Comparative analysis and sensitivity evaluation of metaheuristic algorithms for critical inventory optimization

Comparative analysis and sensitivity evaluation of metaheuristic algorithms for critical inventory optimization | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Comparative analysis and sensitivity evaluation of metaheuristic algorithms for critical inventory optimization Mümün Çalışkan, Murtaza Cicioğlu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7869474/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 15 You are reading this latest preprint version Abstract This study comparatively analyzes the performance of various modern metaheuristic algorithms in addressing the critical stock optimization problem in inventory management. Using a custom-developed software tool, these algorithms were applied to a standardized dataset, and a comprehensive sensitivity analysis was conducted to examine the impact of lead time changes and daily consumption on the optimization results. Findings indicate that certain single-state algorithms demonstrate notable computational efficiency, offering rapid and effective solutions. In terms of overall stock reduction capacity, most algorithms achieved significant decreases in existing stock levels; while some algorithms executed more radical stock reductions, others adopted more conservative approaches. The sensitivity analysis revealed that particular algorithms generated more flexible and robust solutions against uncertainties in lead time and daily consumption, exhibiting less sensitivity to fluctuations in these parameters. Consequently, it is concluded that specific metaheuristic algorithms hold promising potential for developing fast, cost-effective, and efficient inventory policies in such optimization problems. Future research could encompass a more in-depth analysis of cost parameters, explore multi-objective optimization approaches, and investigate the scalability of these algorithms on larger and more complex datasets. Additionally, innovative topics such as real-time data integration and automated algorithm parameter tuning for dynamic inventory management warrant further investigation. Critical Inventory Metaheuristic Algorithms Optimization Sensitivity Evaluation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 1. Introduction In today's competitive market, inventory management is a critical factor that directly impacts a business's operational efficiency and profitability [ 1 , 2 ]. Especially in complex structures like wholesale material sourcing optimization and multi-echelon inventory systems, it holds strategic importance for minimizing stock costs while meeting customer demand [ 3 ]. In this context, determining minimum stock levels based on historical consumption data and lead times for specific products aims to both reduce the risk of stockouts and minimize the costs associated with overstocking [ 4 ]. Moreover, current approaches like sustainable inventory optimization add new dimensions to this field by considering factors such as price flexibility, green demand, and carbon taxes [ 5 ]. This problem is referred to in the literature as the stock level determination problem and fundamentally falls under the scope of dynamic inventory management. The role of machine learning-based optimization techniques in logistics and inventory management to increase supply chain agility and sustainability is also a focus of research in this field [ 6 ]. The optimization problem at hand considers two main dynamic factors: demand forecasting [ 7 , 8 ] and lead times. Accurately predicting product demand based on historical data plays a central role in determining future stock needs, especially in intermittent and irregular demand scenarios, with the use of machine learning algorithms [ 8 ]. Simultaneously, the lead times of products and the uncertainties within these times directly impact the flexibility and reliability of inventory policies. These dynamics complicate the process of optimally determining critical stock levels. In the literature, inventory control problems are generally classified as NP-hard or NP-complete [ 9 , 10 ]. This difficulty arises from the problem's inherent dynamic demand and lead times, the natural uncertainties in predictions based on historical data, and the need to constantly adapt stock levels to changing demand conditions. While solutions can be found in polynomial time under certain assumptions, such as fixed demand and lead times [ 11 ], the complexity and uncertainties of real-world scenarios often make it difficult to find an efficient solution in polynomial time. Consequently, approximate solution methods like metaheuristic algorithms are frequently used for large-scale and dynamic inventory optimization problems [ 12 , 13 , 14 , 15 ]. Figure 1 illustrates a comprehensive comparative analysis of various metaheuristic algorithms for the complex critical inventory optimization problem. The central "Optimal Stock?" dilemma is addressed by a diverse set of algorithms, including Firefly Algorithm (FA), Hill Climbing (HC), Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), Grey Wolf Optimization (GWO), and Ant Colony Optimization (ACO). These algorithms, applied via a developed software platform to the same dataset and problem definition, feed into a "Performance Evaluation & Comparative Analysis." The lower section presents a bar chart summarizing optimization results and a sensitivity analysis diagram, collectively offering insights into the algorithms' efficacy, robustness, and balance in dynamic inventory environments. This systematic approach supports informed decision-making for optimal inventory management. This study addresses the complex nature of the critical stock optimization problem by aiming to comparatively analyze the performance of various modern metaheuristic algorithms. Using developed software, algorithms such as FA, HC, SA, TS, GA, PSO, DE, GWO and ACO were applied to the same dataset and problem definition. Furthermore, a sensitivity analysis was conducted to examine the impact of changes in critical parameters like lead time and daily consumption on the optimization results. This comprehensive comparative and sensitivity analysis aims to provide valuable insights to decision-makers in the field of inventory management regarding the strengths and weaknesses of different algorithms, thereby contributing to the development of more robust and cost-effective stock policies. 2. Material and Method Problem Definition and Objective Function The problem addressed in this study focuses on determining the optimal stock levels for specific items within a company’s inventory, with the objective of minimizing a predefined cost function. The problem is formulated based on critical threshold values for each item, such as required_min_stock and top_two_demand , aiming to minimize the penalty costs incurred when stock levels fall below these thresholds. Specifically, the goal is to reduce the costs arising from the proposed stock level, represented by F(x i ) , failing to meet these critical demands or minimum stock requirements for each item. The mathematical model of problem can be expressed in Eq. ( 1 ). $$\:{MinF(x}_{i})={C}_{critical,\:\:i\left({x}_{i}\right)}+\:{C}_{top\_demand,\:\:i\left({x}_{i}\right)}+\:{Noise}_{i}$$ 1 The notation x i ​ represents the stock level suggested by the optimization algorithm for the i-th item. This is the decision variable the algorithm seeks to optimize. The Noise i ​ represents a random noise term introduced to the optimization process to reflect real-world uncertainties and the stochastic nature of inventory management problems. This encourages the algorithm to find more robust solutions and helps avoid local optima. Its use is optional. Ccritical,i(xi) can be expressed in Eq. ( 2 ). The penalty cost incurred when the stock level for the i -th item falls below the required_min_stock threshold. This cost generally reflects a critical stockout scenario or an urgent replenishment need. $$\:{C}_{critical,\:\:i\left({x}_{i}\right)}=\left\{\begin{array}{c}\left({R}_{i}-{x}_{i}\right)\times\:{P}_{1}\:\:\:\:\:if\:\:\:\:\:\:\:\:{x}_{i}<{R}_{i}\\\:0\:\:\:\:\:\:\:\:\:\:\:\:else\end{array}\right.\:$$ 2 The notation R i ​ represents the required_min_stock threshold for the i -th item, and P 1 ​ represents the cost multiplier per unit, set to 50 in this study. Ctop_demand,i(xi) can be expressed in Eq. ( 3 ). The penalty cost incurred when the stock level for the i -th item falls below the top_two_demand threshold, representing a more critical or costly stock-out condition. $$\:{C}_{top\_demand,\:\:i\left({x}_{i}\right)}=\left\{\begin{array}{c}\left({T}_{i}-{x}_{i}\right)\times\:{P}_{2}\:\:\:\:\:if\:\:\:\:\:\:\:\:{x}_{i}<{T}_{i}\\\:0\:\:\:\:\:\:\:\:\:\:\:\:else\end{array}\right.$$ 3 The notation T i represents the top_two_demand threshold for the i-th item, and P 2 ​ represents the cost multiplier per unit, set to 100 in this study. The objective function calculates the total penalty cost incurred when the proposed stock level falls below the predefined required_min_stock and top_two_demand thresholds. The additive nature of the top_two_demand -related cost on top of the required_min_stock cost indicates that top_two_demand represents a more critical threshold, with a correspondingly higher unit penalty (P 2 > P 1 ) when violated. The optimization algorithm aims to minimize this function, thereby identifying the stock level ( x i ) that results in the lowest cumulative penalty cost. 2.1 Datasets In this study, a real-world data-driven approach is adopted to address the critical stock optimization problem. The analysis is based on two primary datasets: one containing procurement details and predefined minimum stock levels for the items, and the other including the consumption information of the items. These datasets reflect the historical operational data of a company and provide detailed records of inventory movements and supply activities for each individual item. The dataset of lead time and predefined minimum stock level, a sample excerpt of which is presented in Table 1 , comprises static yet essential parameters for each inventory item, including current stock levels, average daily consumption rates, and lead times. Lead times constitute a critical factor for safety stock calculations and the evaluation of potential stock-out risks. Table 1 Sample excerpt from the content of the lead time and predefined min. stock level file Item Code Lead Time Pre-defined Min. Stock Level 1013 20 4 1014 18 2 1017 45 2 1020 16 2 1023 16 2 Table 2 presents a sample excerpt from the file which includes consumption information for a specific material code. This file records product movements that occurred within a defined time period, including the date and quantity of consumption. It contains time-stamped consumption data for each inventory item and serves as a foundational source for constructing demand profiles. In particular, this dataset is critical for calculating daily consumption rates over a specific period and for identifying consumption variability. Table 2 A sample excerpt from the file which includes consumption information for all material codes Item Code Consumed Quantity Date 1013 1 16.07.2020 1013 1 27.07.2020 1013 1 21.08.2020 1013 1 15.09.2020 1013 1 21.09.2020 1013 1 22.09.2020 1013 1 29.09.2020 1013 1 12.10.2020 1013 1 16.10.2020 1013 1 22.10.2020 1013 2 31.10.2020 1013 1 11.11.2020 1013 1 11.11.2020 1013 1 13.11.2020 Prior to integration into the optimization framework, the raw data were subjected to a specialized preprocessing procedure to construct representative consumption profiles. This procedure consists of two main stages. a) Weighted Consumption Calculation ( calculate_weighted_consumption ): Using historical consumption data from the 'Malzeme_hareket.xls' file, a weighted consumption value was calculated for each inventory item. In this step, consumption records from the most recent 12-month period were given twice the weight of those from earlier periods, in order to more accurately reflect current demand trends. This weighting method accounts for temporal demand shifts and seasonal or periodic fluctuations, thereby providing a more realistic demand estimation. A total of 12,346 consumption events were evaluated in this stage. b) Preparation of Consumption Profiles ( prepare_consumption_profile ): For each inventory item, a tailored consumption profile was constructed to support cost calculations associated with stock level optimization. These profiles incorporate key parameters such as average daily consumption and lead time, extracted from the 'stok&temin.xls' dataset. Additionally, the threshold values used in the objective function— required_min_stock and top_two_demand —are defined within these profiles. The required_min_stock threshold typically reflects a minimum critical inventory level or a service level target, while top_two_demand represents a higher-risk threshold, capturing more severe or costly stockout scenarios. Through this preprocessing pipeline, historical consumption dynamics and current supply parameters were systematically integrated to form a structured dataset suitable for effective optimization via metaheuristic algorithms. In total, this analysis was conducted for 780 individual inventory items. 