Application of the metaheuristic algorithms to quantify the GSI based on the RMR classification

Application of the metaheuristic algorithms to quantify the GSI based on the RMR classification | 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 Article Application of the metaheuristic algorithms to quantify the GSI based on the RMR classification Pouya Koureh Davoodi, Farnusch Hajizadeh, Mohammad Rezaei This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6690480/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 07 Aug, 2025 Read the published version in Scientific Reports → Version 1 posted 12 You are reading this latest preprint version Abstract Accurate classification of rock masses is an essential task in earth sciences applications. Among various classification systems, the Rock Mass Rating (RMR) and Geological Strength Index (GSI) are the most frequently utilized ones. Unlike the RMR, which is a quantitative classification, GSI is a qualitative system and needs to be converted into a quantitative one as well due to its multiple applicability in both mining and civil engineering projects. With this objective, GSI quantification directly from RMR can be an attractive issue as it remains a complex task still due to the limited accuracy and generalizability of existing empirical models under varying geological conditions. This study addresses this challenge by analyzing data from eleven different rock types and employing three metaheuristic optimization algorithms, namely Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Grey Wolf Optimization (GWO), to develop predictive models for quantifying GSI based on the RMR. Accordingly, five mathematical GSI-RMR equations including linear, power, exponential, polynomial and logarithmic types were first developed using each algorithm. The resulting equations were assessed using six statistical indicators: R², RMSE, MAE, ASE, MAPE, and MARE. According to this evaluation, the best-performing equation from each algorithm was selected as the optimum and further evaluated using both graphical and statistical analyses, including comparisons with conventional empirical relationships. The findings revealed that the derived GSI-RMR equation from the SA algorithm achieved superior performance based on the score analysis and the REC curve. However, complementary evaluation using A20, IOA, and IOS metrics showed that the derived equation GSI-RMR equations from the GWO and PSO algorithms outperformed SA in certain aspects. These results demonstrate the unique strengths of all three proposed GSI-RMR equations and highlight the importance of multi-criteria evaluation. Overall, the proposed models provide a more accurate and generalizable framework for quantifying GSI from RMR, improving upon traditional empirical methods and enhancing the required accuracy compared to the qualitative GSI estimation. Physical sciences/Engineering Physical sciences/Mathematics and computing Geological Strength Index Rock Mass Rating Particle Swarm Optimization Simulated Annealing Grey Wolf Optimization Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 1. Introduction Accurate classification of rock masses is a crucial aspect of geotechnical and mining engineering, as it directly impacts excavation design, tunnel stability assessment, and the planning of underground support systems such as tunnels 1 . Among the various rock mass classification systems, RMR and GSI play a significant role in evaluating rock mass quality and predicting its mechanical behavior. The application of these classification systems includes slope stability analysis, excavation design, and underground support system planning 2 . The RMR classification system, introduced by Bieniawski (1979), classifies rock masses based on six key parameters, including uniaxial compressive strength (UCS), rock duality designation (RQD), spacing of discontinuities, condition of discontinuities, groundwater conditions, and orientation of discontinuities 3 , 4 . This system provides a quantitative approach for evaluating rock masses, which has led to its widespread acceptance in engineering projects. Nevertheless, because it is based on empirical observations, its effectiveness may vary across different geological conditions, making its application in diverse environments occasionally restrictive 5 . To address some of the limitations of the RMR classification system, Hoek et al. (1995) developed a new classification system known as the GSI. In fact, GSI was introduced as an extension of RMR to provide a better characterization of jointed rock masses. By considering rock structure and discontinuity conditions, GSI offers a more realistic representation of in-situ rock mass behavior, particularly in weak and fractured rock formations. Since its introduction, GSI has been widely used in rock engineering to estimate the strength and deformation properties of rock masses 6 . Estimating GSI from RMR has always been one of the primary challenges in rock engineering. Given the strong correlation between these two indices and their dependence on geological conditions, researchers in recent years have developed empirical relationships between them using regression models. 7 , 8 , as listed in Table 1 . However, many of these existing relationships have been developed based on a limited number of rock types. This limitation can reduce their applicability across diverse geological conditions. Furthermore, conventional regression models may lack the capability to fully capture the complex and nonlinear relationships between RMR and GSI 9 . Table 1 Existing relationships for estimating GSI based on RMR. No. Equations R 2 Reference 1 GSI = 0.7394RMR + 0.57 0.57 10 2 GSI = 0.9934RMR-4.913 0.84 11 3 GSI = 0.739RMR + 12.097 0.759 12 4 GSI = 1.265RMR-21.49 - 7 5 GSI = RMR ± 5 - 13 6 GSI = 1.2092RMR-18.6143 - 5 7 GSI = 0.793RMR + 2.001 0.736 8 8 GSI = 0.876RMR + 0.935 0.876 9 RMR = 0.42GSI + 33 - 9 To overcome the limitations of previous studies, this research employs metaheuristic optimization algorithms, including PSO, SA, and GWO, to develop optimized relationships between RMR and GSI. These algorithms demonstrate high efficiency in solving complex optimization problems by balancing global and local search processes. They can effectively explore high-dimensional parameter spaces without getting trapped in local optima, resulting in more accurate and robust predictive models 14 – 17 . The primary objective of this research is to develop five predictive models, including linear, power, exponential, quadratic polynomial, and logarithmic relationships, and to identify the most accurate equation among them to establish the optimal relationship between RMR and GSI. This study aims to enhance the generalizability and accuracy of the proposed models by utilizing 14 different rock types, including shale, slate, phyllite, chert, conglomerate, limestone, marl, dolomite, sandstone, mudstone, andesite, serpentinite, mylonite, and tuff, collected from the Beheshtabad water transfer tunnel site. The performance of the developed models, optimized using meta-heuristic algorithms, is evaluated through multiple model performance metrics, including R², RMSE, MAE, ASE, MAPE, and MARE, to ensure their reliability. Moreover, to enhance the reliability and credibility of the results, the developed models were compared with existing empirical relationships from previous studies using the coefficient of determination (R²). Sensitivity analysis was also conducted to assess the stability and robustness of the models. In addition to these evaluations, complementary statistical indicators including the REC curve, score analysis, index of agreement (IOA), index of structured overlap (IOS), and the A20 metric were employed to provide a comprehensive assessment of model performance. This study is structured into six sections. Section 1 provides an introduction to the significance of the topic, a review of previous research conducted by other scholars, existing challenges, and the objectives of this study. Section 2 describes the characteristics of the study area, introduces the dataset, and explains the data collection process. Section 3 focuses on the mathematical modeling process and the implementation of meta-heuristic algorithms (SA, PSO, and GWO) to optimize the relationship between RMR and GSI parameters. Section 4 evaluates the performance of the developed models and compares them with existing models. In section 5 , the advantages and limitations of this study are discussed, and recommendations for future research are provided. Finally, section 6 summarizes the findings of this study and emphasizes the importance of meta-heuristic algorithms in rock mass classification. 2. Material and methods 2.1. Data preparation In this research, data collected from the construction site of the Beheshtabad water transfer tunnel were used, and the characteristics of this tunnel will be explained in the following subsections. 