Application of Stork Optimization Algorithm for Solving Sustainable Lot Size Optimization

1  Introduction

Supply Chain Management (SCM) stands as the cornerstone of modern business operations, orchestrating the seamless flow of information, services, and goods from raw material suppliers to end consumers. SCM encompasses a wide array of activities, including procurement, production, inventory management, logistics, and distribution, all aimed at optimizing the overall efficiency and effectiveness of the supply chain network [1]. In today’s highly competitive and globalized marketplace, effective SCM has become a strategic imperative for businesses seeking to gain a competitive edge, enhance customer satisfaction, and drive sustainable growth [2].

Ensuring quality throughout the supply chain is crucial for businesses aiming to enhance efficiency, lower expenses, and swiftly respond to the ever-changing market demands [3,4]. Consequently, SCM applications are often framed as optimization problems that require suitable techniques for resolution. Optimization problem-solving methods are generally categorized into deterministic and stochastic approaches [5]. Deterministic methods, which include gradient-based and non-gradient-based techniques, are effective in solving convex, linear, continuous, differentiable, and low-dimensional problems [6,7]. Traditional optimization methods, such as mathematical modeling and linear programming, have long been employed for supply chain optimization [8]. Nevertheless, these approaches frequently encounter challenges in dealing with the intricate nature and unpredictability of real-world supply chains. Hence, there is a growing enthusiasm for devising metaheuristic algorithms, drawing inspiration from natural processes, to effectively tackle supply chain optimization problems [9].

Metaheuristic algorithms represent widely used stochastic approaches capable of producing effective solutions for optimization problems through random search within the solution space [10]. Although these algorithms do not guarantee finding the global optimum, the solutions they generate are usually close enough to be considered quasi-optimal. The continuous pursuit of more effective optimization solutions has driven researchers to develop numerous metaheuristic algorithms [11].

A significant research question arises: given the existing metaheuristic algorithms, is it still necessary to design new ones? The No Free Lunch (NFL) [12] provides an answer, explaining that due to the random search nature of metaheuristic algorithms, no single algorithm can be the best optimizer for all optimization applications. This inherent diversity in optimization challenges encourages researchers to explore and design innovative metaheuristic algorithms that can address specific problem characteristics, improve performance, and adapt to changing requirements. The pursuit of new algorithms enables the optimization community to push the boundaries of problem-solving capabilities, enhance efficiency, and provide more tailored solutions for a wide array of real-world applications.

Based on extensive literature review, no metaheuristic algorithm inspired by the natural behavior of storks has been previously designed. The intelligent strategies of storks during hunting and their activities during winter migration present a unique potential for developing a new metaheuristic algorithm. To fill this research gap, this paper introduces a novel metaheuristic algorithm inspired by the intelligent behavior of storks in nature, which is detailed in the subsequent sections.

Although similar studies are mentioned in the literature review, the innovative aspects of this paper compared to several of these high-repeated studies are as follows:

In paper [13], the butterfly algorithm is employed for the green lot size optimization problem and it is compared with three methods: genetic algorithm, particle swarm optimization, and firefly algorithm. Although all these algorithms have been widely used, they have not been able to properly address the challenge of green lot size optimization. This is due to the fact that with the progress of science, optimization problems become more complex and existing algorithms may not have the necessary efficiency to effectively solve new optimization challenges. Therefore, the innovation of the proposed approach of this paper, compared to the mentioned source, is to achieve more effective solutions for the green lot size optimization problem by introducing a new algorithm that has separate attitudes to manage exploration and exploitation.

In paper [14], a new optimization approach called Wombat Optimization Algorithm is introduced to solve optimization problems. In that article, it is mentioned that Wombat Optimization Algorithm can be employed as a problem-solving tool to address the green lot size optimization problem in future studies. However, no simulations have been performed, and this issue is raised as a research proposal for future studies. Therefore, the innovation of the proposed approach in this article, compared to the mentioned source, is that the application of a new meta-heuristic algorithm called stork optimization algorithm has been specifically studied on the challenge of green lot size optimization and the results obtained are compared with twelve famous algorithms.

In paper [15], a new metaheuristic algorithm called Technical and Vocational Education and Training-Based Optimizer (TVETBO) is proposed, aimed at handling optimization tasks across various sciences. TVETBO is a human-based algorithm inspired by the process of teaching work-related skills to candidates in technical and vocational education and training schools. The innovation of the proposed SOA approach compared to TVETBO lies in both the source of design inspiration and the mathematical modeling process. SOA is proposed as a swarm-based approach inspired by the natural behavior of storks in the wild.

In general, the innovation of a new metaheuristic algorithm compared to existing algorithms lies in the main idea of its design, its mathematical modeling, and the advantages of managing the exploration and exploitation processes. As evident in the literature review, numerous optimization algorithms have been designed so far. In fact, this raises the central question of research: is there a need to design a new algorithm despite the existence of established ones?

Several reasons serve as primary motivations for the introduction of novel metaheuristic algorithms, as described below. The first motivation stems from the stochastic nature of metaheuristic algorithms, which lack certainty in achieving the global optimum. Therefore, the introduction of a new algorithm that effectively manages the search process may lead to superior solutions for optimization problems.

As a second motivation, we can refer to the concept of the NFL theorem, which states: in no way can it be said that a particular metaheuristic algorithm is the best optimizer for all optimization problems. Therefore, the NFL theorem serves as a main motivation for researchers to design newer metaheuristic algorithms to achieve better solutions.

The third motivation stems from the fact that as science progresses, more complex optimization problems arise, which require more precise optimization techniques for resolution. Therefore, older and existing algorithms may not be well-equipped to handle emerging optimization problems, and researchers can achieve suitable solutions for these types of challenges by designing metaheuristic algorithms with more recent perspectives.

This paper introduces the Stork Optimization Algorithm (SOA), a new approach to optimization problems, highlighting several key contributions:

•   SOA is intricately crafted by emulating the natural behavior of storks in their wild habitat.

•   The foundational inspiration for SOA is drawn from two sources: (i) the stork’s strategy during hunting and (ii) the migration of storks during the winter season.

•   The implementation process of SOA is elucidated, with a mathematical model detailing two essential phases, namely exploration and exploitation. These phases are based on the simulation of storks’ behaviors in nature.

•   The effectiveness of SOA to address Supply Chain Management (SCM) tasks is assessed particularly for sustainable lot size optimization.

•   A comprehensive comparison is conducted, pitting the performance of SOA against twelve well-known metaheuristic algorithms.

The paper is structured as follows: Section 2 presents the literature review. Section 3 introduces and models the proposed Stork Optimization Algorithm. Section 4 evaluates the application of SOA in SCM optimization tasks. Section 5 discusses managerial insights. Section 6 concludes the paper and provides suggestions for future research.

2  Literature Review

Metaheuristic algorithms are recognized as powerful optimization techniques that have garnered considerable interest across diverse domains owing to their capacity to effectively address intricate problems. Unlike conventional optimization approaches like linear programming, which might grapple with the intricacies and uncertainties inherent in real-world situations, metaheuristic algorithms provide a versatile and adjustable method for optimization [16,17]. Based on the source of inspiration in the design of metaheuristic algorithms, they are classified into four groups: swarm-based, evolutionary-based, physics-based, and human-based approaches [18].