2.2 Metaheuristic Algorithms In this study, nine distinct metaheuristic algorithms were employed to address the complexity of the critical stock optimization problem and to compare algorithmic performance across various optimization scenarios. These algorithms were selected due to their demonstrated effectiveness in solving NP-hard problems and their capability to find solutions close to the global optimum. Each algorithm was specifically designed to minimize the defined objective function. The metaheuristic algorithms used in this study, along with their fundamental operating principles, are summarized below. ACO is inspired by the foraging behavior of real ants, which are capable of discovering the shortest paths to food sources via pheromone trails. In computational terms, artificial "pheromone" values are used to probabilistically guide the search through the solution space. Ants that discover more optimal solutions deposit greater quantities of pheromones, thereby increasing the likelihood that subsequent ants will follow these paths. In inventory optimization problems, ants explore various stock level combinations, reinforcing those associated with lower total costs [ 16 ]. The balance between exploration and exploitation in ACO is primarily governed by its key parameters. These parameters influence how ants adapt to changing cost dynamics, how strongly they are guided by previous experiences, and how long the algorithm refines its solutions. Together, they determine the algorithm's efficiency in identifying cost-effective inventory strategies within a complex and dynamic solution space. Please refer the parameters which are used in this study on the Table 3 . Table 3 Parameters of ACO Parameter Description Value α Phermone importance factor 0.97 β Heuristic importance factor 0.25 ρ Phermone evaporation rate 0.1 Q Phermone update constant 1.0 m Number of ants 30 t_max Max i 100 The FA is based on the bioluminescent communication behavior of fireflies, wherein individuals are attracted to brighter ones. In the algorithmic model, brightness is associated with solution quality—typically, lower-cost solutions are considered more "luminous." This attractiveness mechanism facilitates exploration and exploitation within the solution space, allowing convergence toward optimal or near-optimal inventory levels [ 17 ]. The performance of the Firefly Algorithm in solving inventory optimization problems is strongly influenced by a set of key parameters. These parameters govern how fireflies move within the solution space, how strongly they are attracted to one another, how randomness contributes to exploration, and how light intensity decays with distance. The standard parameters used in this study are summarized in Table 4 . Table 4 Parameters of FA Parameter Description Value α Randomness parameter (α) 0.97 β₀ Initial attractiveness (β₀) 0.25 γ Light absorption coefficient (γ) 0.5 δ Randomization scaling factor 0.9 n Number of fireflies (n) 30 t_max Maximum number of iterations (t_max) 100 HC is a simple yet effective local search algorithm that iteratively moves from the current solution to a better neighboring solution. The process continues until no better neighbors can be found. Despite its efficiency in some scenarios, HC is susceptible to becoming trapped in local optima. Within the context of inventory optimization, it evaluates adjacent stock levels (i.e., slightly increasing or decreasing current stock) and selects the option that minimizes cost [ 18 ]. The performance and convergence behavior of the Hill Climbing algorithm are influenced by several key parameters, such as the size of the neighborhood, the step size used to generate new solutions, and the termination criteria. The standard parameters employed in this study are presented in Table 5 . Table 5 Parameters of HC Parameter Description Value t_max Maximum number of iterations 100 Neighborhood Size Neighborhood size 2 Δ (Delta) Step size (Δ) 0.1 * daily_consumption Initial Solution Initial solution Random.uniform (lower, uper) SA is inspired by the annealing process in metallurgy, which involves heating and controlled cooling to alter material properties. Initially, the algorithm allows the acceptance of inferior solutions with a certain probability, which decreases over time (i.e., as the "temperature" cools). This probabilistic acceptance helps escape local minimum and enables convergence toward a global optimum in complex solution landscapes [ 19 ]. The effectiveness and convergence of the Simulated Annealing algorithm depend on several crucial parameters, including the initial temperature, cooling schedule, and stopping conditions. The standard parameters used in this study are summarized in Table 6 . Table 6 Parameters of HC Parameter Description Value T₀ Initial temperature (T₀) 1000 t_max Maximum number of iteration 100 α (alpha) Cooling rate / cooling schedule (α) 0.95 Δ (Neighbor Step Size) Size of random change applied to current solution np.random.uniform(-1, 1) * daily_consumption Initial Solution Starting point of the search lead_time * daily_consumption * 3 Stopping Criterion Stopping criterion Probability of accepting worse solutions (Metropolis criterion) exp(-ΔE / T) TS extends local search methods by introducing a memory structure known as the "tabu list," which stores previously visited solutions or solution attributes that should be avoided. This mechanism prevents cycling and encourages the exploration of unvisited regions of the search space. In inventory management, TS effectively navigates away from suboptimal stock configurations by utilizing this adaptive memory structure [ 20 ]. The effectiveness of Tabu Search relies on several critical parameters that control the memory structure, neighborhood exploration, and stopping conditions. The key parameters used in this study are detailed in Table 7 . Table 7 Parameters of TS Parameter Description Value Tabu List Size Tabu list size 5 Neighborhood Size Neighborhood size max(1, np.ceil(0.1 * daily_consumption)) Aspiration Criteria Aspiration criteria none t_max Maximum number of iterations 100 Intensification Strategy Intensification strategy random.uniform(lower, upper) Diversification Strategy Diversification strategy Random jump GA is based on the principles of natural selection and genetic evolution. It operates on a population of candidate solutions, each encoded as a chromosome. Through genetic operators such as crossover and mutation, new offspring solutions are generated. The fittest individuals, those that yield the most cost-effective inventory strategies, are retained across generations, facilitating the discovery of optimal or near-optimal solutions [ 21 ]. The performance of the Genetic Algorithm depends on several key parameters, including population size, crossover and mutation rates, selection method, and the number of generations. The standard parameters used in this study are summarized in Table 8 . Table 8 Parameters of GA Parameter Description Value N Population size (N) 30 Pc Crossover rate (Pc) 0.7 Pm Mutation rate (Pm) 0.1 Selection Method Selection method Top 50% by fitness (elitist truncation selection) Gmax Number of generations (Gmax) 100 Elitism Rate Elitism rate 2 PSO draws inspiration from the collective behavior observed in bird flocks and fish schools. Each particle in the swarm represents a candidate solution and adjusts its position in the solution space based on its own experience (personal best, or pbest ) and the experience of the swarm (global best, or gbest ). This dynamic leads to a balance between exploration and exploitation, enhancing the efficiency of the search in inventory optimization contexts [ 22 ]. The performance of the Particle Swarm Optimization (PSO) algorithm depends on several key parameters, including inertia weight, cognitive and social acceleration coefficients, velocity limits, and the number of particles. The standard parameters used in this study are summarized in Table 9 . Table 9 Parameters of PSO Parameter Description Value N Population size 30 w Inertia weight 0.5 c1 Cognitive acceleration coefficient 1.5 c2 Social acceleration coefficient 1.5 V0 Initial velocity range Uniform in [-daily_consumption, daily_consumption] X0 Initial position range Uniform in [lower, upper] Gmax Number of generations 100 DE is a population-based optimization technique closely related to GA. It relies on vector differences between existing population members to create new candidate solutions (mutant vectors). This mechanism enables more diverse exploration of the search space, which is particularly useful in high-dimensional and nonlinear optimization problems such as those found in inventory planning [ 23 ]. The performance of the Differential Evolution (DE) algorithm is influenced by several key parameters, such as population size, mutation factor, crossover rate, and the chosen mutation strategy. The standard parameters used in this study are summarized in Table 10 . Table 10 Parameters of DE Parameter Description Value N Population size 30 F Mutation factor 0.7 CR Crossover rate 0.7 Gmax Number of generations 100 Selection Method The method used to select between target and trial vectors If trial_fitness < fitness[i] → replace Strategy Mutation strategy used a + F * (b - c) (classic DE/rand/1 scheme) GWO simulates the leadership hierarchy and hunting strategies of grey wolves in nature. Wolves are categorized into alpha, beta, delta, and omega levels, reflecting their dominance and role in the pack. The optimization process models the wolves' approach to encircling and attacking prey, which corresponds to the algorithm's convergence toward the global optimum solution. GWO has shown promising performance in complex inventory optimization problems due to its effective balance of exploration and exploitation [ 24 ]. The performance of the Grey Wolf Optimizer (GWO) depends primarily on the population size and a control parameter that governs the balance between exploration and exploitation. The standard parameters utilized in this study are summarized in Table 11 . Table 11 Parameters of GWO Parameter Description Value N Population size 30 Gmax Number of generations 100 a Control parameter between exploration and exploitation a = 2 - generation * (2 / max_generations) Alpha (α) The best candidate solution The best fitness Beta (β) The second-best solution Top 2 fitness Delta (δ) The third-best solution Top 3 fitness Position The current position C = 2 * r (with r ∈ [0,1]) To ensure a fair comparison of algorithmic performance and to promote convergence toward optimal solutions, carefully selected parameter settings were applied to each metaheuristic algorithm. Common control parameters—such as the maximum number of iterations ( max_iter ) and the total number of trials ( num_trials )—were standardized across all algorithms. Additionally, algorithm-specific parameters were determined based on values reported in the literature or through preliminary calibration experiments. 