2.1.1. Location of the study area The Beheshtabad water transfer tunnel is located within the provinces of Isfahan and Chaharmahal-va-Bakhtiari in Iran. Designed to supply water to the country’s central plateau, the tunnel originates at the terminus of the Beheshtabad River, positioned at 50°38′ E longitude and 32°02′ N latitude. Extending approximately 65 kilometers in a northeastern direction, it concludes upstream of the Cham Asman Dam along the Zayandeh Rud River, at coordinates 51°12′ E longitude and 32°22′ N latitude 18 . The geographical situation of the studied area and tunnel, which are parts of the Beheshtabad water transmission project, are shown in Fig. 1 . In this figure, the location of the Beheshtabad tunnel is shown as the dashed blue line. 2.1.2. Lithological and geomechanial properties of the study area The entrance segment of the Beheshtabad water transfer tunnel is located within the Folded Zagros tectonic belt, while its outlet section is positioned in the Sanandaj–Sirjan structural zone. To identify the dominant lithological units along the tunnel alignment, comprehensive field surveys and analyses of borehole data were carried out. These investigations revealed that the principal rock types present in the area include conglomerate, sandstone, mudstone, limestone, marly limestone, carbonate limestone, and andesite. The tectonic zoning along the tunnel’s trajectory is illustrated in Fig. 2 . Furthermore, a geological cross-section covering the kilometers 33 to 37 of the tunnel path is provided in Fig. 3 . Based on the data shown in this figure, it can be inferred that the predominant lithologies in this section consist of limestone, dolomite, conglomerate, and marly limestone interbedded with shale layers 18 , 19 . One of the most critical challenges faced during the tunneling operations was the presence of multiple fault zones throughout the study area. From a structural perspective, the Beheshtabad water transfer tunnel is divided into eight segments, each extending over approximately eight kilometers. Detailed geological and geophysical investigations have revealed that each of these segments exhibits distinct structural features, underscoring the complexity and variability of the subsurface conditions along the tunnel alignment 20 . To provide a clearer understanding, Fig. 4 schematically illustrates the structural features of the section 2 (8–16 km), section 4 (24–32 km), and section 7 (48–56 km) of the tunnel length based on the integration of geophysical findings with the geological profile. Based on this figure, it can be observed that the section spanning from kilometer 58 to 64, particularly within the Sanandaj-Sirjan tectonic zone and at its boundary with the Zagros zone, is significantly influenced by faults and folds. 2.1.3. Statistical characteristics of the database Based on the boreholes drilled along the route of the Beheshtabad water transfer tunnel, 150 datasets were measured to determine the required parameters of GSI and RMR. These parameters were measured according to the methods suggested by existing standards. To calculate the RMR, its six essential parameters were first evaluated through field surveys and laboratory experiments. In this study, the UCS was measured according to the standard approach recommended by the International Society for Rock Mechanics (ISRM 1981) 21 . A servo-controlled axial compression testing device was used to determine the UCS value. The device used during the test, along with the fractured sample, is shown in Fig. 5 . In this test, stress was applied at a rate of 0.5 to 1 MPa per second until the sample fractured. The stress value at the point of failure indicates the UCS. The RQD was measured according to the methods recommended by the American Society for Testing and Materials (ASTM 2017) 22 . Additionally, the spacing of discontinuities was precisely calculated through mapping operations. The condition of the joints, including assessments of infilling, roughness, and joint frequency, was also evaluated according to the procedure outlined in the ISRM suggested method for the quantitative description of rock mass discontinuities 23 . Groundwater conditions, which provide an index for determining the rock mass behavior under the influence of groundwater, were evaluated by examining moisture content and seepage on the rock surface. To provide a general perspective on groundwater levels in the Beheshtabad water transfer tunnel, a schematic diagram of groundwater levels along the tunnel from kilometer 0 to 6 is illustrated in Fig. 6 . For each of the six parameters mentioned above, a score is assigned, and the RMR value is calculated by summing the scores of the first five parameters. If, in a specific structure, the angle of discontinuities relative to the tunnel is significant, the final RMR score is adjusted accordingly. Hoek and colleagues proposed a classification system entitled the GSI 6 . This index was developed to address some of the limitations of previous classification systems, such as RMR. Specifically, it deals with those characteristics of the rock mass that influence its strength and deformability. The characteristics of a jointed rock mass depend on the characteristics of the intact rock parts and the sliding and rotational conditions of these pieces under stress fields. These conditions are controlled by the shape and condition of the joint surfaces that separate these blocks. Based on this, the GSI classifies the rock mass according to two parameters: the rock structure, which indicates the degree of blockiness and interlocking of the rock pieces, and the condition of the discontinuity surfaces. According to 6 , the GSI can be calculated and expressed based on these parameters using Fig. 7 . Based on the above procedures, 150 valid series of RMR and GSI measurements were collected along the 64 km length of the Beheshtabad tunnel with different rock types. Accordingly, a powerful database was prepared for the desired meta-heuristics modeling in this research. The statistical characteristics and symbols of the modeling variables are given in Table 2 . Also, the boxplot of statistical information of used data is shown in Fig. 8 . Table 2 Statistical characteristics and symbols of the parameters. Parameter Min Max Mean Std. Dev. RMR 19 68 42.23 12.66 GSI 22 73 46.89 13.29 2.2. Used methods In this section of the present study, three metaheuristic algorithms, including SA, PSO, and GWO, have been employed to develop an optimal relationship between the GSI and RMR parameters. The aim of this section is to introduce these algorithms and explain their applications in this research. The rationale for selecting these algorithms lies in their high speed, extensive search capability within the solution space, and ability to escape local optima 16 , 17 , 24 , 25 , making them suitable choices for modeling the relationship between GSI and RMR. In the following, each algorithm and its application in the present study are explained in detail. 2.2.1. PSO The PSO algorithm is inspired by the social behavior of groups of organisms such as birds or fish. This algorithm, widely used for solving optimization problems, is considered one of the most popular metaheuristic algorithms. In PSO, a group of particles moves within the search space and updates their positions based on two key factors: personal experience (the best local solution) and collective experience (the best global solution). Each particle has two main characteristics: velocity and position, which are improved using motion equations. Simplicity in implementation, low requirement for parameter tuning, and high convergence speed are the main advantages of the PSO algorithm. These features have made the application of PSO successful in various engineering problems. In the present study, the PSO algorithm is used to model the relationship between the RMR and GSI parameters. Specifically, this algorithm searches the solution space to provide optimal coefficients for the proposed model. PSO demonstrates a strong ability to explore different regions of the search space and avoids being trapped in local optima. Moreover, this algorithm performs effectively when dealing with complex variables 26 such as RMR and GSI. Considering these attributes, the use of the PSO algorithm in the present study is deemed appropriate. The flowchart of the PSO algorithm is presented in Fig. 9 . Based on this figure, the particles are initially initialized with random values. Their performance is then evaluated using model performance evaluation criteria such as R², RMSE, MARE, MAPE, MAE and ASE. Subsequently, the best individual and collective solutions are identified, and the velocity of the particles is adjusted to improve the coefficients. This process continues until the optimization criteria are satisfied. Finally, the equation with the optimized coefficients and the model performance evaluation metrics are presented as the output. 2.2.2. SA The SA is another metaheuristic algorithm commonly used to solve optimization problems. This algorithm is inspired by the annealing process of metals, which encompasses progressively reducing the temperature to reach a low-energy equilibrium state. In SA, the search space is explored randomly, and with a certain probability, worse solutions are also accepted. This feature enables SA to discover a wide range of possible solutions. The SA algorithm is simple and flexible in parameter tuning, making it highly adaptable for various optimization tasks 15 . In the present study, SA has been employed to optimize the relationship between the RMR and GSI parameters. This algorithm can provide the best coefficients for the proposed model. The optimization is achieved by gradually reducing the temperature and effectively exploring the search space 15 . By accepting worse solutions in the initial stages, SA can establish a highly accurate and stable relationship between the RMR and GSI parameters. This aspect is particularly important in scenarios where the data exhibits significant variability. In Fig. 10 , the flowchart of the SA algorithm for optimizing the coefficients of the relationship between RMR and GSI parameters is presented. According to this figure, the initial coefficients are generated, and the objective function is designed. In each iteration, new coefficients are randomly generated, and the objective function is recalculated. It is then evaluated whether the new coefficients improve the objective function. If they do, the coefficients are accepted. Otherwise, they are accepted with a certain probability based on the Metropolis rule. This probability depends on two factors: the change in the objective function and the present temperature. The temperature is gradually reduced after each iteration to decrease the probability of accepting mediocre solutions. This process continues until one of the stopping conditions, such as accomplishment a predefined number of iterations or minimal changes in the objective function, is met. Finally, the coefficients optimized by SA, which best represent the relationship between RMR and GSI parameters, are displayed as the output. 