Swarm-based metaheuristic algorithms are conceptualized by emulating the natural behaviors and strategies observed in animals, aquatic organisms, insects, and other living entities within their natural habitats. Among these algorithms, Particle Swarm Optimization (PSO) stands out as a widely adopted approach, drawing inspiration from the collective movement of birds and fish during their search for food [19]. Another noteworthy example is the Ant Colony Optimization (ACO), which mimics the efficient route-finding ability of ants between their nest and food sources [20]. The various behaviors and strategies employed by wildlife, such as foraging, hunting, migration, and ground digging, have been pivotal inspirations for designing a multitude of algorithms. Examples include the Whale Optimization Algorithm (WOA) [21], Aquila Optimizer (AO) [22], White Shark Optimizer (WSO) [23], Dwarf Mongoose Optimization Algorithm (DMOA) [24], Tunicate Swarm Algorithm (TSA) [25], Ebola Optimization Search Algorithm (EOSA) [26], Marine Predator Algorithm (MPA) [27], Prairie Dog Optimization (PDO) [28], African Vultures Optimization Algorithm (AVOA) [29], Grey Wolf Optimizer (GWO) [30], and Reptile Search Algorithm (RSA) [31].

Evolutionary-based metaheuristic algorithms are crafted by incorporating principles from biology, genetics, and the concepts of survival of the fittest, natural selection, and evolutionary operators. Among the most well-known and widely used algorithms in this category are the Genetic Algorithm (GA) and Differential Evolution (DE). These algorithms are inspired by the biological processes of generation and evolution, as outlined in Darwin’s theory. They employ genetic principles and natural selection, alongside evolutionary operators like random crossover, mutation, and selection, to explore and exploit the search space effectively.

Physics-based metaheuristic algorithms are conceived by incorporating models derived from physics phenomena, forces, transformations, laws, and concepts. Simulated Annealing (SA) stands out as one of the widely embraced physics-based metaheuristic algorithms, drawing inspiration from the phenomenon of metal annealing [32]. Algorithms like the Black Hole Algorithm (BHA) [33] and Multi-Verse Optimizer (MVO) [34] utilize concepts from cosmology in their design. Additionally, physical forces and Newton’s laws of motion serve as sources of inspiration for Gravitational Search Algorithm (GSA) [35].

Human-based metaheuristic algorithms are intricately crafted by mimicking the communication patterns, social interactions, decision-making processes, and strategic behaviors observed in both individual and collective human activities. One prominent example of such an algorithm is Teaching-Learning Based Optimization (TLBO). TLBO derives its fundamental concepts from the educational dynamics within a classroom, emphasizing the interactions between teachers imparting knowledge and students assimilating that knowledge. This algorithm is widely recognized and adopted for its efficacy [36]. Another innovative algorithm in this category is the Mother Optimization Algorithm (MOA). MOA is inspired by the nurturing and developmental phases that a mother, named Eshrat, provides to her children. It models the phases of education, where foundational knowledge is imparted; advice, where guidance and support are given; and upbringing, which encompasses the overall growth and development of the child [18]. Doctor and Patient Optimization (DPO) is another human-based algorithm conceptualized by emulating the therapeutic interactions and communication that occur between patients and their doctors. This algorithm captures the essence of diagnostic and treatment processes, reflecting the critical decision-making and trust-based relationship inherent in medical care [37]. Additionally, the Election-Based Optimization Algorithm (EBOA) takes inspiration from the electoral processes observed in democratic societies. It incorporates the mechanisms of voting, candidate selection, and election procedures to solve optimization problems, reflecting the strategic decision-making and collective choices made during elections [38]. Each of these human-based metaheuristic algorithms leverages the complexity and nuance of human behaviors and social systems, providing robust frameworks for addressing and solving intricate optimization challenges.

The utilization of metaheuristic algorithms in Supply Chain Management (SCM) spans a broad spectrum and includes numerous domains such as inventory control, facility siting, routing of vehicles, scheduling production, and designing supply chain networks. For instance, these algorithms can optimize inventory restocking strategies, reduce transportation expenditures, equalize production capabilities, and craft resilient supply chain infrastructures [39]. Table 1 provides a summary of the various applications of metaheuristic algorithms in addressing Supply Chain Management (SCM) challenges.

3  Stork Optimization Algorithm (SOA)

Within this section, the origin and theoretical underpinnings of the novel Stork Optimization Algorithm (SOA) approach are expounded upon. Subsequently, the procedural steps for its implementation are meticulously formulated in mathematical terms, aiming to provide a structured framework for the resolution of optimization problems.

3.1 Inspiration of SOA

Storks are long-necked, long-legged, large wading birds with stout, long bills. Storks have a nearly cosmopolitan distribution; however, they are mostly seen in sub-Saharan Africa and tropical Asia. There is a difference between the male and female species in terms of size, such that males are larger, but they do not have significant differences in their appearance. The bill size of storks is very large compared to its body size and it has different sizes among different genera. The shape of the bill in different species is also different and related to the diet. In some species, the shape of the beak has evolved to hunt fish in shallow water. In some, it is in the form of massive daggers to feed on carrion, fight scavengers and hunt.

Storks are carnivorous predators whose diet includes fish, insects, small mammals, amphibians, reptiles, and other small invertebrates. The common hunting strategy of storks is to walk or stalk in shallow water and grasslands while watching for prey. One of the natural behaviors of storks is their tendency to long annual migrations in winter. In order to avoid long travel and flights, storks move through water routes. Studies and observations show that unlike passerine migrants, migration routes are learned for storks.

Among the natural behaviors of storks, their strategy when hunting prey and their movement during the annual winter migration is much more significant. These natural activities of storks are intelligent processes that are the basic inspiration in designing the proposed SOA approach.

3.2 Algorithm Initialization

The presented SOA methodology is a metaheuristic algorithm grounded in population dynamics, with storks constituting its individual members. Each member of the SOA embodies specific values for decision variables, determined by its spatial location within the search space. Consequently, every SOA member serves as a potential solution to the problem at hand, and its characteristics can be accurately represented mathematically through a vector. In this vector representation, each element corresponds to a distinct decision variable. The collective assembly of these SOA members establishes the algorithm’s population, and this assembly can be mathematically portrayed as a matrix, as per Eq. (1). The initial positioning of storks within the search space is achieved through a random initialization process, governed by Eq. (2).

X=[X1⋮Xi⋮XN]N×m=[x1,1⋯x1,d⋯x1,m⋮⋱⋮⋮xi,1⋯xi,d⋯xi,m⋮⋮⋱⋮xN,1⋯xN,d⋯xN,m]N×m(1)

xi,d=lbd+r⋅(ubd−lbd)(2)

In this context, the notation X denotes the matrix representing the SOA population, where Xi designates the ith stork, denoted as a candidate solution. The element xi,d within this matrix represents the stork’s position in the dth dimension of the search space, signifying a decision variable. Parameters N and m respectively denote the number of storks and the count of decision variables. The variable r takes on a random value within the range [0,1], while lbd and ubd stand for the lower and upper bounds of the dth decision variable.

To assess the problem’s objective function based on the proposed decision variable values for each stork, an evaluation is conducted. This yields a set of computed values for the objective function, succinctly captured in a vector, as outlined in Eq. (3).