2.3 Evaluation Metrics To rigorously assess and compare the performance of the selected metaheuristic algorithms in solving the critical inventory optimization problem, a set of well-established evaluation metrics was employed. These metrics are designed to capture both the solution quality and the computational efficiency of each algorithm, offering a holistic view of their practical effectiveness. Computational Time This metric refers to the total duration (measured in seconds) required by an algorithm to complete a single optimization run. It is essential for evaluating the scalability and applicability of an algorithm in real-world, time-sensitive decision-making environments. Algorithms that yield high-quality solutions with shorter computation times are generally preferred in operational contexts [ 16 , 25 , 27 ]. Inventory Adjustment Statistics This metric evaluates the adjustments an algorithm proposes by comparing its suggested inventory levels against a baseline. It quantifies the number of items recommended for a stock increase, a decrease, or no change at all. The resulting statistics help to reveal the strategic character of an algorithm—whether it is fundamentally aggressive, conservative, or balanced. This insight is critical for assessing the operational consequences of implementing a particular optimization approach, especially in dynamic inventory systems [ 26 , 28 ]. 2.4 Sensitivity Analysis Methodology Understanding the impact of external factors on solutions in inventory optimization problems is crucial for real-world applications. Therefore, a comprehensive sensitivity analysis was conducted to test the robustness of the proposed optimization solutions. Sensitivity analysis examines how variations in specific critical input parameters affect the performance of the algorithms and the resulting optimal inventory levels. The sensitivity analysis focused primarily on the following two key parameters: Lead Time The effect of changes in product procurement lead times (e.g., unexpected delays or accelerations) on the optimal inventory levels and total costs was investigated. Since lead time directly influences safety stock calculations, fluctuations in this parameter can significantly alter inventory-related costs. Daily Consumption The impact of increases or decreases in the average daily consumption of products on the determined stock levels and associated costs was analyzed. Given that demand uncertainty is one of the greatest challenges in inventory management, testing the system’s sensitivity to consumption changes is critical. The sensitivity analysis was performed by applying ± 40% perturbations to each critical parameter. For each perturbation scenario, the selected metaheuristic algorithms were re-executed, and the newly obtained optimal inventory levels and cost values were recorded. This approach enabled the evaluation of the algorithms' solution robustness and adaptability under varying parameter conditions. The analysis results provided valuable insights into the algorithms’ behavior under uncertainty. 2.5 Software Environment and Hardware All optimization algorithms and data processing procedures presented in this study were developed using the Python programming language. The core libraries employed during development and testing include NumPy for scientific computing, Pandas for data manipulation, and SciPy for statistical analyses. The software was executed on a standard desktop computer equipped with an 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80 GHz processor, 16 GB DDR4 memory, and a 64-bit Windows 10 Pro operating system. This configuration provided sufficient computational power for simulation and performance evaluation of the optimization algorithms. 3. Performance Evaluations 3.1 Comparison of Algorithm Performance The outputs of various metaheuristic algorithms compared in this study (FA, HC, SA, TS, GA, PSO, DE, GWO, ACO) are provided by the program as a separate file. Table 12 presents a sample excerpt from this file. Table 12 A sample excerpt from output file Item Code Current min. stock Lead Time Daily Consumption Annual Consumption FA HC SA TS GA PSO DE GWO ACO 1000 1 16 0.01 4 1 1 1 1 1 1 1 1 2 1001 1 16 0.01 3 1 1 1 1 1 1 1 1 2 1005 1 16 0 1 1 1 1 1 1 1 1 1 2 1007 1 16 0.03 11 2 2 2 2 2 2 2 2 2 1009 1 30 0.01 3 1 1 1 1 1 1 1 1 2 1012 1 17 0.04 13 2 2 2 2 2 2 2 2 2 1013 4 20 0.14 52 6 7 9 4 7 5 8 5 5 1014 2 18 0.03 11 2 2 2 2 2 2 2 2 2 1015 1 16 0.02 6 1 1 1 1 1 1 1 1 2 1016 1 16 0.01 4 1 1 1 1 1 1 1 1 2 1017 2 45 0.01 3 2 1 2 1 2 2 2 2 2 Figure 2 presents the execution times of all algorithms. It can be observed that the single-state algorithms (HC, SA, TS), due to the discrete nature of the problem, are able to find solutions in a very short amount of time and exhibit very similar runtime performances. In contrast, the population-based algorithms require significantly longer computation times to reach their best solutions compared to single-state algorithms. Figure 3 illustrates the distribution of optimized stock levels produced by each algorithm across all 780 inventory items. In this visualization, the optimized values are compared against the empirically determined minimum stock levels defined prior to the execution of the algorithms, allowing for a clear depiction of how each algorithm adjusts inventory levels in relation to the baseline. Figure 4 presents a comparison of the total minimum stock quantities optimized by each algorithm. It is observed that most algorithms reduced the total stock quantity to approximately 900 units, while two algorithms produced solutions exceeding 1,000 units. Nevertheless, all algorithms generated solutions with lower total stock quantities compared to the initial baseline of 1,425 units. Among them, the most aggressive stock reduction was achieved by the Tabu Search (TS) algorithm with 849 units, whereas the most conservative result was obtained by the Ant Colony Optimization (ACO) algorithm, which maintained a total of 1,241 units. 3.2 Sensitivity Analysis Results Each algorithm's output was tested for sensitivity to potential fluctuations in daily consumption and variability in lead times, based on ± 40% perturbations. This analysis aimed to assess the robustness of the solutions generated by the algorithms under uncertain conditions and to determine which algorithm provides more reliable results in the face of such external changes. As illustrated in Fig. 5 , the impact of lead time on the average minimum optimized inventory levels was evaluated within a ± 40% deviation range. Assuming that the 0% point represents the baseline solution produced by each algorithm, it is observed that HC and TS are the algorithms that recommend the lowest inventory levels under a -40% lead time scenario. Moreover, these algorithms also require the smallest increase in inventory when facing a + 40% delay in lead time. Apart from SA and ACO, HC and TS yield results that are relatively close to other algorithms but exhibit greater flexibility. In contrast, SA and ACO are observed to produce solutions that demand more significant changes in inventory levels in response to lead time fluctuations, indicating a lower level of robustness under such conditions. As presented in Fig. 6 , the effect of lead time variability on the average minimum optimized stock quantity was assessed within a ± 40% deviation range. Assuming that the 0% point represents the baseline solution generated by each algorithm, it is observed that HC and TS once again recommend the lowest inventory levels under a -40% consumption scenario. Similarly, in the case of a + 40% increase in consumption, these algorithms require the smallest increase in inventory, demonstrating robustness against demand fluctuations. These two algorithms (HC and TS) produce results that are comparable to most other algorithms—excluding SA and ACO—yet exhibit greater flexibility in adapting to consumption changes. In contrast, SA and ACO tend to generate solutions that require significantly higher adjustments in inventory levels, indicating lower sensitivity resilience compared to the others. To derive meaningful insights from sensitivity analyses and to correctly identify the most appropriate algorithm, it is essential to clearly define the underlying inventory management strategy. This includes specifying whether stockouts for spare parts are acceptable, whether cost minimization is prioritized, and whether storage volume constraints are critical. These strategic considerations directly influence the interpretation and applicability of the results. Nonetheless, when evaluating the outputs of the model holistically—taking into account both performance metrics and sensitivity analyses—it becomes evident that TS, and to a similar extent HC, offer fast convergence, cost-effective, and operationally efficient solutions under varying uncertainty conditions. 4. Conclusion This study presents a comparative analysis of nine metaheuristic algorithms—Ant Colony Optimization (ACO), Firefly Algorithm (FA), Hill Climbing (HC), Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Grey Wolf Optimization (GWO)—applied to the critical inventory optimization problem. In addition to performance comparisons, a comprehensive sensitivity analysis was conducted to evaluate the robustness of the proposed solutions under changes in key external factors, namely lead time and daily consumption. The results demonstrate that single-state algorithms, particularly TS and HC, excel in computational efficiency by generating high-quality solutions in significantly shorter runtimes. While population-based algorithms generally required longer processing times, most were still able to reduce the total minimum inventory level from 1425 items to approximately 900 items. TS achieved the most aggressive reduction with 849 items, whereas ACO was the most conservative, with a total of 1241 items. Sensitivity analysis revealed that TS and HC produced more robust and flexible solutions under ± 40% perturbations in lead time and daily demand, showing minimal increases in inventory levels in both scenarios. On the other hand, SA and ACO were more sensitive to these variations, leading to greater fluctuations in the optimized stock levels. Overall, TS and HC emerged as strong candidates due to their speed, cost-effectiveness, and adaptability. However, selecting the optimal algorithm should be aligned with the organization’s specific inventory management strategy, including factors such as tolerance for stockouts, storage constraints, and cost prioritization. A clearer definition of these strategic considerations will enhance the relevance and accuracy of the chosen optimization approach. Future research may delve into a more granular analysis of cost components, including holding, ordering, and shortage costs, to enhance the economic realism of the models. Additionally, the development of multi-objective optimization frameworks could offer valuable insights into the trade-offs between conflicting performance metrics. As the complexity and scale of real-world problems increase, assessing the scalability of the proposed algorithms on larger and more intricate datasets becomes essential. Moreover, integrating real-time data streams could enable dynamic inventory control, allowing for more responsive and adaptive decision-making. Finally, the implementation of automated parameter tuning mechanisms may further refine algorithm performance, reducing manual intervention and improving overall system efficiency. Declarations Author Contribution M.Ç. and M.C. wrote the main manuscript text and prepared all figures. All authors reviewed the manuscript. References Villegas-Ch, W., Navarro, A.M., Sanchez-Viteri, S.: Optimization of inventory management through computer vision and machine learning technologies. Intell. Syst. 