2.2.3. GWO The GWO introduced an inventive metaheuristic algorithm in 2014 by Mirjalili and colleagues 17 . This algorithm inducements stimulus from the social hierarchy and cooperative hunting strategies of grey wolves in the wild. In the GWO framework, wolves are divided into four distinct groups: Alpha wolves, which signify the finest answers; Beta and Delta wolves, which assist in guiding the search process; Omega wolves, representing the rest of the population. The positions of the wolves are iteratively updated, guided by the Alpha, Beta, and Delta wolves. This hierarchical approach allows the algorithm to effectively balance exploration searching for new solutions and exploitation refining the finest answers found so far. Such a balance helps the algorithm avoid becoming trapped in local optima. a, A, and C are three key parameters in GWO that help optimize the positions. The parameter a can assists of the adjustment of the search range and gradually decreases from its initial value to zero. Parameters A and C are coefficient vectors used to calculate the distance and direction of the wolves' movement towards the best positions. These parameters adjust the direction and movement of the wolves in the search space to help the algorithm find better solutions 17 . In this study, the GWO algorithm has been applied to optimize the coefficients of the relationship between RMR and GSI parameters. The development commences with the generation of an original population of random positions for the wolves. For each position, the objective function is evaluated, and the three best positions are selected as the Alpha, Beta, and Delta wolves, completing the initialization phase. During the position update phase, the positions of the Omega wolves are adjusted based on the positions of the Alpha, Beta, and Delta wolves. This adjustment is governed by equations that mimic the direction and distance of the wolves' movements toward their prey, which corresponds to the optimal solution. This iterative procedure endures pending a predefined stopping standard is met, such as attainment of the peak number of repetitions or observing negligible improvements in the objective function. Due to its adaptability and its ability to avoid local optima, the GWO algorithm is highly suitable for deriving accurate coefficients for the relationship between RMR and GSI parameters. The flowchart in Fig. 11 illustrates the steps of the GWO algorithm as applied to optimize the coefficients of the relationship between RMR and GSI parameters, providing a visual representation of the procedure described above. 3. GSI-RMR modeling Hoek and colleagues announced a classification system identified as the GSI. The GSI specifically addresses the characteristics of rock mass discontinuities that affect their strength and deformability 6 . The characteristics of the rock mass depend on the sliding and rotational characteristics of the blocks under stress fields. These conditions are controlled by the shape and surface characteristics of the blocks, which differentiate them from one another. The GSI classification establishes a relationship based on two parameters of the rock mass: the rock structure, which indicates the degree of blockiness and interconnecting of rock fragments, and the surface conditions of discontinuities 27 . Based on this, Hoek and colleagues suggested that the GSI can be calculated and expressed using a diagram (Fig. 7 ) derived from these two parameters. The use of this diagram to determine the GSI value requires specialized expertise with a high level of geological knowledge and field observations, which demands considerable time and relatively high costs. In contrast to this method, researchers in recent years have developed relationships to calculate GSI from other rock mass classification parameters. As a result, by having one of the rock mass classification parameters and using the developed models, the GSI parameter can be easily calculated 5 , 8 , 13 . As shown in Table 1 , these relationships are mostly linear. Additionally, the coefficient of determination (R²) for these relationships indicates the average accuracy of the corresponding relationships in calculating the GSI. Furthermore, the type of rocks examined for estimating GSI using these existing models is very limited. This limitation can restrict the applicability of the existing relationships in different geological conditions, where a wider variety of rock types are present. To address the shortcomings of the existing models, this study aims to develop five different relationships between GSI and RMR, including linear, power, exponential, quadratic polynomial, and logarithmic relationships, using three different metaheuristic algorithms: PSO, SA, and GWO. To achieve this goal, five initial equations, including linear, power, exponential, quadratic polynomial, and logarithmic equations, are first defined for the model (Table 3 ). The model then optimizes the coefficients of each equation and determines the best coefficients to achieve the equation with the highest accuracy for estimating GSI. By substituting these coefficients into the initial equations, the final equations between the RMR and GSI parameters are developed. Each of these five equations, which has the highest R² value and the lowest MARE, MAE, MAPE, RMSE, and ASE values, is selected as the proposed relationship developed by the algorithm used for its development. Then, the selected equations of each algorithm are compared with the selected equations of the other two algorithms according to the aforementioned performance evaluation indices. Thus, the relationship with the highest accuracy for estimating GSI based on RMR is determined. In addition, this study utilizes 14 different types of rocks to develop the models under discussion. These rocks include shale, slate, phyllite, chert, conglomerate, limestone, marl, dolomite, sandstone, mudstone, andesite, serpentinite, mylonite, and tuff. This can expand the generalizability of the established models in this research. In the continuation of this section, the modeling of relationships between the RMR and GSI parameters is carried out based on the metaheuristic algorithms PSO, SA, and GWO. Table 3 Initial equations for modeling the GSI-RMR relationship. R Equation Type 1 GSI=(a*RMR) + b Linear 2 GSI = a*RMR b Power 3 GSI = a*e (b*RMR) Exponential 4 GSI=(a*RMR 2 )+(b*RMR) + c Polynomial 5 GSI = a*ln(RMR) + b Logarithmic 3.1. Modeling using the PSO In this section, the PSO algorithm is employed to model the relationship between the GSI and RMR parameters. The objective of this effort is to develop an optimized relationship for estimating the GSI parameter using the RMR. The accuracy of this relationship is evaluated using performance indices, including R 2 , RMSE, MARE, MAPE, ASE, and MAE. In the current PSO modeling, the number of particles in the initial population was set to 30 to maintain a balance between accuracy and computational cost, as an excessive increase in the number of particles could lead to longer processing times. Additionally, the maximum number of iterations was set to 300 to provide sufficient opportunity for the algorithm to converge while avoiding unnecessary executions. These values were determined based on previous studies and an analysis of the algorithm's convergence behavior. In this model, the coefficients of the initial equations referenced in Table 3 were optimized using the PSO algorithm. A list of these coefficients for each of the five models including linear, power, exponential, quadratic polynomial, and logarithmic model is provided in Table 4 . By substituting these coefficients into their corresponding equations from Table 3 , the final equations describing the relationship between GSI and RMR are obtained. These relationships, along with the performance evaluation indices for each equation, are presented in Table 5 . The graphs of these five equations are also plotted in Fig. 12 . Based on Table 4 , by evaluating and comparing the performance assessment parameters of the models developed using the PSO algorithm, the linear model is selected as the best relationship among the five equations developed by PSO for GSI prediction. This is because, in this model, the R² value is higher than that of other models, while the values of RMSE, MARE, MAPE, MAE, and ASE are lower than those of other models, indicating a relatively higher accuracy of this model compared to the other four models. Table 4 The optimized coefficients calculated using the PSO algorithm. R Coefficient Equation type a b c 1 1.0703 1.6760 - Linear 2 26.2852 -50 - Logarithmic 3 22.2040 0.2057 - Power 4 19.6419 0.0197 - Exponential 5 0.0143 -0.2615 30.3932 Polynomial Table 5 The developed equations between RMR and GSI using the PSO algorithm. R Equation Model performance evaluation parameters R 2 RMSE MARE MAPE MAE ASE 1 GSI = 1.0703 * RMR + 1.6760 0.8785 4.6563 0.0814 8.9980 3.8298 21.6808 2 GSI = 26.2852 * log(RMR) − 50 0.7813 6.2486 0.1053 12.3518 4.9544 39.0444 3 GSI = 22.2040 * RMR 0.2057 0.3782 10.5356 0.0864 8.9497 8.8907 110.9993 4 GSI = 19.6419 * exp(0.0197 * RMR) 0.8522 5.1362 0.0623 6.2941 4.1747 26.3811 5 GSI = 0.0143 * RMR 2 − 0.2615*RMR + 30.3932 0.8326 4.6563 0.0814 8.9980 3.8298 21.6808 3.2. Modeling using the SA The SA algorithm was applied to establish an optimized relationship between the GSI and RMR parameters. This modeling effort aimed to accurately estimate GSI based on RMR values by enhancing the predictive capability of the proposed equations. The accurateness of the established models was assessed using several