F=[F1⋮Fi⋮FN]N×1=[F(X1)⋮F(Xi)⋮F(XN)]N×1(3)

In this context, the variable F represents the vector encapsulating the evaluated objective function, with Fi denoting the specific assessment of the objective function based on the ith stork.

The evaluated objective function values are pivotal in determining the quality of population members as they present candidate solutions. The highest quality solution is indicated by the most favorable objective function value, representing the best-performing member, whereas the least favorable value denotes the worst-performing member. During each iteration of the Stork Optimization Algorithm (SOA), the positions of the storks within the search space are updated, which subsequently affects the objective function values. This iterative process demands the continuous updating of the best-performing member by comparing the newly acquired objective function values in each iteration. Through this process, the algorithm ensures that the optimal solution is progressively refined.

3.3 Mathematical Modelling of SOA

The proposed Stork Optimization Algorithm (SOA) functions as an iterative process, designed to update the positions of population members through two primary phases: exploration and exploitation. This methodology is inspired by the natural behaviors exhibited by storks. In the exploration phase, the algorithm simulates the migratory patterns of storks, promoting a broad search across the solution space to identify diverse potential solutions. In contrast, the exploitation phase mimics the hunting strategies of storks, focusing on refining and improving the existing solutions to achieve optimal results. The following section elaborates on the detailed procedure for updating the storks’ positions within the search space, ensuring a comprehensive and methodical approach to solving optimization problems. This dual-phase strategy allows SOA to effectively balance the exploration of new regions with the exploitation of known high-quality areas, enhancing its overall performance and robustness in finding optimal solutions.

3.3.1 Phase 1: Migration Strategy (Exploration)

One of the key behaviors observed in storks is their annual migration during the winter season, where they navigate to more favorable habitats. This migration strategy, replicated in the Stork Optimization Algorithm (SOA), forms the foundation of the algorithm’s first phase for updating population members within the search space. By simulating the migratory journey of storks, SOA induces significant movement in the positions of population members, facilitating extensive exploration and global search capabilities. Within the SOA framework, each member identifies potential migration destinations based on the superior objective function values of other population members. Utilizing Eq. (4), these candidate destinations are determined, guiding the migration process for each stork. This approach enables SOA to leverage the collective intelligence of the population, fostering effective exploration of the solution space and enhancing the algorithm’s capacity to discover optimal solutions.

CDi={Xk:Fk<Fiandk≠i},i=1,2,…,Nandk∈{1,2,…,N}(4)

Here, CDi is the set of candidate destinations for migration of the ith stork, Xk is the stork with a better objective function value than ith stork, and Fk is the its objective function value.

Within the framework of SOA, the algorithm posits that every individual stork autonomously selects a migration destination from the pool of potential options in a random manner before embarking on its journey towards the chosen destination. Drawing inspiration from the intricate movements of storks during migration, the algorithm computes a novel position for each stork as it progresses towards its designated migration destination, as defined by Eq. (5). Subsequently, upon reaching the new position, the algorithm evaluates the objective function value. Should this evaluation yield an improvement in the objective function value, the new position effectively supersedes the previous position of the respective stork, as outlined in Eq. (6). This iterative process enables the algorithm to iteratively refine the positions of individual storks based on their movement towards migration destinations, fostering continual optimization and enhancing the algorithm’s capacity to converge towards optimal solutions.

xi,dP1=xi,d+(1−2r)⋅(SCDi,d−I⋅xi,d),i=1,2,…,N,andd=1,2,…,m(5)

Xi={XiP1,FiP1<FiXi,else(6)

Here, XiP1 is the new suggested position of ith stork based on first phase of SOA, xi,dP1 is its dth dimension, FiP1 is its objective function value, r is a random number with a normal distribution in the range of [0,1], SCDi,d is the dth dimension of selected candidate destination for migration of the ith stork, I is a random number from set {1,2}, N is the number of storks, and m is the number of decision variables.

3.3.2 Phase 2: Hunting Strategy (Exploitation)

One of the distinctive behaviors exhibited by storks is their hunting strategy, characterized by a meticulous approach to tracking and capturing prey in grasslands and shallow waters. Storks, being carnivorous birds, employ a combination of surveillance, pursuit, and stealth techniques to stalk and seize their prey. In the context of the SOA framework, the algorithm integrates the simulation of storks’ hunting behaviors into its second phase, which governs the updating of population members within the search space. The deliberate and calculated movements executed by storks during the hunting process induce subtle adjustments in their positions within the search space, thereby augmenting the algorithm’s capacity for local search and exploitation. Within the SOA paradigm, each stork is envisaged to have a prey located proximately to its position, mimicking the natural hunting scenario. Leveraging the simulated dynamics of the stork’s predatory assault on the prey, the algorithm computes a novel position for the stork utilizing Eq. (7). Subsequently, upon evaluating the objective function value associated with the new position, the algorithm ascertains if an enhancement in performance has been attained. Should an improvement in the objective function value be discerned, the stork is then relocated to the new position in accordance with Eq. (8). This iterative process of updating stork positions based on simulated hunting maneuvers fosters localized refinement and exploitation of the search space, thereby facilitating the algorithm’s ability to converge towards optimal solutions.

xi,dP2=(1+1−2rt+1)⋅xi,d,i=1,2,…,N,d=1,2,…,m,andt=1,2,…,T(7)

Xi={XiP2,FiP2<FiXi,else(8)

Here, XiP2 is the new suggested position of the ith stork based on second phase of SOA, xi,dP2 is its dth dimension, FiP2 is its objective function value, t is the iteration counter of the algorithm, and T is the maximum number of algorithm iterations.

3.4 Repetition Process, Pseudocode, and Flowchart of SOA

Once all storks’ positions within the search space undergo updates orchestrated by both the exploration and exploitation phases, the initial iteration of the SOA culminates. Following this, the algorithm seamlessly transitions into subsequent iterations, armed with freshly adjusted values, perpetuating the process of refining the storks’ positions within the search space using Eqs. (4) to (8) until reaching the ultimate iteration. With each iteration, meticulous attention is given to revising and storing the best candidate solution unearthed thus far. Upon the completion of the SOA’s iterative journey, the most promising candidate solution uncovered throughout the algorithm’s rigorous iterations emerges as the definitive resolution to the problem at hand. The detailed steps governing the execution of the SOA are meticulously encapsulated in the form of pseudo-code presented in Algorithm 1.

3.5 Computational Complexity of SOA

In this particular subsection, we venture into a detailed examination of the computational intricacies entailed by the proposed SOA methodology. The preparatory and initialization phase inherent to SOA demonstrate a complexity level quantified at O(Nm), with N signifying the total count of storks involved, while m embodies the number of variables associated with the problem under scrutiny. Within the framework of SOA’s design, the iterative process involves the systematic updating of storks’ positions across two pivotal phases: exploration and exploitation. Consequently, the computational intricacies affiliated with this iterative position updating mechanism are aptly encapsulated within a complexity framework denoted by O(2NmT), where T symbolizes the maximum number of iterations stipulated by the algorithm. Henceforth, in light of these meticulous considerations, the overarching computational complexity attributed to the proposed SOA methodology is succinctly delineated as O(Nm(1 + 2T)).