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Berlin, Heidelberg: Springer Berlin Heidelberg, (2009) Russell, S.J.: and Peter Norvig. Artificial Intelligence: A Modern Approach, Global Edition 4e. (2021) Kirkpatrick, Scott, C., Daniel Gelatt Jr., Mario, P.: Vecchi Optim. simulated annealing science. 220 , 671–680 (1983) Glover, F.: Tabu search∗. In: Handbook of combinatorial optimization, pp. 3261–3362. Springer, New York, NY (2013) Holland, J.H.: Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. MIT Press (1992) Kennedy, J., Eberhart, R.: Particle swarm optimization. Proceedings of ICNN'95-international conference on neural networks . Vol. 4. ieee, (1995) Storn, R.: Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11 (4), 341–359 (1997) Mirjalili, S.: Mirjalili, and Andrew Lewis. Grey wolf optimizer. Adv. Eng. Softw. 69 , 46–61 (2014) Talbi, E.-G.: Metaheuristics: from design to implementation. Wiley (2009) Michalewicz, Z., Fogel, D.B.: How to solve it: modern heuristics. Springer Science & Business Media (2013) Gendreau, M., Potvin, J.-Y. (eds.): Handbook of metaheuristics, vol. 2. Springer, New York (2010) Chopra, S.: Peter Meindl, and Dharam Vir Kalra. In: Supply chain management by Pearson. Pearson Education India (2007) Additional Declarations No competing interests reported. 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Cicioğlu","email":"data:image/png;base64,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","orcid":"","institution":"Bursa Uludağ University","correspondingAuthor":true,"prefix":"","firstName":"Murtaza","middleName":"","lastName":"Cicioğlu","suffix":""}],"badges":[],"createdAt":"2025-10-15 14:53:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7869474/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7869474/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101648642,"identity":"cdef1f90-263b-4793-a8a3-1b91b57071c4","added_by":"auto","created_at":"2026-02-02 08:59:35","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":79046,"visible":true,"origin":"","legend":"\u003cp\u003eProposed metaheuristic approaches for critical inventory optimization\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7869474/v1/57155ee4fed9b5eb5c59c607.jpg"},{"id":101648714,"identity":"803d6271-728c-4121-a5b3-94ce0f446ba4","added_by":"auto","created_at":"2026-02-02 08:59:43","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":67967,"visible":true,"origin":"","legend":"\u003cp\u003eProcessing time of each algorithm\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7869474/v1/32b50e55d5013739e2cceb8b.jpg"},{"id":101648645,"identity":"5483b243-4172-4e95-b585-fe763a566999","added_by":"auto","created_at":"2026-02-02 08:59:35","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":106644,"visible":true,"origin":"","legend":"\u003cp\u003eInventory variation according to algorithms\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7869474/v1/50aadf391da54465a4dcefc5.jpg"},{"id":101648717,"identity":"40640bb6-0924-4e14-b248-38105399450b","added_by":"auto","created_at":"2026-02-02 08:59:43","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":76498,"visible":true,"origin":"","legend":"\u003cp\u003eA comparison of the total minimum stock quantities optimized by each algorithm\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7869474/v1/440a68bb9b013ac30e9b979f.jpg"},{"id":101648644,"identity":"72be0838-28ea-4f2f-a061-f50ad4d53f2b","added_by":"auto","created_at":"2026-02-02 08:59:35","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":80127,"visible":true,"origin":"","legend":"\u003cp\u003eThe impact of lead time on the average minimum optimized inventory levels\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7869474/v1/c499b1c4fd181fef307a72d3.jpg"},{"id":101648684,"identity":"4481510f-cf1b-4f90-a334-abdc3b33710c","added_by":"auto","created_at":"2026-02-02 08:59:40","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":83346,"visible":true,"origin":"","legend":"\u003cp\u003eThe impact of lead time variability on the average minimum optimized inventory levels\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7869474/v1/b89cb3e84fc0346aebb117c6.jpg"},{"id":101648795,"identity":"3eb067dc-30c1-4f55-99f5-a457327041c3","added_by":"auto","created_at":"2026-02-02 08:59:50","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1638895,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7869474/v1/79dd46cc-a4c7-4d80-99de-ef79062e78be.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Comparative analysis and sensitivity evaluation of metaheuristic algorithms for critical inventory optimization","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn today's competitive market, inventory management is a critical factor that directly impacts a business's operational efficiency and profitability [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Especially in complex structures like wholesale material sourcing optimization and multi-echelon inventory systems, it holds strategic importance for minimizing stock costs while meeting customer demand [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. In this context, determining minimum stock levels based on historical consumption data and lead times for specific products aims to both reduce the risk of stockouts and minimize the costs associated with overstocking [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Moreover, current approaches like sustainable inventory optimization add new dimensions to this field by considering factors such as price flexibility, green demand, and carbon taxes [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. This problem is referred to in the literature as the stock level determination problem and fundamentally falls under the scope of dynamic inventory management. The role of machine learning-based optimization techniques in logistics and inventory management to increase supply chain agility and sustainability is also a focus of research in this field [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe optimization problem at hand considers two main dynamic factors: demand forecasting [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and lead times. Accurately predicting product demand based on historical data plays a central role in determining future stock needs, especially in intermittent and irregular demand scenarios, with the use of machine learning algorithms [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. Simultaneously, the lead times of products and the uncertainties within these times directly impact the flexibility and reliability of inventory policies. These dynamics complicate the process of optimally determining critical stock levels.\u003c/p\u003e \u003cp\u003eIn the literature, inventory control problems are generally classified as NP-hard or NP-complete [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. This difficulty arises from the problem's inherent dynamic demand and lead times, the natural uncertainties in predictions based on historical data, and the need to constantly adapt stock levels to changing demand conditions. While solutions can be found in polynomial time under certain assumptions, such as fixed demand and lead times [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], the complexity and uncertainties of real-world scenarios often make it difficult to find an efficient solution in polynomial time. Consequently, approximate solution methods like metaheuristic algorithms are frequently used for large-scale and dynamic inventory optimization problems [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates a comprehensive comparative analysis of various metaheuristic algorithms for the complex critical inventory optimization problem. The central \"Optimal Stock?\" dilemma is addressed by a diverse set of algorithms, including Firefly Algorithm (FA), Hill Climbing (HC), Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), Grey Wolf Optimization (GWO), and Ant Colony Optimization (ACO). These algorithms, applied via a developed software platform to the same dataset and problem definition, feed into a \"Performance Evaluation \u0026amp; Comparative Analysis.\" The lower section presents a bar chart summarizing optimization results and a sensitivity analysis diagram, collectively offering insights into the algorithms' efficacy, robustness, and balance in dynamic inventory environments. This systematic approach supports informed decision-making for optimal inventory management.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThis study addresses the complex nature of the critical stock optimization problem by aiming to comparatively analyze the performance of various modern metaheuristic algorithms. Using developed software, algorithms such as FA, HC, SA, TS, GA, PSO, DE, GWO and ACO were applied to the same dataset and problem definition. Furthermore, a sensitivity analysis was conducted to examine the impact of changes in critical parameters like lead time and daily consumption on the optimization results. This comprehensive comparative and sensitivity analysis aims to provide valuable insights to decision-makers in the field of inventory management regarding the strengths and weaknesses of different algorithms, thereby contributing to the development of more robust and cost-effective stock policies.\u003c/p\u003e"},{"header":"2. Material and Method","content":"\u003cp\u003e \u003cstrong\u003eProblem Definition and Objective Function\u003c/strong\u003e \u003cp\u003eThe problem addressed in this study focuses on determining the optimal stock levels for specific items within a company\u0026rsquo;s inventory, with the objective of minimizing a predefined cost function. The problem is formulated based on critical threshold values for each item, such as \u003cem\u003erequired_min_stock\u003c/em\u003e and \u003cem\u003etop_two_demand\u003c/em\u003e, aiming to minimize the penalty costs incurred when stock levels fall below these thresholds. Specifically, the goal is to reduce the costs arising from the proposed stock level, represented by \u003cb\u003eF(x\u003c/b\u003e\u003csub\u003e\u003cb\u003ei\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e)\u003c/b\u003e, failing to meet these critical demands or minimum stock requirements for each item. The mathematical model of problem can be expressed in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{MinF(x}_{i})={C}_{critical,\\:\\:i\\left({x}_{i}\\right)}+\\:{C}_{top\\_demand,\\:\\:i\\left({x}_{i}\\right)}+\\:{Noise}_{i}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe notation \u003cb\u003ex\u003c/b\u003e\u003csub\u003e\u003cb\u003ei\u003c/b\u003e​\u003c/sub\u003e represents the stock level suggested by the optimization algorithm for the i-th item. This is the decision variable the algorithm seeks to optimize. The \u003cb\u003eNoise\u003c/b\u003e\u003csub\u003e\u003cb\u003ei\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e​\u003c/b\u003e represents a random noise term introduced to the optimization process to reflect real-world uncertainties and the stochastic nature of inventory management problems. This encourages the algorithm to find more robust solutions and helps avoid local optima. Its use is optional. \u003cb\u003eCcritical,i(xi)\u003c/b\u003e can be expressed in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The penalty cost incurred when the stock level for the \u003cb\u003ei\u003c/b\u003e-th item falls below the required_min_stock threshold. This cost generally reflects a critical stockout scenario or an urgent replenishment need.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{C}_{critical,\\:\\:i\\left({x}_{i}\\right)}=\\left\\{\\begin{array}{c}\\left({R}_{i}-{x}_{i}\\right)\\times\\:{P}_{1}\\:\\:\\:\\:\\:if\\:\\:\\:\\:\\:\\:\\:\\:{x}_{i}\u0026lt;{R}_{i}\\\\\\:0\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:else\\end{array}\\right.