performance metrics, including the coefficient of determination (R²), root mean square error (RMSE), mean absolute relative error (MARE), mean absolute percentage error (MAPE), average squared error (ASE), and mean absolute error (MAE). To ensure effective convergence and reliable outcomes, the maximum number of function evaluations for the SA algorithm was set to 2000, and the convergence tolerance was defined as 1e-4. These parameters were selected through iterative testing to minimize error and enhance model performance. In this study, the SA algorithm was employed to optimize the coefficients of the five fundamental equation types presented in Table 3 , including linear, power, exponential, quadratic polynomial, and logarithmic forms. The resulting optimized coefficients are listed in Table 6 . By incorporating these coefficients into their respective equations, the final predictive models for GSI estimation were formulated. These models, along with their corresponding performance indices, are summarized in Table 7 . Additionally, graphical representations of the five equations are illustrated in Fig. 13 . According to the comparative evaluation presented in Table 7 , the linear model outperforms the other equations developed using the SA algorithm. This conclusion is supported by its higher R² value and lower RMSE, MARE, MAPE, MAE, and ASE values, indicating superior accuracy and robustness in predicting GSI from RMR. Table 6 The optimized coefficients calculated using the SA algorithm. R Coefficient Equation type a b c 1 1.0296 3.3090 - Linear 2 36.2266 -86.9522 - Logarithmic 3 1.6820 0.8903 - Power 4 8.6264 0.0352 - Exponential 5 -0.0086 1.5968 -2.7388 Polynomial Table 7 The developed equations between RMR and GSI using the SA algorithm. R Equation Model performance evaluation parameters R 2 RMSE MARE MAPE MAE ASE 1 GSI = 1.0296 * RMR + 3.3090 0.9542 2.8201 0.0483 5.4549 2.2672 8.0663 2 GSI = 36.2266 * log(RMR) -86.9522 0.9244 3.6426 0.0645 7.0924 3.0229 13.2682 3 GSI = 1.6820 * RMR 0.8903 0.9533 2.8615 0.0492 5.5966 2.3086 8.1879 4 GSI = 8.6264 * exp(0.0352 * RMR) 0.5511 8.8755 0.1673 18.1669 7.8457 78.7748 5 GSI = -0.0086 * RMR 2 + 1.5968 * RMR − 2.7388 0.9139 3.8870 0.0673 7.7855 3.1562 15.1085 3.3. Modeling using the GWO In a similar approach, the GWO method was applied to model the relationship between the GSI and RMR parameters. The aim of this modeling was to develop a more accurate estimation of GSI using RMR. The estimation capability of the established models was measured using th standard evaluation metrics including R², RMSE, MAE, MAPE, MARE, and ASE. Based on trial-and-error procedure, the number of wolves was set to 20 and the number of iterations was set to 500 in order to maintain a balance between computational efficiency and solution accuracy. The coefficients of five models, namely linear, power, exponential, quadratic polynomial, and logarithmic, introduced earlier in Table 3 , were optimized using this method. The updated coefficients are presented in Table 8 . By substituting these values into their respective equations, the final relationships between GSI and RMR were obtained. These equations and their corresponding performance metrics are provided in Table 9 . Graphical representations of the five models are illustrated in Fig. 14 . According to the results, both the linear and power models demonstrated the highest R² values, and their error indices showed only minor differences. However, given the simplicity and practical applicability of the linear model, it is proposed as the most suitable equation among the five models developed using the GWO method for GSI estimation. Table 8 The optimized coefficients calculated using the GWO algorithm. R Coefficient Equation type a b c 1 1.0254 3.4845 - Linear 2 10 10 - Logarithmic 3 1.5524 0.9110 - Power 4 18.6041 0.0210 - Exponential 5 -0.0013 1.1384 1.2749 Polynomial Table 9 The developed equations between RMR and GSI using the GWO algorithm. R Equation Model Performance Evaluation Parameters R 2 RMSE MARE MAPE MAE ASE 1 GSI = 1.0254 * RMR + 3.4845 0.9541 2.8396 0.0484 5.4644 2.2676 8.8396 2 GSI = 10 * log(RMR) + 10 0.4071 10.2010 0.1869 21.4527 8.7637 104.0597 3 GSI = 1.5524 * RMR 0.9110 0.9541 2.8371 0.0484 5.4958 2.2789 8.0491 4 GSI = 18.6041 * exp(0.0210 * RMR) 0.9329 3.4309 0.0592 6.7951 2.7751 11.7708 5 GSI = -0.0013 * RMR 2 + 1.1384 * RMR + 1.2749 0.9543 2.8329 0.0484 5.4431 2.2618 8.0255 4. Assessment of the suggested models 4.1. Assessment of predictive accuracy using standard performance metrics. The prediction capability of the PSO, GWO, and SA models is further assessed through the standard metrics, including R², RMSE, MAE, MAPE, ASE, and MARE. Ideally, a model with an R² value of 1 and zero values for the remaining error indices is considered perfect. The equations used to calculate these metrics are provided in Equations ( 1 ) to ( 6 ). Table 10 presents the calculated values of these performance indices, reflecting the predictive capability of each model in estimating GSI from RMR. As shown, the PSO model performs less accurately than GWO and SA across all six metrics and can thus be considered the least effective among the three. The SA model, on the other hand, demonstrates superior performance in five out of the six criteria. Although the difference between SA and GWO in the MARE value is minimal, SA generally shows the best overall accuracy in this study. While SA is identified as the most accurate model, it is noteworthy that all three methods exhibit acceptable predictive capability, as supported by the presented evaluation indices. $${R^2}=1 - \frac{{\sum\limits_{{i=1}}^{n} {{{(yi - \hat {y}i)}^2}} }}{{\sum\limits_{{i=1}}^{n} {{{(yi - \bar {y}i)}^2}} }}$$ 1 $$MAPE=\frac{{100\% }}{n} - \sum\limits_{{i=1}}^{n} {\left| {\frac{{yi - \hat {y}i}}{{yi}}} \right|}$$ 2 $$RMSE=\sqrt {\frac{1}{n}\sum\limits_{{i=1}}^{n} {(yi - \hat {y}} {)^2}}$$ 3 $$MARE=\frac{1}{n}\sum\limits_{{i=1}}^{n} {\frac{{\left| {yi - \hat {y}} \right|}}{{\left| {yi} \right|}}}$$ 4 $$MAE=\frac{1}{n}\sum\limits_{{i=1}}^{n} {{{\left| {yi - \hat {y}} \right|}^{}}}$$ 5 $$ASE=\frac{1}{n}\sum\limits_{{i=1}}^{n} {{{(yi - \hat {y}i)}^2}}$$ 6 In the above equations, y i signifies the i th amount of actual data (observed), ŷ i donates the i th amount of estimated data (predicted), n is the datasets number, ȳ is the average amount of the actual data (observed). Table 10 The obtained values of performance evaluation metrics for the proposed models. Model Index R 2 RMSE MAE MAPE ASE MARE PSO 0.8785 4.6563 0.0814 8.9980 3.8298 21.6808 SA 0.9542 2.8201 0.0483 5.4549 2.2672 8.0663 GWO 0.9541 2.8371 0.0484 5.4958 2.2789 8.0491 4.2. Scatter plot analysis of predicted vs. actual GSI values To further assess the predictive capabilities of the proposed models, scatter plots comparing the actual GSI values with the predicted ones are illustrated in Fig. 15 . In this figure, each sub-plot demonstrates the performance of one of the three models: PSO, SA, and GWO. The diagonal line (1:1 line) represents the ideal case where predicted values perfectly match the actual observations. According to the plotted results and performance metrics summarized earlier, the SA model yields the highest prediction accuracy with R² = 0.9542 and RMSE = 2.8201. Its scatter points are densely clustered along the 1:1 line, indicating close agreement with the actual data. The GWO model also shows a similar level of accuracy (R² = 0.9541 and RMSE = 2.8371), with only minor deviations from the ideal line. In contrast, the PSO model performs comparatively worse, with a lower coefficient of determination (R² = 0.8785) and a higher RMSE (4.6563), as its scatter points display greater dispersion from the diagonal. Despite these differences, the scatter plots reveal that all three models demonstrate reasonable accuracy in estimating GSI values from the RMR input, supporting their applicability in geomechanical predictive modeling. 4.3. Sensitivity analysis Figure 16 illustrates the error distribution of the SA, GWO, and PSO models under two different conditions: utilizing the entire dataset and after randomly removing 20% of the data. Sensitivity analysis is crucial in evaluating the robustness of predictive models, as it helps determine how changes in the input data affect model performance. In this case, the removal of a portion of the data allows us to assess whether the models remain stable and reliable when faced with incomplete datasets. As shown in Fig. 16 , the overall error distribution has remained nearly unchanged after data reduction, maintaining an approximately normal shape. This suggests that the examined models exhibit high stability, as their performance does not deteriorate significantly with reduced data availability. In other words, removing 20% of the data has not caused a sudden increase in prediction error or introduced substantial variations in model behavior. Among the three models, PSO appears to have the least deviation in its error distribution, indicating greater robustness against data reduction. 4.4. Error distribution analysis Figure 17 illustrates the error distributions of the GWO, SA, and PSO models. The histograms represent the frequency of error values, while the red curves depict the fitted normal distribution for each model. The overall shape of the distributions indicates that the prediction errors are symmetrically centered around zero, suggesting that the models exhibit unbiased performance. Additionally, the error distributions closely resemble a normal pattern, which implies that the residuals are well-behaved and do not show significant skewness or irregularities. This characteristic is particularly important in predictive modeling, as it suggests that the models do not exhibit systematic bias and that their errors follow a stable and predictable pattern. Such stability enhances confidence in the reliability of the employed optimization algorithms. 