4  SOA for Sustainable Lot Size Optimization

Within this particular section, the adeptness and efficacy of SOA in navigating the intricacies of optimization tasks within the realm of Supply Chain Management (SCM) are rigorously examined and put to the test. To fulfill this objective, the prowess of SOA is harnessed and applied to the domain of sustainable lot size optimization, serving as a litmus test for its applicability and effectiveness in real-world SCM scenarios.

4.1 Sustainable Lot Size Optimization

Supply chain management involves optimizing the flow of products to meet customer demands efficiently. It requires strategic planning, cooperation among partners, and effective procurement and distribution. Inventory management is crucial, ensuring sustainable and profitable relationships throughout the supply chain. Lot size, the quantity ordered for procurement or production, plays a key role in balancing customer demands with supply. Managing inventories is complex, especially with variable and unclear demand, but it helps coordinate cycles and mitigate risks. The size of the lot impacts customer satisfaction and company profits. Achieving supply chain objectives requires understanding the timing, cost, parameters, and strategies for lot sizing, and optimizing it to improve service levels. Lot sizing optimization is essential for companies to efficiently manage inventory levels and daily consumption coverage [67].

The burgeoning importance of supply chain management has sparked elevated aspirations for advancement within prominent enterprises. While endeavors aimed at cost reduction, such as procurement optimization, lean manufacturing practices, and the externalization of logistical operations, have bolstered the synchronization of both physical and informational flows, a novel paradigm has emerged within supply chain frameworks. This paradigm shift entails a strategic emphasis on the global optimization of networks to mitigate interface losses, curtail inventory stockpiles, and augment customer satisfaction [68]. Consequently, this holistic approach has furnished a competitive edge, particularly in today’s fiercely contested and globally interconnected commercial landscape, typified by discerning consumers and rapid inventory turnover. Nonetheless, this strategic orientation has precipitated environmental repercussions, notably in the form of heightened emission levels. Consequently, escalating public awareness concerning climate change and corporate social responsibility, encompassing concerns such as greenhouse gas emissions, quality of life enhancements, and employment generation, has catalyzed the ascent of sustainable supply chain management [69]. His study endeavors to craft an integrated inventory-emission CO2 model aimed at cost minimization in lot size inventory management, while concurrently addressing the imposition of carbon levies stemming from CO2 emissions incurred during transit under conditions of demand uncertainty. A myriad of research undertakings have sought to delineate optimal emission models, with considerations spanning standard taxation expenses, transit distances, and average CO2 emission rates per kilometer. The emission cost, denoted as CE is delineated as follows [70]:

CE=ECO2⋅dist⋅tCO2(9)

where ECO2 is the average CO2 emission per kilometer; dist is the total distance separation between the supplier and the warehouse; tCO2 is the CO2 emission tax/gr.

4.2 Model and Parameter Setting

Sustainable lot size optimization refers to the process of determining the most environmentally and socially responsible production batch sizes while balancing economic considerations within a supply chain context. Traditional lot sizing models primarily focus on minimizing costs such as setup costs, inventory holding costs, and ordering costs. However, sustainable lot size optimization expands the scope to include environmental impacts, resource utilization, and social considerations.

In sustainable lot size optimization, factors such as energy consumption, raw material usage, waste generation, emissions, and social impacts are taken into account alongside economic factors. The objective is to find lot sizes that not only minimize costs but also minimize negative environmental impacts and promote social responsibility throughout the supply chain.

The formulation of a mathematical model for sustainable lot size optimization necessitates the comprehensive incorporation of both economic considerations and environmental impacts. The primary objective entails the identification of the most optimal lot size at every juncture within the supply chain, with a dual focus on mitigating CO2 emissions and overall expenses. This intricate model encompasses a plethora of constraints, encompassing production constraints, inventory capacity limitations, and the imperative of meeting demand requisites. Furthermore, meticulous attention has been directed towards integrating specific sustainability benchmarks aimed at capping CO2 emissions associated with manufacturing processes, transportation activities, and warehousing operations. The evaluation process mirrors a deterministic framework, wherein the harmonization of economic feasibility with environmental stewardship assumes paramount importance.

The company seeks to minimize shortages, optimize surplus inventory, and determine the ideal lot size. Upon receiving customer demand, shortages in inventory prompt decisions on initiating production or placing orders. Any remaining inventory constitutes backlog, requiring careful monitoring to prevent surplus and devise strategies for reduction if necessary.

The mathematical model of sustainable lot size optimization is defined as follows [13]:

TC=Cc⋅DQ+Cp⋅P⋅Q+SS2+p⋅A⋅DQ+CE⋅DQ(10)

In the provided equation, TC represents the total cost, serving as the objective function to be optimized. The components of this equation include Cc, representing the order cost per unit; Cp, denoting the holding cost per unit; P, indicating the price; p, representing the shortage cost per unit; A, representing the expected shortage per cycle; D, denoting the annual demand; CE, representing the footprint emission cost; Q, indicating the quantity; and SS, representing the shortage. Each of these variables contributes to the overall cost calculation within the supply chain management context, encompassing aspects such as ordering, holding, shortage, demand, emissions, and pricing considerations.

The performance evaluation of SOA is conducted through a rigorous comparison against twelve widely recognized metaheuristic algorithms, encompassing a diverse range of methodologies. These algorithms include the Genetic Algorithm (GA) [71], Particle Swarm Optimization (PSO) [19], Gravitational Search Algorithm (GSA) [35], Teaching-Learning Based Optimization (TLBO) [36], Multi-Verse Optimizer (MVO) [34], Grey Wolf Optimizer (GWO) [30], Whale Optimization Algorithm (WOA) [21], Marine Predator Algorithm (MPA) [27], Tunicate Search Algorithm (TSA) [25], Reptile Search Algorithm (RSA) [31], African Vultures Optimization Algorithm (AVOA) [29], and White Shark Optimizer (WSO) [23]. It is imperative to underscore that, to ensure a fair and equitable comparison, the original formulations of the competing algorithms, as presented by their primary developers, have been utilized in the simulation studies. Moreover, in the case of PSO and GA, the standard iterations, as delineated by Professor Ali Mirjalili, have been employed. The culmination of these comprehensive evaluations is presented in Table 2, wherein the results of implementing SOA and its counterparts in sustainable lot size optimization are meticulously documented. The discerning analysis of these outcomes unequivocally underscores the superior optimization prowess of SOA, as evidenced by its remarkable efficacy in optimizing the objective function and furnishing substantially improved values for TC.

4.3 Statistical Analysis and Discussion

The comparison of metaheuristic algorithms through statistical metrics like mean, best, worst, std, median, and rank yields insightful insights into their respective performances. However, to delve deeper into the statistical significance of the proposed approach compared to its counterparts, a more rigorous statistical analysis is imperative. For this purpose, the Wilcoxon rank sum test [72] is employed, a non-parametric statistical method adept at discerning significant differences between two datasets. By leveraging the p-value index, this test determines whether a notable divergence exists in the performance of two algorithms. The results of the Wilcoxon rank sum test comparing SOA’s performance against that of other competing algorithms are detailed in Table 3. In instances where the p-value index falls below 0.05, it indicates that the proposed approach demonstrates a noteworthy statistical advantage over its corresponding competitors. Based on the outcomes gleaned from the statistical analysis, SOA exhibits significant statistical superiority across all twelve competing algorithms for sustainable lot size optimization.