\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe notation \u003cb\u003eR\u003c/b\u003e\u003csub\u003e\u003cb\u003ei\u003c/b\u003e​\u003c/sub\u003e represents the required_min_stock threshold for the \u003cb\u003ei\u003c/b\u003e-th item, and \u003cb\u003eP\u003c/b\u003e\u003csub\u003e\u003cb\u003e1\u003c/b\u003e\u003c/sub\u003e​ represents the cost multiplier per unit, set to 50 in this study. \u003cb\u003eCtop_demand,i(xi)\u003c/b\u003e can be expressed in Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). The penalty cost incurred when the stock level for the \u003cb\u003ei\u003c/b\u003e-th item falls below the top_two_demand threshold, representing a more critical or costly stock-out condition.\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{C}_{top\\_demand,\\:\\:i\\left({x}_{i}\\right)}=\\left\\{\\begin{array}{c}\\left({T}_{i}-{x}_{i}\\right)\\times\\:{P}_{2}\\:\\:\\:\\:\\:if\\:\\:\\:\\:\\:\\:\\:\\:{x}_{i}\u0026lt;{T}_{i}\\\\\\:0\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:else\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe notation \u003cb\u003eT\u003c/b\u003e\u003csub\u003e\u003cb\u003ei\u003c/b\u003e\u003c/sub\u003e represents the top_two_demand threshold for the i-th item, and \u003cb\u003eP\u003c/b\u003e\u003csub\u003e\u003cb\u003e2\u003c/b\u003e\u003c/sub\u003e​ represents the cost multiplier per unit, set to 100 in this study. The objective function calculates the total penalty cost incurred when the proposed stock level falls below the predefined \u003cem\u003erequired_min_stock\u003c/em\u003e and \u003cem\u003etop_two_demand\u003c/em\u003e thresholds. The additive nature of the \u003cem\u003etop_two_demand\u003c/em\u003e-related cost on top of the \u003cem\u003erequired_min_stock\u003c/em\u003e cost indicates that \u003cem\u003etop_two_demand\u003c/em\u003e represents a more critical threshold, with a correspondingly higher unit penalty (P\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;\u0026gt;\u0026thinsp;P\u003csub\u003e1\u003c/sub\u003e) when violated. The optimization algorithm aims to minimize this function, thereby identifying the stock level (\u003cb\u003ex\u003c/b\u003e\u003csub\u003ei\u003c/sub\u003e) that results in the lowest cumulative penalty cost.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Datasets\u003c/h2\u003e \u003cp\u003eIn this study, a real-world data-driven approach is adopted to address the critical stock optimization problem. The analysis is based on two primary datasets: one containing procurement details and predefined minimum stock levels for the items, and the other including the consumption information of the items. These datasets reflect the historical operational data of a company and provide detailed records of inventory movements and supply activities for each individual item.\u003c/p\u003e \u003cp\u003eThe dataset of lead time and predefined minimum stock level, a sample excerpt of which is presented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, comprises static yet essential parameters for each inventory item, including current stock levels, average daily consumption rates, and lead times. Lead times constitute a critical factor for safety stock calculations and the evaluation of potential stock-out risks.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSample excerpt from the content of the lead time and predefined min. stock level file\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eItem Code\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLead Time\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePre-defined Min. Stock Level\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1017\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1020\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1023\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents a sample excerpt from the file which includes consumption information for a specific material code. This file records product movements that occurred within a defined time period, including the date and quantity of consumption. It contains time-stamped consumption data for each inventory item and serves as a foundational source for constructing demand profiles. In particular, this dataset is critical for calculating daily consumption rates over a specific period and for identifying consumption variability.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eA sample excerpt from the file which includes consumption information for all material codes\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eItem Code\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eConsumed Quantity\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eDate\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16.07.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e27.07.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e21.08.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e15.09.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e21.09.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e22.09.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e29.09.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e12.10.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16.10.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e22.10.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e31.10.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e11.11.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e11.11.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e13.11.2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ePrior to integration into the optimization framework, the raw data were subjected to a specialized preprocessing procedure to construct representative consumption profiles. This procedure consists of two main stages.\u003c/p\u003e \u003cp\u003e \u003cp\u003ea) Weighted Consumption Calculation (\u003cem\u003ecalculate_weighted_consumption\u003c/em\u003e): Using historical consumption data from the \u003cem\u003e'Malzeme_hareket.xls'\u003c/em\u003e file, a weighted consumption value was calculated for each inventory item. In this step, consumption records from the most recent 12-month period were given twice the weight of those from earlier periods, in order to more accurately reflect current demand trends. This weighting method accounts for temporal demand shifts and seasonal or periodic fluctuations, thereby providing a more realistic demand estimation. A total of 12,346 consumption events were evaluated in this stage.\u003c/p\u003e \u003cp\u003eb) Preparation of Consumption Profiles (\u003cem\u003eprepare_consumption_profile\u003c/em\u003e): For each inventory item, a tailored \u003cem\u003econsumption profile\u003c/em\u003e was constructed to support cost calculations associated with stock level optimization. These profiles incorporate key parameters such as average daily consumption and lead time, extracted from the \u003cem\u003e'stok\u0026amp;temin.xls'\u003c/em\u003e dataset. Additionally, the threshold values used in the objective function\u0026mdash;\u003cem\u003erequired_min_stock\u003c/em\u003e and \u003cem\u003etop_two_demand\u003c/em\u003e\u0026mdash;are defined within these profiles. The \u003cem\u003erequired_min_stock\u003c/em\u003e threshold typically reflects a minimum critical inventory level or a service level target, while \u003cem\u003etop_two_demand\u003c/em\u003e represents a higher-risk threshold, capturing more severe or costly stockout scenarios.\u003c/p\u003e \u003c/p\u003e \u003cp\u003eThrough this preprocessing pipeline, historical consumption dynamics and current supply parameters were systematically integrated to form a structured dataset suitable for effective optimization via metaheuristic algorithms. In total, this analysis was conducted for 780 individual inventory items.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Metaheuristic Algorithms\u003c/h2\u003e \u003cp\u003eIn this study, nine distinct metaheuristic algorithms were employed to address the complexity of the critical stock optimization problem and to compare algorithmic performance across various optimization scenarios. These algorithms were selected due to their demonstrated effectiveness in solving NP-hard problems and their capability to find solutions close to the global optimum. Each algorithm was specifically designed to minimize the defined objective function. The metaheuristic algorithms used in this study, along with their fundamental operating principles, are summarized below.\u003c/p\u003e \u003cp\u003eACO is inspired by the foraging behavior of real ants, which are capable of discovering the shortest paths to food sources via pheromone trails. In computational terms, artificial \"pheromone\" values are used to probabilistically guide the search through the solution space. Ants that discover more optimal solutions deposit greater quantities of pheromones, thereby increasing the likelihood that subsequent ants will follow these paths. In inventory optimization problems, ants explore various stock level combinations, reinforcing those associated with lower total costs [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The balance between exploration and exploitation in ACO is primarily governed by its key parameters. These parameters influence how ants adapt to changing cost dynamics, how strongly they are guided by previous experiences, and how long the algorithm refines its solutions. Together, they determine the algorithm's efficiency in identifying cost-effective inventory strategies within a complex and dynamic solution space. Please refer the parameters which are used in this study on the Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of ACO\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eα\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePhermone importance factor\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.97\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eβ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHeuristic importance factor\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eρ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePhermone evaporation rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eQ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePhermone update constant\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003em\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of ants\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003et_max\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMax i\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe FA is based on the bioluminescent communication behavior of fireflies, wherein individuals are attracted to brighter ones. In the algorithmic model, brightness is associated with solution quality\u0026mdash;typically, lower-cost solutions are considered more \"luminous.