4.5. Model assessment through REC criterion To further assessment of the proposed models accurateness, the Regression Error Characteristic (REC) curves were employed. Figure 18 illustrates the REC curves for the SA, GWO, and PSO models based on the absolute error between the predicted and actual GSI values. The area under the curve (AUC) was also computed for each model, resulting in values of 5.3618 for SA, 5.3530 for GWO, and 5.3168 for PSO. A higher AUC value reflects a better predictive performance across a range of error thresholds 28 – 31 . As shown in Fig. 18 , the SA model achieved the highest AUC, indicating the most accurate performance in estimating GSI. The GWO and PSO models ranked second and third, respectively. However, the relatively close AUC values among all three models suggest that each algorithm provides an acceptable level of accuracy in GSI prediction. 4.6. Model Assessment through score analysis In this study, in addition to the REC curve, score analysis was also employed to more precisely evaluate the performance of the GWO, SA, and PSO models. While the REC curve presents model performance in a continuous manner based on the percentage of accurate predictions across different error thresholds, score analysis enables a discrete comparison of the models based on six statistical indices: R², RMSE, MAE, MAPE, ASE, and MARE. In this approach, for each of the six mentioned indices, a score ranging from 1 (for the weakest model) to 3 (for the best model) was assigned to the models under evaluation. The total score of each model was then visualized using a stacked bar chart, as shown in Fig. 19 , to facilitate easier comparison. In this figure, the height of each column represents the overall score of each model, while the colored segments reflect the contribution of each individual index 28 – 31 . As illustrated, the SA model, with a total score of 17, achieved the best overall prediction capability, followed by the GWO and PSO models with scores of 13 and 6, respectively. It is worth noting that the results obtained from this score-based comparison are in full agreement with those derived from the REC curve, further confirming the superior accuracy of the SA model. 4.7. Assessment of models performance using A20, IOA, and IOS metrics To complete the evaluation of the proposed models, three additional indices including Acceptable 20% Accuracy Index (A20), Index of Agreement (IOA), and Index of Symmetric Agreement (IOS) were calculated and presented in Table 11 . The a20 represents the percentage of data points with a relative error below 20%. This metric is determined using Eq. (7) 28,32 . The IOA assesses how closely the predicted values match the observed ones, with scores ranging from 0 to 1. A value nearer to 1 signifies stronger agreement. It is calculated using Eq. ( 8 ). The IOS is an enhanced version of IOA that additionally accounts for the average of predicted values. It is computed using Eq. ( 9 ). In Equations (7) to ( 9 ), Pi, Qi, and N refer to the estimated value, the real value, and the total data points number, respectively 32 – 34 . According to Table 11 , the PSO technique concluded the maximum relative accuracy in the low-error range with an A20 value of 0.985612. On the other hand, the GWO model outperformed the others in terms of IOA and IOS, with values of 0.987845 and 0.797099, respectively. The SA model yielded A20, IOA, and IOS values of 0.978417, 0.987726, and 0.796304, respectively, indicating that although its performance is comparable to the other two models, it ranks slightly below the GWO model. These results are graphically depicted in Fig. 20 . While the SA model demonstrated superior performance over the GWO and PSO models in both the REC curve and score analysis, the findings based on these three complementary indices suggest that the GWO model holds a slight advantage in certain aspects of predictive accuracy. This observation highlights the importance of utilizing a diverse set of evaluation criteria to gain a more comprehensive and precise understanding of model performance. \(a20=\frac{1}{N}\sum\limits_{{i=1}}^{N} {\delta i}\) Where \(\delta i=1if\left| {\frac{{Pi - Qi}}{{Qi}}} \right| \leqslant 0.2\) , otherwise \(\delta i=0\) (7) $$IOA=1 - \frac{{\sum\limits_{{i=1}}^{N} {{{(Pi - Qi)}^2}} }}{{\sum\limits_{{i=1}}^{N} {{{(\left| {Pi - \overline {Q} } \right|+\left| {Qi - \overline {Q} } \right|)}^2}} }}$$ 8 $$IOS=1 - \frac{{\sum\limits_{{i=1}}^{N} {{{(Pi - Qi)}^2}} }}{{\sum\limits_{{i=1}}^{N} {{{(\left| {Pi - \overline {P} } \right|+\left| {Qi - \overline {Q} } \right|)}^2}} }}$$ 9 Table 11 The obtained amounts of performance assessment metrics for the proposed models. Model Index A20 IOA IOS SA 0.978417 0.987726 0.986304 GWO 0.978417 0.987845 0.797099 PSO 0.985612 0.987815 0.793174 4.8. Taylor diagram analysis To provide a comprehensive comparison of model performance, a Taylor diagram was employed, as shown in Fig. 21 . This graphical representation simultaneously visualizes the correlation coefficient, standard deviation, and centered root mean square error (RMSE) of the PSO, GWO, and SA models in predicting GSI values from RMR input data. As depicted in the figure, all three models are located in close proximity to the reference point, which represents perfect agreement with the observed data. This close clustering indicates that the proposed metaheuristic models exhibit strong predictive performance, with high correlation and minimal deviation from actual GSI values. Moreover, the convergence of the three models in the Taylor space reinforces the findings obtained from other evaluation metrics discussed in this study, including the REC curve, score-based analysis, A20 index, IOA, and IOS. These consistent results confirm that all three algorithms, despite slight variations in individual indices, are highly reliable for estimating GSI. The differences between their performances are considered statistically negligible for practical applications. 5. Comparative analysis with similar researches In this section of the research, to further evaluate the proposed relationships between RMR and GSI parameters developed by the SA, GWO, and PSO algorithms, the relationships obtained from these algorithms are compared with the results of models from previous studies based on various quantitative and qualitative criteria. To achieve this goal, first, the R² index obtained from the proposed equations of each of the three models used in this research is compared with the R² values of the proposed equations of existing relationships developed by other researchers. Additionally, the types of rocks studied are examined as a basis for a qualitative comparison of the proposed relationships of this research and existing relationships. Based on the comparison of R² values of existing relationships between RMR and GSI parameters (Table 1 ) and the proposed models in this study (Table 10 ), it can be concluded that the R² values obtained from the proposed relationships in this study are higher than the R² values of existing models, which indicates the relatively better performance of the proposed models in this study compared to existing models. Furthermore, In Fig. 22 , the proposed equations in this study are compared with the equations presented by other researchers. Based on this figure, the proposed equations in this research demonstrate higher accuracy in predicting GSI compared to the equations developed by various researchers. This figure indicates that the equations developed in this study exhibit a better fit with the utilized data, which can be attributed to the use of metaheuristic algorithms such as SA, GWO, and PSO. These methods possess greater capability compared to traditional optimization techniques in modeling complex relationships between parameters such as RMR and GSI. Reviews show that in studies conducted by other researchers, a more limited variety of rock types has been investigated, which is considered a major weakness because it significantly reduces the generalizability of existing relationships. To address this problem, this study has attempted to increase the generalizability of the proposed relationships as much as possible by examining a wider range of rocks, including: shale, slate, phyllite, chert, conglomerate, limestone, marl, dolomite, sandstone, mudstone, andesite, serpentinite, mylonite, and tuff. These comparisons confirm the qualitative and quantitative superiority of the proposed relationships in this study over existing relationships. However, despite the high capability of the developed models in estimating GSI and its relatively lower error compared to other models, further research on other rock types, especially igneous and metamorphic rocks, is necessary to further increase the generalizability of existing relationships. 6. Research benefits, restrictions and future directions 6.1. Benefits This study presents optimized mathematical relationships between RMR and GSI using SA, GWO, and PSO algorithms. By formulating separate equations for each algorithm and assessing their accuracy, the effectiveness of these models has been validated. The results demonstrate that all three models provide high accuracy and efficiency in capturing the relationship between RMR and GSI, confirming their reliability in estimating GSI based on RMR. Moreover, the inclusion of 14 different rock types in this study enhances the generalizability of the proposed models, making them applicable to a wide range of geological formations. These findings can significantly contribute to geomechanical modeling and have practical implications for rock mass classification, tunnel stability analysis, and drilling optimization. 