As it is evident from the results of simulation and statistical analysis, the proposed SOA approach has a significant statistical superiority compared to the competing algorithms so that in all ten case studies it has provided better results as the first best optimizer. This superiority is due to the advantages that the proposed SOA approach has.

One of the reasons for the superiority of SOA is that two separate phases are considered in this algorithm to deal with exploration and exploitation. The exploration phase, focusing on global search, has resulted in the ability of SOA to discover the original optimal region and avoid getting stuck in local optima. The exploitation phase, focusing on local search, has resulted in the ability of SOA to guide the algorithm towards better solutions and converge towards the global optimum.

On the other hand, in SOA design, as seen in Eq. (7), the (1+1−2rt+1) term is considered. This term is well designed for SOA to be able to establish a suitable balance between exploration and exploitation during algorithm iterations. In this way, in the initial iterations where the values of “t” are still small, the priority of the search process is exploration and global search so that the algorithm is able to scan all areas of the problem solving space well with the aim of identifying the main optimal area. After that, with the advancement of the algorithm and the increase of “t” values, the priority of the search process is given by exploitation and local search so that the algorithm can converge towards better and even global optimal solutions with small and accurate displacements near the discovered solutions. Therefore, what has specifically led to the superiority of the proposed SOA approach over competing algorithms is the ability of SOA to manage exploration, exploitation, and balancing them during the search process during algorithm iterations.

5  Managerial Insights

The research conducted on sustainable lot size optimization unveils a myriad of strategic insights beneficial for supply chain managers navigating the complexities of modern-day logistics. By intricately intertwining economic considerations with environmental imperatives, the model furnishes a robust decision-making framework that adeptly juggles efficiency and sustainability goals. An essential managerial takeaway gleaned from this analysis revolves around the nuanced trade-offs between cost reduction strategies and CO2 emission mitigation efforts. Through the model’s lens, decision-makers gain the capacity to scrutinize the impacts of various production and transportation approaches on both economic viability and environmental responsibility. By navigating these intricate balances, managers can pinpoint optimal lot sizes across the supply chain spectrum, thus mitigating CO2 emissions while simultaneously optimizing costs and advancing sustainable development objectives. Furthermore, the model underscores the paramount importance of collaborative synergies, particularly among supply chain stakeholders. Through collaborative mechanisms such as information exchange, harmonized production workflows, and shared sustainability objectives, partners can streamline lot size production processes and minimize environmental footprints. This collaborative ethos not only yields significant cost efficiencies and environmental dividends but also bolsters overall supply chain effectiveness. The insights distilled from this model serve as a compass for managers, empowering them to make informed decisions concerning transportation strategies, inventory management, and production scheduling. By adopting the innovative Stork Optimization Approach (SOA) proposed in this study and integrating it with the sustainable lot size optimization model, supply chain managers can infuse sustainable practices into their operations. This approach enables them to realize cost savings while simultaneously improving environmental performance indicators. Through the utilization of SOA, managers can optimize various aspects of their supply chain, including production, inventory management, and distribution, with a focus on sustainability. By considering environmental factors in decision-making processes, such as minimizing waste and reducing carbon emissions, organizations can align their operations with sustainable objectives. Ultimately, the integration of SOA and sustainable lot size optimization offers a strategic pathway for businesses to enhance both their economic and environmental sustainability profiles within their supply chain operations.

6  Conclusion

This paper introduced an innovative bio-inspired metaheuristic algorithm, coined the Stork Optimization Algorithm (SOA), which takes inspiration from the natural behaviors exhibited by storks. The conceptual foundation of SOA was rooted in the strategic hunting techniques and migratory patterns observed in stork populations. The theoretical underpinnings of SOA where meticulously elucidated, with its implementation procedures mathematically modeled across two distinct phases: (i) exploration, simulating the winter migration of storks, and (ii) exploitation, mirroring the strategic hunting maneuvers of storks. To evaluate the efficacy of SOA within the realm of Supply Chain Management (SCM), the algorithm was applied to sustainable lot size optimization. The optimization results underscored the algorithm’s proficiency in both exploration and exploitation, effectively maintaining a delicate balance between the two throughout the iterative search process. Comparative analysis against twelve other metaheuristic algorithms highlighted the superior performance of SOA, consistently outperforming its competitors across various case studies.

Furthermore, this study unveils a myriad of avenues for future research endeavors. Chief among these is the development of multi-objective and binary versions of SOA, aimed at broadening the algorithm’s applicability and versatility. Additionally, future investigations could explore the potential applications of SOA in addressing optimization challenges spanning diverse scientific domains and real-world engineering contexts.

Acknowledgement: The researchers would like to express their gratitude to the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan for funding this work.

Funding Statement: This research is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan, Grant No. AP19674517.

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Tareq Hamadneh, Khalid Kaabneh, Mohammad Dehghani, Omar Alssayed; data collection: Gulnara Bektemyssova, Zeinab Monrazeri, Dauren Umutkulov, Galymzhan Shaikemelev, Zoubida Benmamoun; analysis and interpretation of results: Zeinab Monrazeri, Gulnara Bektemyssova, Omar Alssayed, Tareq Hamadneh, Zoubida Benmamoun, Khalid Kaabneh; draft manuscript preparation: Tareq Hamadneh, Khalid Kaabneh, Omar Alssayed, Gulnara Bektemyssova, Dauren Umutkulov, Galymzhan Shaikemelev, Mohammad Dehghani. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The authors confirm that the data supporting the findings of this study are available within the article.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

References

1. A. M. Hmouda, G. Orzes, and P. C. Sauer, “Sustainable supply chain management in energy production: A literature review,” Renew. Sustain. Energ. Rev., vol. 191, no. 16, pp. 114085, 2024. doi: 10.1016/j.rser.2023.114085. [Google Scholar] [CrossRef]

2. I. J. Chen and A. Paulraj, “Towards a theory of supply chain management: The constructs and measurements,” J. Oper. Manag., vol. 22, no. 2, pp. 119–150, 2004. doi: 10.1016/j.jom.2003.12.007. [Google Scholar] [CrossRef]

3. I. Jebbor, Z. Benmamoun, and H. Hachimi, “Optimizing manufacturing cycles to improve production: Application in the traditional shipyard industry,” Processes, vol. 11, no. 11, pp. 3136, 2023. doi: 10.3390/pr11113136. [Google Scholar] [CrossRef]

4. K. Khlie and A. Abouabdellah, “Identification of the patient requirements using lean six sigma and data mining,” Int. J. Eng., vol. 30, no. 5, pp. 691–699, 2017. [Google Scholar]

5. L. Liberti and S. Kucherenko, “Comparison of deterministic and stochastic approaches to global optimization,” Int. Trans. Oper. Res., vol. 12, no. 3, pp. 263–285, 2005. doi: 10.1111/j.1475-3995.2005.00503.x. [Google Scholar] [CrossRef]

6. W. G. Alshanti, I. M. Batiha, Ma’mon A. Hammad, and R. Khalil, “A novel analytical approach for solving partial differential equations via a tensor product theory of banach spaces,” Partial Differ. Equ. Appl. Math., vol. 8, no. 2, pp. 100531, Dec. 01, 2023. doi: 10.1016/j.padiff.2023.100531. [Google Scholar] [CrossRef]