\" This attractiveness mechanism facilitates exploration and exploitation within the solution space, allowing convergence toward optimal or near-optimal inventory levels [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. The performance of the Firefly Algorithm in solving inventory optimization problems is strongly influenced by a set of key parameters. These parameters govern how fireflies move within the solution space, how strongly they are attracted to one another, how randomness contributes to exploration, and how light intensity decays with distance. The standard parameters used in this study are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of FA\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eα\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRandomness parameter (α)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.97\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eβ₀\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInitial attractiveness (β₀)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eγ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLight absorption coefficient (γ)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eδ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRandomization scaling factor\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003en\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of fireflies (n)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003et_max\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMaximum number of iterations (t_max)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eHC is a simple yet effective local search algorithm that iteratively moves from the current solution to a better neighboring solution. The process continues until no better neighbors can be found. Despite its efficiency in some scenarios, HC is susceptible to becoming trapped in local optima. Within the context of inventory optimization, it evaluates adjacent stock levels (i.e., slightly increasing or decreasing current stock) and selects the option that minimizes cost [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. The performance and convergence behavior of the Hill Climbing algorithm are influenced by several key parameters, such as the size of the neighborhood, the step size used to generate new solutions, and the termination criteria. The standard parameters employed in this study are presented in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of HC\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003et_max\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMaximum number of iterations\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eNeighborhood Size\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNeighborhood size\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eΔ (Delta)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStep size (Δ)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1 * daily_consumption\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eInitial Solution\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInitial solution\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRandom.uniform (lower, uper)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eSA is inspired by the annealing process in metallurgy, which involves heating and controlled cooling to alter material properties. Initially, the algorithm allows the acceptance of inferior solutions with a certain probability, which decreases over time (i.e., as the \"temperature\" cools). This probabilistic acceptance helps escape local minimum and enables convergence toward a global optimum in complex solution landscapes [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. The effectiveness and convergence of the Simulated Annealing algorithm depend on several crucial parameters, including the initial temperature, cooling schedule, and stopping conditions. The standard parameters used in this study are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of HC\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eT₀\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInitial temperature (T₀)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003et_max\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMaximum number of iteration\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eα (alpha)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCooling rate / cooling schedule (α)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.95\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eΔ (Neighbor Step Size)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSize of random change applied to current solution\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003enp.random.uniform(-1, 1) * daily_consumption\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eInitial Solution\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStarting point of the search\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003elead_time * daily_consumption * 3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eStopping Criterion\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStopping criterion\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eProbability of accepting worse solutions (Metropolis criterion) exp(-ΔE / T)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTS extends local search methods by introducing a memory structure known as the \"tabu list,\" which stores previously visited solutions or solution attributes that should be avoided. This mechanism prevents cycling and encourages the exploration of unvisited regions of the search space. In inventory management, TS effectively navigates away from suboptimal stock configurations by utilizing this adaptive memory structure [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. The effectiveness of Tabu Search relies on several critical parameters that control the memory structure, neighborhood exploration, and stopping conditions. The key parameters used in this study are detailed in Table\u0026nbsp;\u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab7\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of TS\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eTabu List Size\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTabu list size\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eNeighborhood Size\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNeighborhood size\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003emax(1, np.ceil(0.1 * daily_consumption))\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eAspiration Criteria\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAspiration criteria\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003enone\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003et_max\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMaximum number of iterations\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eIntensification Strategy\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eIntensification strategy\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003erandom.uniform(lower, upper)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eDiversification Strategy\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDiversification strategy\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRandom jump\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eGA is based on the principles of natural selection and genetic evolution. It operates on a population of candidate solutions, each encoded as a chromosome. Through genetic operators such as crossover and mutation, new offspring solutions are generated. The fittest individuals, those that yield the most cost-effective inventory strategies, are retained across generations, facilitating the discovery of optimal or near-optimal solutions [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. The performance of the Genetic Algorithm depends on several key parameters, including population size, crossover and mutation rates, selection method, and the number of generations. The standard parameters used in this study are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab8\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 8\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of GA\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eN\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePopulation size (N)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePc\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCrossover rate (Pc)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePm\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMutation rate (Pm)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eSelection Method\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSelection method\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTop 50% by fitness (elitist truncation selection)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eGmax\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of generations (Gmax)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eElitism Rate\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eElitism rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ePSO draws inspiration from the collective behavior observed in bird flocks and fish schools. Each particle in the swarm represents a candidate solution and adjusts its position in the solution space based on its own experience (personal best, or \u003cem\u003epbest\u003c/em\u003e) and the experience of the swarm (global best, or \u003cem\u003egbest\u003c/em\u003e). This dynamic leads to a balance between exploration and exploitation, enhancing the efficiency of the search in inventory optimization contexts [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. The performance of the Particle Swarm Optimization (PSO) algorithm depends on several key parameters, including inertia weight, cognitive and social acceleration coefficients, velocity limits, and the number of particles. The standard parameters used in this study are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab9\" class=\"InternalRef\"\u003e9\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab9\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 9\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of PSO\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePopulation size\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ew\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInertia weight\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ec1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCognitive acceleration coefficient\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ec2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSocial acceleration coefficient\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eV0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInitial velocity range\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eUniform in [-daily_consumption, daily_consumption]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eX0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInitial position range\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"1\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eUniform in [lower, upper]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGmax\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of generations\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eDE is a population-based optimization technique closely related to GA. It relies on vector differences between existing population members to create new candidate solutions (mutant vectors). This mechanism enables more diverse exploration of the search space, which is particularly useful in high-dimensional and nonlinear optimization problems such as those found in inventory planning [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. The performance of the Differential Evolution (DE) algorithm is influenced by several key parameters, such as population size, mutation factor, crossover rate, and the chosen mutation strategy. The standard parameters used in this study are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab10\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab10\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 10\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of DE\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePopulation size\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMutation factor\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCR\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCrossover rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGmax\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of generations\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSelection Method\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eThe method used to select between target and trial vectors\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eIf trial_fitness\u0026thinsp;\u0026lt;\u0026thinsp;fitness[i] \u0026rarr; replace\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStrategy\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMutation strategy