6.2. Restrictions Despite the promising results, this study is limited to sedimentary rocks, which may affect the applicability of the developed models to igneous and metamorphic formations. Another limitation arises from the nature of metaheuristic algorithms, which rely on exploratory search mechanisms and do not guarantee finding the absolute optimal solution. Additionally, hyper parameter tuning and model optimization, especially for large datasets, can be time-consuming and computationally demanding. 6.3. Future Directions To enhance the robustness and applicability of the models, future studies should expand the dataset to include igneous and metamorphic rocks, improving the models' generalizability across diverse geological settings. Exploring hybrid approaches that combine multiple metaheuristic algorithms may lead to improved accuracy and model stability. A promising avenue for further research is the integration of metaheuristic algorithms with machine learning techniques or neural networks, which could enhance predictive accuracy while reducing dependency on extensive parameter tuning. Furthermore, validating the proposed models with real-world geotechnical data, such as exploratory drilling results or slope stabilization projects, would strengthen their practical applicability. 7. Conclusions Three metaheuristic algorithms, including PSO, SA, and GWO, were employed to evaluate GSI in 14 different rock types and to develop reliable relationships for determining GSI based on RMR. RMR was used as the input for the models to predict GSI, as these two parameters exhibit a high correlation. In total, 150 datasets were measured and analyzed to achieve the study's objective, which is predicting GSI using RMR. The results of this study and the analyses performed led to the following key findings: The performance evaluation parameters R², RMSE, MAPE, MAE, ASE, and MARE for the PSO model were calculated as 0.8785, 4.6563, 8.9980, 3.8298, 21.6808, and 0.0814, respectively. These results indicate a reasonable predictive capability of the PSO model in estimating GSI. For the GWO model, the performance evaluation parameters R², RMSE, MAPE, MAE, ASE, and MARE were calculated as 0.9541, 2.8371, 5.4958, 2.2789, 8.0491, and 0.0484, respectively. These results indicate that the GWO model outperforms the PSO model in predicting GSI. However, its accuracy is still lower than that of the SA model. Among the three models used for GSI prediction, the SA model exhibited the highest accuracy with R² = 0.9542, RMSE = 2.8201, MAPE = 5.4549, MAE = 2.2672, ASE = 8.0663, and MARE = 0.0483. These values indicate that the SA model is the most reliable and the best-performing model among the applied models. The results of scatter plot analysis and sensitivity analysis, confirm the statistical findings and indicate that the SA model outperforms the GWO and PSO models in predicting GSI based on RMR. The REC curve analysis showed that the SA model had the highest AUC, confirming its superior prediction performance across different error thresholds. This result was also supported by score analysis, where the SA model achieved the highest total score among the models, highlighting its overall effectiveness. Complementary evaluation using A20, IOA, and IOS metrics revealed that while PSO had the best low-error accuracy (A20), the GWO model slightly outperformed others in IOA and IOS. However, the SA model showed balanced and competitive results, confirming its robustness and reliability in different evaluation criteria. The comparison of the developed models in this research with existing empirical models showed that the proposed models exhibit higher accuracy, greater generalizability, and better alignment with real data obtained from the Beheshtabad water transfer tunnel site. The use of metaheuristic models has significantly improved the accuracy of GSI prediction based on RMR compared to existing empirical models. Considering the obtained results, it can be stated that this research involved several innovations, comprising a high diversity of the studied rock types, application of three new algorithms (SA, GWO, and PSO) for GSI estimation with high accuracy. Also, developing a new GSI prediction model based on the RMR is a main novelty of the current study which can be considered as a non-destructive approach. Abbreviations ASE Average squared error ASTM American society for testing and material GSI Geological strength index GWO Grey Wolf Optimization ISRM International society for rock mechanics MAE Mean absolute error MAPE Mean absolute percentage error MARE Mean absolute relative error PSO Particle Swarm Optimization R 2 Coefficient of determination RMR Rock mass rating RMSE Root mean squared error RQD Rock quality designation SA Simulated Annealing UCS Uniaxial Compressive Strength REC Regression Error Characteristic curve AUC Area Under the Curve IOA Index of Agreement IOA Index of Symmetric Agreement A20 Acceptable 20% Accuracy Index Pi Predicted value Qi Observed value N Total number of data points Declarations Author contributions P. Koureh Davoodi conducted the data collection, wrote the first draft of the manuscript, and participated in the research ideation and analysis of the results as a PhD candidate. F. Hajizadeh proposed the research idea, supervised all data collection activities, and guided the development of the manuscript as the primary supervisor. In addition, M. Rezaei proposed the research idea, supervised all data collection activities, guided the development of the manuscript, played a significant role in data analysis and manuscript revision, and manuscript editing as the secondary supervisor. Funding This research did not receive any specific grant from funding agencies in public, commercial, or not-for-profit sectors. Competing interests The authors declare no competing interests. Permission for land study The authors declare that all land experiments and studies were carried out according to authorized rules. Data availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. References Ranasooriya, J. 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10:00:42","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":2711255,"visible":true,"origin":"","legend":"\u003cp\u003eRepresentation of the structural zones in the study area along the Beheshtabad water transfer tunnel.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/184301c07c70b028a56acba5.png"},{"id":83825524,"identity":"c834d637-a76a-4a43-bc7b-acae4ed44770","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":484668,"visible":true,"origin":"","legend":"\u003cp\u003eThe geological cross-section of the kilometers 29 to 37 of the Beheshtabad tunnel route.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/23016d95dcfa95034f855890.png"},{"id":83825551,"identity":"9a9960b4-5dfc-48f0-8071-86f443d8aa7e","added_by":"auto","created_at":"2025-06-03 10:00:43","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":3419693,"visible":true,"origin":"","legend":"\u003cp\u003eStructural characteristics of the section 2 (8-16 km), section 4 (24-32 km), and section 7 (48-56 km) of the tunnel length based on the integration of geophysical findings with the geological profile.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/fde841af52973917d985879d.png"},{"id":83825532,"identity":"540a6bf4-b62f-48f8-8709-5540a87f4e4f","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":166670,"visible":true,"origin":"","legend":"\u003cp\u003eThe servo-controlled axial compression testing device to measure the UCS.\u003c/p\u003e","description":"","filename":"5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/7959a770580242a5c22d43fe.jpeg"},{"id":83826180,"identity":"82056c82-e8fa-4b77-95dd-18383dd4d70e","added_by":"auto","created_at":"2025-06-03 10:08:42","extension":"jpeg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":236837,"visible":true,"origin":"","legend":"\u003cp\u003eProfile of groundwater level of the first 6 km of the tunnel length.\u003c/p\u003e","description":"","filename":"6.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/1039a689678c6b8b2e6cad79.jpeg"},{"id":83826177,"identity":"69fd88cb-db4b-427e-97d2-9223cbfc18e3","added_by":"auto","created_at":"2025-06-03 10:08:42","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":1129181,"visible":true,"origin":"","legend":"\u003cp\u003eA qualitative presentation to determine the GSI rock mass containing joints \u003csup\u003e6\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/5a2aa393dcb3f975ab6d37a8.png"},{"id":83825534,"identity":"02cf38ac-b3fc-4194-803c-67cf6fcbf74c","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"jpeg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":77693,"visible":true,"origin":"","legend":"\u003cp\u003eBox plot with statistical information.\u003c/p\u003e","description":"","filename":"8.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/93cdeed7500633b3d3114723.jpeg"},{"id":83826181,"identity":"e5274ffe-4995-4e59-b495-dfcfd39a0138","added_by":"auto","created_at":"2025-06-03 10:08:43","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":181485,"visible":true,"origin":"","legend":"\u003cp\u003ePSO flowchart.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/680394e8cc369584f3a2b85c.png"},{"id":83826175,"identity":"2b85c684-62f1-4e54-9475-83fa5d886ce8","added_by":"auto","created_at":"2025-06-03 10:08:42","extension":"jpeg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":215001,"visible":true,"origin":"","legend":"\u003cp\u003eSA flowchart.\u003c/p\u003e","description":"","filename":"10.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/4b78b11ce9b78ee8e71888a2.jpeg"},{"id":83826178,"identity":"ab951c13-37ae-433e-afe6-5f5c4f87b545","added_by":"auto","created_at":"2025-06-03 10:08:42","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":75429,"visible":true,"origin":"","legend":"\u003cp\u003eGWO flowchart.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/397e2dec02cba4d7db469bb5.png"},{"id":83825544,"identity":"e7eed436-c0c3-4894-9b37-326e4539c1ad","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"jpeg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":122648,"visible":true,"origin":"","legend":"\u003cp\u003eGraphs of the equations developed using the PSO algorithm.