7. A. A. Al-Nana, I. M. Batiha, and S. Momani, “A numerical approach for dealing with fractional boundary value problems,” Mathematics, vol. 11, no. 19, pp. 4082, 2023. doi: 10.3390/math11194082. [Google Scholar] [CrossRef]

8. Y. Raouf, Z. Benmamoun, H. Hachimi, I. Jebbor, M. Haqqi and M. Akikiz, “Towards a smart and sustainable industry: Cycle time optimization,” in 2023 3rd Int. Conf. Innov. Res. Appl. Sci., Eng. Technol. (IRASET), Mohammedia, Morocco, 2023, pp. 1–7. [Google Scholar]

9. S. Bouazza, Z. Benmamoun, and H. Hachimi, “Optimization of logistics to support the African’s development,” in 2019 5th Int. Conf. Optim. Appl. (ICOA), Kenitra, Morocco, 2019, pp. 1–5. [Google Scholar]

10. J. de Armas, E. Lalla-Ruiz, S. L. Tilahun, and S. Voß, “Similarity in metaheuristics: A gentle step towards a comparison methodology,” Nat. Comput., vol. 21, no. 2, pp. 265–287, 2022. doi: 10.1007/s11047-020-09837-9. [Google Scholar] [CrossRef]

11. M. Dehghani et al., “Binary spring search algorithm for solving various optimization problems,” Appl. Sci., vol. 11, no. 3, pp. 1286, 2021. doi: 10.3390/app11031286. [Google Scholar] [CrossRef]

12. D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput., vol. 1, no. 1, pp. 67–82, 1997. doi: 10.1109/4235.585893. [Google Scholar] [CrossRef]

13. Z. Benmamoun, W. Fethallah, M. Ahlaqqach, I. Jebbor, M. Benmamoun and M. Elkhechafi, “Butterfly algorithm for sustainable lot size optimization,” Sustainability, vol. 15, no. 15, pp. 11761, 2023. doi: 10.3390/su151511761. [Google Scholar] [CrossRef]

14. Z. Benmamoun, K. Khlie, M. Dehghani, and Y. Gherabi, “WOA: Wombat optimization algorithm for solving supply chain optimization problems,” Mathematics, vol. 12, no. 7, pp. 1059, 2024. doi: 10.3390/math12071059. [Google Scholar] [CrossRef]

15. M. Hubalovska and S. Major, “A new human-based metaheuristic algorithm for solving optimization problems based on technical and vocational education and training,” Biomimetics, vol. 8, no. 6, pp. 508, 2023. doi: 10.3390/biomimetics8060508. [Google Scholar] [PubMed] [CrossRef]

16. S. B. Pandya, K. Kalita, P. Jangir, R. K. Ghadai, and L. Abualigah, “Multi-objective geometric mean optimizer (MOGMOA novel metaphor-free population-based math-inspired multi-objective algorithm,” Int. J. Comput. Intell. Syst., vol. 17, no. 1, pp. 91, Apr. 11, 2024. doi: 10.1007/s44196-024-00420-z. [Google Scholar] [CrossRef]

17. L. Abualigah, A. Diabat, S. Mirjalili, M. Abd Elaziz, and A. H. Gandomi, “The arithmetic optimization algorithm,” Comput. Methods Appl. Mech. Eng., vol. 376, pp. 113609, 2021. [Google Scholar]

18. I. Matoušová, P. Trojovský, M. Dehghani, E. Trojovská, and J. Kostra, “Mother optimization algorithm: A new human-based metaheuristic approach for solving engineering optimization,” Sci. Rep., vol. 13, no. 1, pp. 10312, Jun. 26, 2023. doi: 10.1038/s41598-023-37537-8. [Google Scholar] [PubMed] [CrossRef]

19. J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. ICNN’95—Int. Conf. Neural Netw., Perth, WA, Australia, 1995, vol. 4, pp. 1942–1948. [Google Scholar]

20. M. Dorigo, V. Maniezzo, and A. Colorni, “Ant system: Optimization by a colony of cooperating agents,” IEEE Trans. Syst., Man, Cybern., Part B (Cybern.), vol. 26, no. 1, pp. 29–41, 1996. doi: 10.1109/3477.484436. [Google Scholar] [PubMed] [CrossRef]

21. S. Mirjalili and A. Lewis, “The whale optimization algorithm,” Adv. Eng. Softw., vol. 95, no. 12, pp. 51–67, 2016. doi: 10.1016/j.advengsoft.2016.01.008. [Google Scholar] [CrossRef]

22. L. Abualigah, D. Yousri, M. Abd Elaziz, A. A. Ewees, M. A. Al-qaness and A. H. Gandomi, “Aquila optimizer: A novel meta-heuristic optimization Algorithm,” Comput. Ind. Eng., vol. 157, no. 11, pp. 107250, 2021. doi: 10.1016/j.cie.2021.107250. [Google Scholar] [CrossRef]

23. M. Braik, A. Hammouri, J. Atwan, M. A. Al-Betar, and M. A. Awadallah, “White shark optimizer: A novel bio-inspired meta-heuristic algorithm for global optimization problems,” Knowl.-Based Syst., vol. 243, pp. 108457, 2022. doi: 10.1016/j.knosys.2022.108457. [Google Scholar] [CrossRef]

24. J. O. Agushaka, A. E. Ezugwu, and L. Abualigah, “Dwarf mongoose optimization algorithm,” Comput. Methods Appl. Mech. Eng., vol. 391, pp. 114570, 2022. [Google Scholar]

25. S. Kaur, L. K. Awasthi, A. L. Sangal, and G. Dhiman, “Tunicate swarm algorithm: A new bio-inspired based metaheuristic paradigm for global optimization,” Eng. Appl. Artif. Intell., vol. 90, no. 2, pp. 103541, Apr. 01, 2020. doi: 10.1016/j.engappai.2020.103541. [Google Scholar] [CrossRef]

26. O. N. Oyelade, A. E. S. Ezugwu, T. I. Mohamed, and L. Abualigah, “Ebola optimization search algorithm: A new nature-inspired metaheuristic optimization algorithm,” IEEE Access, vol. 10, no. 2, pp. 16150–16177, 2022. doi: 10.1109/ACCESS.2022.3147821. [Google Scholar] [CrossRef]

27. A. Faramarzi, M. Heidarinejad, S. Mirjalili, and A. H. Gandomi, “Marine predators algorithm: A nature-inspired metaheuristic,” Expert. Syst. Appl., vol. 152, no. 4, pp. 113377, 2020. doi: 10.1016/j.eswa.2020.113377. [Google Scholar] [CrossRef]

28. A. E. Ezugwu, J. O. Agushaka, L. Abualigah, S. Mirjalili, and A. H. Gandomi, “Prairie dog optimization algorithm,” Neural Comput. Appl., vol. 34, no. 22, pp. 20017–20065, 2022. doi: 10.1007/s00521-022-07530-9. [Google Scholar] [CrossRef]

29. B. Abdollahzadeh, F. S. Gharehchopogh, and S. Mirjalili, “African vultures optimization algorithm: A new nature-inspired metaheuristic algorithm for global optimization problems,” Comput. Ind. Eng., vol. 158, no. 4, pp. 107408, 2021. doi: 10.1016/j.cie.2021.107408. [Google Scholar] [CrossRef]