used\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea\u0026thinsp;+\u0026thinsp;F * (b - c) (classic DE/rand/1 scheme)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eGWO simulates the leadership hierarchy and hunting strategies of grey wolves in nature. Wolves are categorized into alpha, beta, delta, and omega levels, reflecting their dominance and role in the pack. The optimization process models the wolves' approach to encircling and attacking prey, which corresponds to the algorithm's convergence toward the global optimum solution. GWO has shown promising performance in complex inventory optimization problems due to its effective balance of exploration and exploitation [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. The performance of the Grey Wolf Optimizer (GWO) depends primarily on the population size and a control parameter that governs the balance between exploration and exploitation. The standard parameters utilized in this study are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab11\" class=\"InternalRef\"\u003e11\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab11\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 11\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of GWO\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePopulation size\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGmax\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of generations\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ea\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eControl parameter between exploration and exploitation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea\u0026thinsp;=\u0026thinsp;2 - generation * (2 / max_generations)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAlpha (α)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eThe best candidate solution\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eThe best fitness\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBeta (β)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eThe second-best solution\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTop 2 fitness\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDelta (δ)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eThe third-best solution\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTop 3 fitness\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePosition\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eThe current position\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eC\u0026thinsp;=\u0026thinsp;2 * r (with r \u0026isin; [0,1])\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTo ensure a fair comparison of algorithmic performance and to promote convergence toward optimal solutions, carefully selected parameter settings were applied to each metaheuristic algorithm. Common control parameters\u0026mdash;such as the maximum number of iterations (\u003cem\u003emax_iter\u003c/em\u003e) and the total number of trials (\u003cem\u003enum_trials\u003c/em\u003e)\u0026mdash;were standardized across all algorithms. Additionally, algorithm-specific parameters were determined based on values reported in the literature or through preliminary calibration experiments.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Evaluation Metrics\u003c/h2\u003e \u003cp\u003eTo rigorously assess and compare the performance of the selected metaheuristic algorithms in solving the critical inventory optimization problem, a set of well-established evaluation metrics was employed. These metrics are designed to capture both the solution quality and the computational efficiency of each algorithm, offering a holistic view of their practical effectiveness.\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eComputational Time\u003c/strong\u003e \u003cp\u003eThis metric refers to the total duration (measured in seconds) required by an algorithm to complete a single optimization run. It is essential for evaluating the scalability and applicability of an algorithm in real-world, time-sensitive decision-making environments. Algorithms that yield high-quality solutions with shorter computation times are generally preferred in operational contexts [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e].\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eInventory Adjustment Statistics\u003c/strong\u003e \u003cp\u003eThis metric evaluates the adjustments an algorithm proposes by comparing its suggested inventory levels against a baseline. It quantifies the number of items recommended for a stock increase, a decrease, or no change at all. The resulting statistics help to reveal the strategic character of an algorithm\u0026mdash;whether it is fundamentally aggressive, conservative, or balanced. This insight is critical for assessing the operational consequences of implementing a particular optimization approach, especially in dynamic inventory systems [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e].\u003c/p\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Sensitivity Analysis Methodology\u003c/h2\u003e \u003cp\u003eUnderstanding the impact of external factors on solutions in inventory optimization problems is crucial for real-world applications. Therefore, a comprehensive sensitivity analysis was conducted to test the robustness of the proposed optimization solutions. Sensitivity analysis examines how variations in specific critical input parameters affect the performance of the algorithms and the resulting optimal inventory levels. The sensitivity analysis focused primarily on the following two key parameters:\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eLead Time\u003c/strong\u003e \u003cp\u003eThe effect of changes in product procurement lead times (e.g., unexpected delays or accelerations) on the optimal inventory levels and total costs was investigated. Since lead time directly influences safety stock calculations, fluctuations in this parameter can significantly alter inventory-related costs.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eDaily Consumption\u003c/strong\u003e \u003cp\u003eThe impact of increases or decreases in the average daily consumption of products on the determined stock levels and associated costs was analyzed. Given that demand uncertainty is one of the greatest challenges in inventory management, testing the system\u0026rsquo;s sensitivity to consumption changes is critical.\u003c/p\u003e \u003c/p\u003e \u003cp\u003eThe sensitivity analysis was performed by applying\u0026thinsp;\u0026plusmn;\u0026thinsp;40% perturbations to each critical parameter. For each perturbation scenario, the selected metaheuristic algorithms were re-executed, and the newly obtained optimal inventory levels and cost values were recorded. This approach enabled the evaluation of the algorithms' solution robustness and adaptability under varying parameter conditions. The analysis results provided valuable insights into the algorithms\u0026rsquo; behavior under uncertainty.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5 Software Environment and Hardware\u003c/h2\u003e \u003cp\u003eAll optimization algorithms and data processing procedures presented in this study were developed using the Python programming language. The core libraries employed during development and testing include NumPy for scientific computing, Pandas for data manipulation, and SciPy for statistical analyses. The software was executed on a standard desktop computer equipped with an 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80 GHz processor, 16 GB DDR4 memory, and a 64-bit Windows 10 Pro operating system. This configuration provided sufficient computational power for simulation and performance evaluation of the optimization algorithms.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Performance Evaluations","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Comparison of Algorithm Performance\u003c/h2\u003e \u003cp\u003eThe outputs of various metaheuristic algorithms compared in this study (FA, HC, SA, TS, GA, PSO, DE, GWO, ACO) are provided by the program as a separate file. Table\u0026nbsp;\u003cspan refid=\"Tab12\" class=\"InternalRef\"\u003e12\u003c/span\u003e presents a sample excerpt from this file.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab12\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 12\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eA sample excerpt from output file\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"14\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c11\" colnum=\"11\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c12\" colnum=\"12\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c13\" colnum=\"13\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c14\" colnum=\"14\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eItem Code\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCurrent min. stock\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eLead Time\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDaily Consumption\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eAnnual Consumption\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eFA\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eHC\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eSA\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eTS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\"\u003e \u003cp\u003eGA\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c11\"\u003e \u003cp\u003ePSO\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c12\"\u003e \u003cp\u003eDE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c13\"\u003e \u003cp\u003eGWO\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c14\"\u003e \u003cp\u003eACO\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1009\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1012\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e52\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1016\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1017\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the execution times of all algorithms. It can be observed that the single-state algorithms (HC, SA, TS), due to the discrete nature of the problem, are able to find solutions in a very short amount of time and exhibit very similar runtime performances. In contrast, the population-based algorithms require significantly longer computation times to reach their best solutions compared to single-state algorithms.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e illustrates the distribution of optimized stock levels produced by each algorithm across all 780 inventory items. In this visualization, the optimized values are compared against the empirically determined minimum stock levels defined prior to the execution of the algorithms, allowing for a clear depiction of how each algorithm adjusts inventory levels in relation to the baseline.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e presents a comparison of the total minimum stock quantities optimized by each algorithm. It is observed that most algorithms reduced the total stock quantity to approximately 900 units, while two algorithms produced solutions exceeding 1,000 units. Nevertheless, all algorithms generated solutions with lower total stock quantities compared to the initial baseline of 1,425 units. Among them, the most aggressive stock reduction was achieved by the Tabu Search (TS) algorithm with 849 units, whereas the most conservative result was obtained by the Ant Colony Optimization (ACO) algorithm, which maintained a total of 1,241 units.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Sensitivity Analysis Results\u003c/h2\u003e \u003cp\u003eEach algorithm's output was tested for sensitivity to potential fluctuations in daily consumption and variability in lead times, based on \u0026plusmn;\u0026thinsp;40% perturbations. This analysis aimed to assess the robustness of the solutions generated by the algorithms under uncertain conditions and to determine which algorithm provides more reliable results in the face of such external changes.