\u003c/p\u003e","description":"","filename":"12.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/b36813636c3c94884e92e97d.jpeg"},{"id":83825533,"identity":"da2cd472-2344-4506-88d7-d8716f521f67","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"jpeg","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":109283,"visible":true,"origin":"","legend":"\u003cp\u003eGraphs of the equations developed using the SA algorithm.\u003c/p\u003e","description":"","filename":"13.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/ddc8b5ed2ef895bb46c0842f.jpeg"},{"id":83825546,"identity":"87eab31c-c431-4d8b-81aa-e28cba408ab9","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"jpeg","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":116694,"visible":true,"origin":"","legend":"\u003cp\u003eGraphs of the equations developed using the GWO algorithm.\u003c/p\u003e","description":"","filename":"14.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/4f37518aa2fc849c9a362a55.jpeg"},{"id":83825535,"identity":"265cbf2a-e145-4caa-9b6d-4f915f3d9c37","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":28885,"visible":true,"origin":"","legend":"\u003cp\u003eThe scatter plots of the GWO, SA, and PSO models.\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/51f2a605c40b791497deb88d.png"},{"id":83825545,"identity":"726179c6-76b3-493e-9a0b-e3bbea731e3a","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":293856,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of error distributions before and after data reduction.\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/ecd0f2d8499c193d5d41cebf.png"},{"id":83825541,"identity":"0069f638-848a-4712-9061-134f1bc57f5d","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"jpeg","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":154316,"visible":true,"origin":"","legend":"\u003cp\u003eError distributions of GWO, SA, and PSO models.\u003c/p\u003e","description":"","filename":"17.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/0413f2d27805af379fcc7384.jpeg"},{"id":83825549,"identity":"86d34242-ef13-4bad-a8d9-122876679d7f","added_by":"auto","created_at":"2025-06-03 10:00:43","extension":"png","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":117064,"visible":true,"origin":"","legend":"\u003cp\u003eREC curves for the PSO, SA, and GWO models based on absolute prediction error.\u003c/p\u003e","description":"","filename":"18.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/047244e54dc12702f5bca93d.png"},{"id":83826466,"identity":"84892b90-abec-44c2-80e1-e9758b35ff57","added_by":"auto","created_at":"2025-06-03 10:16:42","extension":"jpeg","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":79917,"visible":true,"origin":"","legend":"\u003cp\u003eScore-based comparison of PSO, SA, and GWO models.\u003c/p\u003e","description":"","filename":"19.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/53bfc1850bbdef4c94f9773a.jpeg"},{"id":83826176,"identity":"9660a60c-a447-4bb2-a631-ca9f7c845442","added_by":"auto","created_at":"2025-06-03 10:08:42","extension":"jpeg","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":91491,"visible":true,"origin":"","legend":"\u003cp\u003eEvaluation of model accuracy using A20, IOA, and IOS metrics.\u003c/p\u003e","description":"","filename":"20.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/23321a00687e27482bcf1cfb.jpeg"},{"id":83825547,"identity":"b5c68cb2-aa13-4a41-8408-09d868e8988d","added_by":"auto","created_at":"2025-06-03 10:00:43","extension":"jpeg","order_by":21,"title":"Figure 21","display":"","copyAsset":false,"role":"figure","size":235346,"visible":true,"origin":"","legend":"\u003cp\u003eTaylor diagram for comparing the performance of the PSO, GWO, and SA models in predicting GSI from RMR.\u003c/p\u003e","description":"","filename":"21.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/68ee46b76906a925922781f4.jpeg"},{"id":83825543,"identity":"ad6c49b1-797d-4c76-a432-a5c74ea8cfa7","added_by":"auto","created_at":"2025-06-03 10:00:42","extension":"png","order_by":22,"title":"Figure 22","display":"","copyAsset":false,"role":"figure","size":272813,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the graphs of proposed and existing equations for GSI prediction based on RMR.\u003c/p\u003e","description":"","filename":"22.png","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/399cea142964633f737659e1.png"},{"id":88814198,"identity":"9077c28d-5713-4ef2-91be-449376219168","added_by":"auto","created_at":"2025-08-11 16:08:16","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":13709183,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6690480/v1/068db8eb-fde7-40d4-b31d-d41b552848e0.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Application of the metaheuristic algorithms to quantify the GSI based on the RMR classification","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eAccurate classification of rock masses is a crucial aspect of geotechnical and mining engineering, as it directly impacts excavation design, tunnel stability assessment, and the planning of underground support systems such as tunnels\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e. Among the various rock mass classification systems, RMR and GSI play a significant role in evaluating rock mass quality and predicting its mechanical behavior. The application of these classification systems includes slope stability analysis, excavation design, and underground support system planning \u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e. The RMR classification system, introduced by Bieniawski (1979), classifies rock masses based on six key parameters, including uniaxial compressive strength (UCS), rock duality designation (RQD), spacing of discontinuities, condition of discontinuities, groundwater conditions, and orientation of discontinuities \u003csup\u003e\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e,\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e. This system provides a quantitative approach for evaluating rock masses, which has led to its widespread acceptance in engineering projects. Nevertheless, because it is based on empirical observations, its effectiveness may vary across different geological conditions, making its application in diverse environments occasionally restrictive \u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eTo address some of the limitations of the RMR classification system, Hoek et al. (1995) developed a new classification system known as the GSI. In fact, GSI was introduced as an extension of RMR to provide a better characterization of jointed rock masses. By considering rock structure and discontinuity conditions, GSI offers a more realistic representation of in-situ rock mass behavior, particularly in weak and fractured rock formations. Since its introduction, GSI has been widely used in rock engineering to estimate the strength and deformation properties of rock masses \u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e. Estimating GSI from RMR has always been one of the primary challenges in rock engineering. Given the strong correlation between these two indices and their dependence on geological conditions, researchers in recent years have developed empirical relationships between them using regression models. \u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e,\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e, as listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. However, many of these existing relationships have been developed based on a limited number of rock types. This limitation can reduce their applicability across diverse geological conditions. Furthermore, conventional regression models may lack the capability to fully capture the complex and nonlinear relationships between RMR and GSI \u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e.\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\u003eExisting relationships for estimating GSI based on RMR.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\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 \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNo.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEquations\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eR\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eReference\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;0.7394RMR\u0026thinsp;+\u0026thinsp;0.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;0.9934RMR-4.913\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;0.739RMR\u0026thinsp;+\u0026thinsp;12.097\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.759\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;1.265RMR-21.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;RMR\u0026thinsp;\u0026plusmn;\u0026thinsp;5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;1.2092RMR-18.6143\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;0.793RMR\u0026thinsp;+\u0026thinsp;2.