30. S. Mirjalili, S. M. Mirjalili, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw., vol. 69, pp. 46–61, Mar. 01, 2014. doi: 10.1016/j.advengsoft.2013.12.007. [Google Scholar] [CrossRef]

31. L. Abualigah, M. Abd Elaziz, P. Sumari, Z. W. Geem, and A. H. Gandomi, “Reptile search algorithm (RSAA nature-inspired meta-heuristic optimizer,” Expert. Syst. Appl., vol. 191, no. 11, pp. 116158, 2022. doi: 10.1016/j.eswa.2021.116158. [Google Scholar] [CrossRef]

32. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, no. 4598, pp. 671–680, 1983. [Google Scholar] [PubMed]

33. A. Hatamlou, “Black hole: A new heuristic optimization approach for data clustering,” Inf. Sci., vol. 222, pp. 175–184, 2013. doi: 10.1016/j.ins.2012.08.023. [Google Scholar] [CrossRef]

34. S. Mirjalili, S. M. Mirjalili, and A. Hatamlou, “Multi-verse optimizer: A nature-inspired algorithm for global optimization,” Neural Comput. Appl., vol. 27, no. 2, pp. 495–513, 2016. doi: 10.1007/s00521-015-1870-7. [Google Scholar] [CrossRef]

35. E. Rashedi, H. Nezamabadi-Pour, and S. Saryazdi, “GSA: A gravitational search algorithm,” Inf. Sci., vol. 179, no. 13, pp. 2232–2248, 2009. doi: 10.1016/j.ins.2009.03.004. [Google Scholar] [CrossRef]

36. R. V. Rao, V. J. Savsani and D. Vakharia, “Teaching-learning-based optimization: A novel method for constrained mechanical design optimization problems,” Comput.-Aided Des., vol. 43, no. 3, pp. 303–315, 2011. doi: 10.1016/j.cad.2010.12.015. [Google Scholar] [CrossRef]

37. M. Dehghani et al., “A new “Doctor and Patient” optimization algorithm: An application to energy commitment problem,” Appl. Sci., vol. 10, no. 17, pp. 5791, 2020. doi: 10.3390/app10175791. [Google Scholar] [CrossRef]

38. P. Trojovský and M. Dehghani, “A new optimization algorithm based on mimicking the voting process for leader selection,” PeerJ Comput. Sci., vol. 8, no. 4, pp. e976, 2022. doi: 10.7717/peerj-cs.976. [Google Scholar] [PubMed] [CrossRef]

39. L. Abualigah, E. S. Hanandeh, R. A. Zitar, C. L. Thanh, S. Khatir and A. H. Gandomi, “Revolutionizing sustainable supply chain management: A review of metaheuristics,” Eng. Appl. Artif. Intell., vol. 126, no. 6, pp. 106839, 2023. doi: 10.1016/j.engappai.2023.106839. [Google Scholar] [CrossRef]

40. M. Elkhechafi, Z. Benmamoun, H. Hachimi, A. Amine, and Y. Elkettani, “Firefly algorithm for supply chain optimization,” Lobachevskii J. Math., vol. 39, no. 3, pp. 355–367, Apr. 01, 2018. doi: 10.1134/S1995080218030125. [Google Scholar] [CrossRef]

41. K. Patne, N. Shukla, S. Kiridena, and M. K. Tiwari, “Solving closed-loop supply chain problems using game theoretic particle swarm optimisation,” Int. J. Prod. Res., vol. 56, no. 17, pp. 5836–5853, 2018. doi: 10.1080/00207543.2018.1478149. [Google Scholar] [CrossRef]

42. J. Hong, A. Diabat, V. V. Panicker, and S. Rajagopalan, “A two-stage supply chain problem with fixed costs: An ant colony optimization approach,” Int. J. Prod. Econ., vol. 204, no. 3, pp. 214–226, Oct. 01, 2018. doi: 10.1016/j.ijpe.2018.07.019. [Google Scholar] [CrossRef]

43. J. Jiang, D. Wu, Y. Chen, D. Yu, L. Wang and K. Li, “Fast artificial bee colony algorithm with complex network and naive bayes classifier for supply chain network management,” Soft Comput., vol. 23, no. 24, pp. 13321–13337, Dec. 01, 2019. doi: 10.1007/s00500-019-03874-y. [Google Scholar] [CrossRef]

44. A. Cheraghalipour, M. M. Paydar, and M. Hajiaghaei-Keshteli, “Designing and solving a bi-level model for rice supply chain using the evolutionary algorithms,” Comput. Electron. Agric., vol. 162, pp. 651–668, Jul. 01, 2019. doi: 10.1016/j.compag.2019.04.041. [Google Scholar] [CrossRef]

45. W. Yu, G. Hou, and J. Li, “Supply chain joint inventory management and cost optimization based on ant colony algorithm and fuzzy model,” Tehnički Vjesnik, vol. 26, no. 6, pp. 1729–1737, 2019. [Google Scholar]

46. F. Xiong, P. Gong, P. Jin, and J. Fan, “Supply chain scheduling optimization based on genetic particle swarm optimization algorithm,” Cluster Comput., vol. 22, no. Suppl 6, pp. 14767–14775, 2019. doi: 10.1007/s10586-018-2400-z. [Google Scholar] [CrossRef]

47. C. P. Igiri, Y. Singh, D. Bhargava, and S. Shikaa, “Improved African buffalo optimisation algorithm for petroleum product supply chain management,” Int. J. Grid Utility Comput., vol. 11, no. 6, pp. 769–779, 2020. doi: 10.1504/IJGUC.2020.110905. [Google Scholar] [CrossRef]

48. A. K. Sinha and A. Anand, “Optimizing supply chain network for perishable products using improved bacteria foraging algorithm,” Appl. Soft Comput., vol. 86, no. 5, pp. 105921, Jan. 01, 2020. doi: 10.1016/j.asoc.2019.105921. [Google Scholar] [CrossRef]

49. N. Nezamoddini, A. Gholami, and F. Aqlan, “A risk-based optimization framework for integrated supply chains using genetic algorithm and artificial neural networks,” Int. J. Prod. Econ., vol. 225, no. 1, pp. 107569, Jul. 01, 2020. doi: 10.1016/j.ijpe.2019.107569. [Google Scholar] [CrossRef]

50. F. Keshavarz-Ghorbani and S. H. R. Pasandideh, “Optimizing a two-level closed-loop supply chain under the vendor managed inventory contract and learning: Fibonacci, GA, IWO, MFO algorithms,” Neural Comput. Appl., vol. 33, no. 15, pp. 9425–9450, 2021. doi: 10.1007/s00521-021-05703-6. [Google Scholar] [CrossRef]

51. A. M. Fathollahi-Fard, M. A. Dulebenets, M. Hajiaghaei-Keshteli, R. Tavakkoli-Moghaddam, M. Safaeian and H. Mirzahosseinian, “Two hybrid meta-heuristic algorithms for a dual-channel closed-loop supply chain network design problem in the tire industry under uncertainty,” Adv. Eng. Inform., vol. 50, no. 4, pp. 101418, Oct. 01, 2021. doi: 10.1016/j.aei.2021.101418. [Google Scholar] [CrossRef]

