.5764197.8732334.0578
median1647.5233583.1562591.6034033.1132045.6132944.283423.1622436.282388.8392985.1523187.5842617.7982650.95
rank11271328114391056
C17-F17mean1728.12906.3392276.9073111.5121864.4832815.6052529.5082011.9921912.2642088.6432310.6442183.7282062.855
best1718.7612505.6162188.5352859.8581754.0492111.4882211.2041942.2851789.2841944.8412207.1732033.9562002.551
worst1733.6613373.822323.8123576.6981926.0414638.6872713.8822123.9072032.7482253.8282422.942469.8592120.463
std7.3012400.403668.35209363.245582.578141324.987240.952485.26455119.8597140.5146108.6653214.926753.74566
median1729.9892872.962297.643004.7471888.922256.1232596.4721990.8881913.5122077.9512306.2322115.5492064.203
rank11281321110436975
C17-F18mean1825.69719,134,4461,784,00622,000,5171896.58224,464,9333,973,536431,441.6283,040.91,122,150347,272.992,986.632,454,920
best1822.5255,512,424190,522.67,113,2011873.99897,701.21,339,457109,027.953,417.9521,268.4194,965.166,344.711,916,677
worst1828.4237,159,5523,558,88943,221,7831909.63446,361,7338,200,7011,166,914726,273.51,410,592675,556.9110,216.23,598,176
std2.94036415,225,8021,718,03116,663,82717.4394927,476,1163,208,798537,005.1344,619.5444,998.7241,287.120,875.12838,994.8
median1825.92116,932,9041,693,30718,833,5431901.35225,300,1493,176,993224,912.2176,236.11,278,369259,284.897,692.82,152,413
rank11181221310647539
C17-F19mean1910.9893.91 × 10846,201.226.59 × 1081923.7831.98 × 1089,646,131633,054.82,715,8753,872,54455,673.8730,583.081,091,994
best1908.8412.92 × 10810,342.484.76 × 1081921.20224623441,255,74616,582.0848,298.052,010,39230,482.76525.453431,854.3
worst1913.0955.09 × 108102,145.59.99 × 1081928.7095.49 × 10816,655,7461,422,5478,756,3655,504,51274,705.1290,345.841,939,444
std2.1025581.18 × 10843,520.532.52 × 1083.662182.75 × 1087,642,858744,628.84,412,4691,870,46320,035.843,510.04691,979.8
median1911.013.81 × 10836,158.435.81 × 1081922.6111.21 × 10810,336,517546,545.11,029,4193,987,63658,753.8312,730.52998,338
rank11241321110689537
C17-F20mean2065.7882666.7082488.2472703.242176.8912633.5282625.1512467.5922311.5882598.4282741.6572428.3422379.616
best2029.5232605.0812389.1242582.792061.082544.6752514.9372273.5252156.3152559.6472496.6842366.3062309.465
worst2161.1272707.2762610.7112770.1162265.332699.372756.9682771.2582427.8642682.5563094.922526.6752429.376
std69.2664847.36501101.17390.3070493.0314570.35704109.9008232.2741123.569162.23832275.376876.1418455.62911
median2036.2512677.2382476.5772730.0262190.5772645.0332614.3492412.7912331.0872575.7552687.5122410.1932389.813
rank11171221096381354
C17-F21mean2308.4572540.5312417.3342579.4582366.6562481.1462532.2922393.0242382.7042454.2942505.0542412.9672452.339
best2304.0342473.5622270.8132523.0922356.9692323.9932485.022366.9122363.9352442.9382490.7512398.0422428.045
worst2312.9882583.2632523.6592647.8082382.7582571.812574.0882410.4182392.1522465.1642529.1382427.0612491.924
std4.8528357.52589115.389858.9879212.3411120.146249.4110420.095114.301611.4557418.5313613.9277729.96371
median2308.4022552.652437.4312573.4662363.4492514.392535.032397.3822387.3642454.5362500.1632413.3822444.694
rank11261329114381057
C17-F22mean23006159.1734657.7825994.3632302.9356695.4275766.8153423.9932575.1874598.675034.7644055.3442573.944
best23005928.7842302.7865294.9952301.9146538.315117.0912305.372487.1722588.5653455.5372407.6832523.642
worst23006518.3525567.5766696.9462304.6586769.1646346.2524812.8372755.3076827.1225728.7075649.3722612.918
std1.62 × 10−5274.1631711.374655.65631.339246118.2848555.97971425.549133.292511.4711153.2811622.32848.22624
median23006094.7785380.3835992.7572302.5836737.1175801.9593288.8822529.1334489.4965477.4064082.162579.608
rank11281121310547963
C17-F23mean2655.083018.5542837.8323056.0172645.9953021.882919.912708.4832717.862822.1113411.742819.8052870.907
best2653.7422981.5992706.4833034.1022470.2192961.0122820.6092671.5982656.1032751.2663359.5182740.2942794.193
worst2657.3773094.2942977.5023079.4512713.0353176.5473008.0212732.9252747.2022876.3123492.822876.4362935.906
std1.79991455.76503121.722320.26983127.8613112.997685.524731.958646.135156.9940967.6883963.5829363.44241
median2654.6012999.1632833.6713055.2572700.3632974.982925.5062714.7052734.0682830.4343397.312831.2442876.765
rank21071211193461358
C17-F24mean2831.413179.1083080.4863247.4812884.0893155.713043.272899.2842909.6492992.6123212.5923053.2983118.4
best2829.9933148.0562987.5243181.2732868.2793081.8243001.5682868.5872903.3182971.2893182.1372996.2183054.623
worst2832.3673235.1183181.5583355.9262891.0283192.7963062.0562910.7072916.4743019.7553239.3893132.246