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, the impact of lead time on the average minimum optimized inventory levels was evaluated within a\u0026thinsp;\u0026plusmn;\u0026thinsp;40% deviation range. Assuming that the 0% point represents the baseline solution produced by each algorithm, it is observed that HC and TS are the algorithms that recommend the lowest inventory levels under a -40% lead time scenario. Moreover, these algorithms also require the smallest increase in inventory when facing a\u0026thinsp;+\u0026thinsp;40% delay in lead time. Apart from SA and ACO, HC and TS yield results that are relatively close to other algorithms but exhibit greater flexibility. In contrast, SA and ACO are observed to produce solutions that demand more significant changes in inventory levels in response to lead time fluctuations, indicating a lower level of robustness under such conditions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, the effect of lead time variability on the average minimum optimized stock quantity was assessed within a\u0026thinsp;\u0026plusmn;\u0026thinsp;40% deviation range. Assuming that the 0% point represents the baseline solution generated by each algorithm, it is observed that HC and TS once again recommend the lowest inventory levels under a -40% consumption scenario. Similarly, in the case of a\u0026thinsp;+\u0026thinsp;40% increase in consumption, these algorithms require the smallest increase in inventory, demonstrating robustness against demand fluctuations.\u003c/p\u003e \u003cp\u003eThese two algorithms (HC and TS) produce results that are comparable to most other algorithms\u0026mdash;excluding SA and ACO\u0026mdash;yet exhibit greater flexibility in adapting to consumption changes. In contrast, SA and ACO tend to generate solutions that require significantly higher adjustments in inventory levels, indicating lower sensitivity resilience compared to the others.\u003c/p\u003e \u003cp\u003eTo derive meaningful insights from sensitivity analyses and to correctly identify the most appropriate algorithm, it is essential to clearly define the underlying inventory management strategy. This includes specifying whether stockouts for spare parts are acceptable, whether cost minimization is prioritized, and whether storage volume constraints are critical. These strategic considerations directly influence the interpretation and applicability of the results.\u003c/p\u003e \u003cp\u003eNonetheless, when evaluating the outputs of the model holistically\u0026mdash;taking into account both performance metrics and sensitivity analyses\u0026mdash;it becomes evident that TS, and to a similar extent HC, offer fast convergence, cost-effective, and operationally efficient solutions under varying uncertainty conditions.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eThis study presents a comparative analysis of nine metaheuristic algorithms\u0026mdash;Ant Colony Optimization (ACO), Firefly Algorithm (FA), Hill Climbing (HC), Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Grey Wolf Optimization (GWO)\u0026mdash;applied to the critical inventory optimization problem. In addition to performance comparisons, a comprehensive sensitivity analysis was conducted to evaluate the robustness of the proposed solutions under changes in key external factors, namely lead time and daily consumption. The results demonstrate that single-state algorithms, particularly TS and HC, excel in computational efficiency by generating high-quality solutions in significantly shorter runtimes. While population-based algorithms generally required longer processing times, most were still able to reduce the total minimum inventory level from 1425 items to approximately 900 items. TS achieved the most aggressive reduction with 849 items, whereas ACO was the most conservative, with a total of 1241 items. Sensitivity analysis revealed that TS and HC produced more robust and flexible solutions under \u0026plusmn;\u0026thinsp;40% perturbations in lead time and daily demand, showing minimal increases in inventory levels in both scenarios. On the other hand, SA and ACO were more sensitive to these variations, leading to greater fluctuations in the optimized stock levels.\u003c/p\u003e \u003cp\u003eOverall, TS and HC emerged as strong candidates due to their speed, cost-effectiveness, and adaptability. However, selecting the optimal algorithm should be aligned with the organization\u0026rsquo;s specific inventory management strategy, including factors such as tolerance for stockouts, storage constraints, and cost prioritization. A clearer definition of these strategic considerations will enhance the relevance and accuracy of the chosen optimization approach. Future research may delve into a more granular analysis of cost components, including holding, ordering, and shortage costs, to enhance the economic realism of the models. Additionally, the development of multi-objective optimization frameworks could offer valuable insights into the trade-offs between conflicting performance metrics. As the complexity and scale of real-world problems increase, assessing the scalability of the proposed algorithms on larger and more intricate datasets becomes essential. Moreover, integrating real-time data streams could enable dynamic inventory control, allowing for more responsive and adaptive decision-making. Finally, the implementation of automated parameter tuning mechanisms may further refine algorithm performance, reducing manual intervention and improving overall system efficiency.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eM.\u0026Ccedil;. and M.C. wrote the main manuscript text and prepared all figures. All authors reviewed the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eVillegas-Ch, W., Navarro, A.M., Sanchez-Viteri, S.: Optimization of inventory management through computer vision and machine learning technologies. Intell. Syst. Appl. \u003cb\u003e24\u003c/b\u003e, 200438 (2024)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVaez-Alaei, M., et al.: Target stock level and fill rate optimization for worldwide spare parts inventory management: A case study in business aircraft industry. 2018 IEEE International Conference on Technology Management, Operations and Decisions (ICTMOD). IEEE, (2018)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eCui, L., et al.: A multi-location inventory system with occasional allocation under uncertain defective rates. Data Sci. Manage. (2025)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHedenstierna, P., Hilletofth, P., Olli-Pekka, H.: An integrative approach to inventory control. In: Rapid Modelling for Increasing Competitiveness: Tools and Mindset, pp. 105\u0026ndash;118. 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Softw. \u003cb\u003e69\u003c/b\u003e, 46\u0026ndash;61 (2014)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTalbi, E.-G.: Metaheuristics: from design to implementation. Wiley (2009)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMichalewicz, Z., Fogel, D.B.: How to solve it: modern heuristics. Springer Science \u0026amp; Business Media (2013)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGendreau, M., Potvin, J.-Y. (eds.): Handbook of metaheuristics, vol. 2. Springer, New York (2010)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eChopra, S.: Peter Meindl, and Dharam Vir Kalra. In: Supply chain management by Pearson. Pearson Education India (2007)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"cluster-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Cluster Computing](https://www.springer.com/journal/10586)","snPcode":"10586","submissionUrl":"https://submission.nature.com/new-submission/10586/3","title":"Cluster Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Critical Inventory, Metaheuristic Algorithms, Optimization, Sensitivity Evaluation","lastPublishedDoi":"10.21203/rs.3.rs-7869474/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7869474/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study comparatively analyzes the performance of various modern metaheuristic algorithms in addressing the critical stock optimization problem in inventory management. Using a custom-developed software tool, these algorithms were applied to a standardized dataset, and a comprehensive sensitivity analysis was conducted to examine the impact of lead time changes and daily consumption on the optimization results. Findings indicate that certain single-state algorithms demonstrate notable computational efficiency, offering rapid and effective solutions. In terms of overall stock reduction capacity, most algorithms achieved significant decreases in existing stock levels; while some algorithms executed more radical stock reductions, others adopted more conservative approaches. The sensitivity analysis revealed that particular algorithms generated more flexible and robust solutions against uncertainties in lead time and daily consumption, exhibiting less sensitivity to fluctuations in these parameters. Consequently, it is concluded that specific metaheuristic algorithms hold promising potential for developing fast, cost-effective, and efficient inventory policies in such optimization problems. Future research could encompass a more in-depth analysis of cost parameters, explore multi-objective optimization approaches, and investigate the scalability of these algorithms on larger and more complex datasets. Additionally, innovative topics such as real-time data integration and automated algorithm parameter tuning for dynamic inventory management warrant further investigation.\u003c/p\u003e","manuscriptTitle":"Comparative analysis and sensitivity evaluation of metaheuristic algorithms for critical inventory optimization","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-02 08:58:49","doi":"10.21203/rs.3.rs-7869474/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-05-01T11:11:17+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-12T15:35:59+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-09T21:36:10+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-05T09:09:11+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-02T21:57:07+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-01-31T12:29:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"143515166244202537458343876590713867149","date":"2026-01-31T12:19:20+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"46568370699202838774037137394316394931","date":"2026-01-29T11:55:53+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"179054964557022331312769685610217722964","date":"2026-01-29T08:50:35+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"226327980794177961142078753478630476903","date":"2026-01-29T08:43:42+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"302327676793597214351486425830690373106","date":"2026-01-29T08:34:45+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-29T08:23:32+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-11-04T06:44:03+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-10-17T10:19:28+00:00","index":"","fulltext":""},{"type":"submitted","content":"Cluster Computing","date":"2025-10-15T14:45:23+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"cluster-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Cluster Computing](https://www.springer.com/journal/10586)","snPcode":"10586","submissionUrl":"https://submission.nature.com/new-submission/10586/3","title":"Cluster Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"4da92f55-b6c0-4fcd-8607-5ebfc60d9fb8","owner":[],"postedDate":"February 2nd, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-01T11:11:17+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-01T11:24:16+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-02 08:58:49","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7869474","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7869474","identity":"rs-7869474","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}