52. M. Fathi, M. Khakifirooz, A. Diabat, and H. Chen, “An integrated queuing-stochastic optimization hybrid genetic algorithm for a location-inventory supply chain network,” Int. J. Prod. Econ., vol. 237, no. 8, pp. 108139, Jul. 01, 2021. doi: 10.1016/j.ijpe.2021.108139. [Google Scholar] [CrossRef]

53. R. Ehtesham Rasi and M. Sohanian, “A multi-objective optimization model for sustainable supply chain network with using genetic algorithm,” J. Model. Manag., vol. 16, no. 2, pp. 714–727, 2021. doi: 10.1108/JM2-06-2020-0150. [Google Scholar] [CrossRef]

54. C. Lu, L. Gao, J. Yi, and X. Li, “Energy-efficient scheduling of distributed flow shop with heterogeneous factories: A real-world case from automobile industry in China,” IEEE Trans. Ind. Inform., vol. 17, no. 10, pp. 6687–6696, 2021. doi: 10.1109/TII.2020.3043734. [Google Scholar] [CrossRef]

55. C. Lu, Y. Huang, L. Meng, L. Gao, B. Zhang and J. Zhou, “A pareto-based collaborative multi-objective optimization algorithm for energy-efficient scheduling of distributed permutation flow-shop with limited buffers,” Robot. Comput.-Integr. Manuf., vol. 74, no. 4, pp. 102277, Apr. 01, 2022. doi: 10.1016/j.rcim.2021.102277. [Google Scholar] [CrossRef]

56. P. Seydanlou, F. Jolai, R. Tavakkoli-Moghaddam, and A. M. Fathollahi-Fard, “A multi-objective optimization framework for a sustainable closed-loop supply chain network in the olive industry: Hybrid meta-heuristic algorithms,” Expert. Syst. Appl., vol. 203, no. 2, pp. 117566, Oct. 01, 2022. doi: 10.1016/j.eswa.2022.117566. [Google Scholar] [CrossRef]

57. D. Karami, “Supply chain network design using particle swarm optimization (PSO) algorithm,” Int. J. Ind. Eng. Oper. Res., vol. 4, no. 1, pp. 1–8, 2022. [Google Scholar]

58. G. Chaharmahali, D. Ghandalipour, M. Jasemi, and S. Molla-Alizadeh-Zavardehi, “Modified metaheuristic algorithms to design a closed-loop supply chain network considering quantity discount and fixed-charge transportation,” Expert Syst. Appl., vol. 202, no. 4, pp. 117364, Sep. 15, 2022. doi: 10.1016/j.eswa.2022.117364. [Google Scholar] [CrossRef]

59. V. K. Chouhan, F. Goodarzian, M. Esfandiari, and A. Abraham, “Designing a new supply chain network considering transportation delays using meta-heuristics,” Int. Fuzzy Techniques Emerg. Cond. Digit. Transform. (INFUS 2021), vol. 307, pp. 570–579, 2021. [Google Scholar]

60. A. Mohammed, M. S. Al-shaibani, and S. O. Duffuaa, “A meta-heuristic-based algorithm for designing multi-objective multi-echelon supply chain network,” Appl. Soft Comput., vol. 147, no. 1, pp. 110774, Nov. 01, 2023. doi: 10.1016/j.asoc.2023.110774. [Google Scholar] [CrossRef]

61. C. Pérez, L. Climent, G. Nicoló, A. Arbelaez, and M. A. Salido, “A hybrid metaheuristic with learning for a real supply chain scheduling problem,” Eng. Appl. Artif. Intell., vol. 126, no. 2, pp. 107188, Nov. 01, 2023. doi: 10.1016/j.engappai.2023.107188. [Google Scholar] [CrossRef]

62. R. Kuo, M. F. Luthfiansyah, N. A. Masruroh, and F. E. Zulvia, “Application of improved multi-objective particle swarm optimization algorithm to solve disruption for the two-stage vehicle routing problem with time windows,” Expert Syst. Appl., vol. 225, no. 3, pp. 120009, 2023. doi: 10.1016/j.eswa.2023.120009. [Google Scholar] [CrossRef]

63. A. H. Sadeghi, E. A. Bani, A. Fallahi, and R. Handfield, “Grey wolf optimizer and whale optimization algorithm for stochastic inventory management of reusable products in a two-level supply chain,” IEEE Access, vol. 11, pp. 40278–40297, 2023. doi: 10.1109/ACCESS.2023.3269292. [Google Scholar] [CrossRef]

64. A. M. Golmohammadi, H. Abedsoltan, A. Goli, and I. Ali, “Multi-objective dragonfly algorithm for optimizing a sustainable supply chain under resource sharing conditions,” Comput. Ind. Eng., vol. 187, no. 6, pp. 109837, 2024. doi: 10.1016/j.cie.2023.109837. [Google Scholar] [CrossRef]

65. N. Hamidian, M. M. Paydar, and M. Hajiaghaei-Keshteli, “A hybrid meta-heuristic approach to design a Bi-objective cosmetic tourism supply chain: A case study,” Eng. Appl. Artif. Intell., vol. 127, no. 2, pp. 107331, 2024. doi: 10.1016/j.engappai.2023.107331. [Google Scholar] [CrossRef]

66. V. H. S. Pham, V. N. Nguyen, and N. T. Nguyen Dang, “Hybrid whale optimization algorithm for enhanced routing of limited capacity vehicles in supply chain management,” Sci. Rep., vol. 14, no. 1, pp. 793, 2024. doi: 10.1038/s41598-024-51359-2. [Google Scholar] [PubMed] [CrossRef]

67. Z. Benmamoun, H. Hachimi, and A. Amine, “Inventory management optimization using lean six-sigma,” in Proc. Int. Conf. Ind. Eng. Oper. Manag., Rabat, Morocco, Apr. 11–13, 2017, pp. 11–13. [Google Scholar]

68. Z. Benmamoun, W. Fethallah, S. Bouazza, A. A. Abdo, D. Serrou and H. Benchekroun, “A framework for sustainability evaluation and improvement of radiology service,” J. Clean. Prod., vol. 401, no. 1, pp. 136796, 2023. doi: 10.1016/j.jclepro.2023.136796. [Google Scholar] [CrossRef]

69. M. Ahlaqqach, J. Benhra, S. Mouatassim, and S. Lamrani, “Closed loop location routing supply chain network design in the end of life pharmaceutical products,” Supply Chain Forum: An Int. J., vol. 21, no. 2, pp. 79–92, 2020. doi: 10.1080/16258312.2020.1752112. [Google Scholar] [CrossRef]

70. A. Mortazavi and M. Moloodpoor, “Enhanced butterfly optimization algorithm with a new fuzzy regulator strategy and virtual butterfly concept,” Knowl.-Based Syst., vol. 228, pp. 107291, 2021. doi: 10.1016/j.knosys.2021.107291. [Google Scholar] [CrossRef]

71. D. E. Goldberg and J. H. Holland, “Genetic algorithms and machine learning,” Mach. Learn., vol. 3, no. 2, pp. 95–99, Apr. 11–13, 2017. doi: 10.1023/A:1022602019183. [Google Scholar] [CrossRef]

72. F. Wilcoxon, “Individual comparisons by ranking methods,” in Breakthroughs in Statistics: Methodology and Distribution. New York, NY: Springer New York, 1992, pp. 196–202. doi: 10.1007/978-1-4612-4380-9_16. [Google Scholar] [CrossRef]