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.736\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;0.876RMR\u0026thinsp;+\u0026thinsp;0.935\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.876\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRMR\u0026thinsp;=\u0026thinsp;0.42GSI\u0026thinsp;+\u0026thinsp;33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e\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 overcome the limitations of previous studies, this research employs metaheuristic optimization algorithms, including PSO, SA, and GWO, to develop optimized relationships between RMR and GSI. These algorithms demonstrate high efficiency in solving complex optimization problems by balancing global and local search processes. They can effectively explore high-dimensional parameter spaces without getting trapped in local optima, resulting in more accurate and robust predictive models \u003csup\u003e\u003cspan additionalcitationids=\"CR15 CR16\" citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe primary objective of this research is to develop five predictive models, including linear, power, exponential, quadratic polynomial, and logarithmic relationships, and to identify the most accurate equation among them to establish the optimal relationship between RMR and GSI. This study aims to enhance the generalizability and accuracy of the proposed models by utilizing 14 different rock types, including shale, slate, phyllite, chert, conglomerate, limestone, marl, dolomite, sandstone, mudstone, andesite, serpentinite, mylonite, and tuff, collected from the Beheshtabad water transfer tunnel site. The performance of the developed models, optimized using meta-heuristic algorithms, is evaluated through multiple model performance metrics, including R\u0026sup2;, RMSE, MAE, ASE, MAPE, and MARE, to ensure their reliability. Moreover, to enhance the reliability and credibility of the results, the developed models were compared with existing empirical relationships from previous studies using the coefficient of determination (R\u0026sup2;). Sensitivity analysis was also conducted to assess the stability and robustness of the models. In addition to these evaluations, complementary statistical indicators including the REC curve, score analysis, index of agreement (IOA), index of structured overlap (IOS), and the A20 metric were employed to provide a comprehensive assessment of model performance.\u003c/p\u003e \u003cp\u003eThis study is structured into six sections. Section \u003cspan refid=\"Sec1\" class=\"InternalRef\"\u003e1\u003c/span\u003e provides an introduction to the significance of the topic, a review of previous research conducted by other scholars, existing challenges, and the objectives of this study. Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e describes the characteristics of the study area, introduces the dataset, and explains the data collection process. Section \u003cspan refid=\"Sec11\" class=\"InternalRef\"\u003e3\u003c/span\u003e focuses on the mathematical modeling process and the implementation of meta-heuristic algorithms (SA, PSO, and GWO) to optimize the relationship between RMR and GSI parameters. Section \u003cspan refid=\"Sec15\" class=\"InternalRef\"\u003e4\u003c/span\u003e evaluates the performance of the developed models and compares them with existing models. In section \u003cspan refid=\"Sec24\" class=\"InternalRef\"\u003e5\u003c/span\u003e, the advantages and limitations of this study are discussed, and recommendations for future research are provided. Finally, section \u003cspan refid=\"Sec25\" class=\"InternalRef\"\u003e6\u003c/span\u003e summarizes the findings of this study and emphasizes the importance of meta-heuristic algorithms in rock mass classification.\u003c/p\u003e"},{"header":"2. Material and methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Data preparation\u003c/h2\u003e \u003cp\u003eIn this research, data collected from the construction site of the Beheshtabad water transfer tunnel were used, and the characteristics of this tunnel will be explained in the following subsections.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section3\"\u003e \u003ch2\u003e2.1.1. Location of the study area\u003c/h2\u003e \u003cp\u003eThe Beheshtabad water transfer tunnel is located within the provinces of Isfahan and Chaharmahal-va-Bakhtiari in Iran. Designed to supply water to the country\u0026rsquo;s central plateau, the tunnel originates at the terminus of the Beheshtabad River, positioned at 50\u0026deg;38\u0026prime; E longitude and 32\u0026deg;02\u0026prime; N latitude. Extending approximately 65 kilometers in a northeastern direction, it concludes upstream of the Cham Asman Dam along the Zayandeh Rud River, at coordinates 51\u0026deg;12\u0026prime; E longitude and 32\u0026deg;22\u0026prime; N latitude \u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e. The geographical situation of the studied area and tunnel, which are parts of the Beheshtabad water transmission project, are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. In this figure, the location of the Beheshtabad tunnel is shown as the dashed blue line.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e \u003ch2\u003e2.1.2. Lithological and geomechanial properties of the study area\u003c/h2\u003e \u003cp\u003eThe entrance segment of the Beheshtabad water transfer tunnel is located within the Folded Zagros tectonic belt, while its outlet section is positioned in the Sanandaj\u0026ndash;Sirjan structural zone. To identify the dominant lithological units along the tunnel alignment, comprehensive field surveys and analyses of borehole data were carried out. These investigations revealed that the principal rock types present in the area include conglomerate, sandstone, mudstone, limestone, marly limestone, carbonate limestone, and andesite. The tectonic zoning along the tunnel\u0026rsquo;s trajectory is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Furthermore, a geological cross-section covering the kilometers 33 to 37 of the tunnel path is provided in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Based on the data shown in this figure, it can be inferred that the predominant lithologies in this section consist of limestone, dolomite, conglomerate, and marly limestone interbedded with shale layers \u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eOne of the most critical challenges faced during the tunneling operations was the presence of multiple fault zones throughout the study area. From a structural perspective, the Beheshtabad water transfer tunnel is divided into eight segments, each extending over approximately eight kilometers. Detailed geological and geophysical investigations have revealed that each of these segments exhibits distinct structural features, underscoring the complexity and variability of the subsurface conditions along the tunnel alignment \u003csup\u003e\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eTo provide a clearer understanding, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e schematically illustrates the structural features of the section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e (8\u0026ndash;16 km), section \u003cspan refid=\"Sec15\" class=\"InternalRef\"\u003e4\u003c/span\u003e (24\u0026ndash;32 km), and section \u003cspan refid=\"Sec29\" class=\"InternalRef\"\u003e7\u003c/span\u003e (48\u0026ndash;56 km) of the tunnel length based on the integration of geophysical findings with the geological profile. Based on this figure, it can be observed that the section spanning from kilometer 58 to 64, particularly within the Sanandaj-Sirjan tectonic zone and at its boundary with the Zagros zone, is significantly influenced by faults and folds.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.1.3. Statistical characteristics of the database\u003c/h2\u003e \u003cp\u003eBased on the boreholes drilled along the route of the Beheshtabad water transfer tunnel, 150 datasets were measured to determine the required parameters of GSI and RMR. These parameters were measured according to the methods suggested by existing standards. To calculate the RMR, its six essential parameters were first evaluated through field surveys and laboratory experiments. In this study, the UCS was measured according to the standard approach recommended by the International Society for Rock Mechanics (ISRM 1981)\u003csup\u003e\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u003c/sup\u003e. A servo-controlled axial compression testing device was used to determine the UCS value. The device used during the test, along with the fractured sample, is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. In this test, stress was applied at a rate of 0.5 to 1 MPa per second until the sample fractured. The stress value at the point of failure indicates the UCS.\u003c/p\u003e \u003cp\u003eThe RQD was measured according to the methods recommended by the American Society for Testing and Materials (ASTM 2017)\u003csup\u003e\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e. Additionally, the spacing of discontinuities was precisely calculated through mapping operations. The condition of the joints, including assessments of infilling, roughness, and joint frequency, was also evaluated according to the procedure outlined in the ISRM suggested method for the quantitative description of rock mass discontinuities \u003csup\u003e\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e. Groundwater conditions, which provide an index for determining the rock mass behavior under the influence of groundwater, were evaluated by examining moisture content and seepage on the rock surface. To provide a general perspective on groundwater levels in the Beheshtabad water transfer tunnel, a schematic diagram of groundwater levels along the tunnel from kilometer 0 to 6 is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. For each of the six parameters mentioned above, a score is assigned, and the RMR value is calculated by summing the scores of the first five parameters. If, in a specific structure, the angle of discontinuities relative to the tunnel is significant, the final RMR score is adjusted accordingly.\u003c/p\u003e \u003cp\u003eHoek and colleagues proposed a classification system entitled the GSI\u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e. This index was developed to address some of the limitations of previous classification systems, such as RMR. Specifically, it deals with those characteristics of the rock mass that influence its strength and deformability. The characteristics of a jointed rock mass depend on the characteristics of the intact rock parts and the sliding and rotational conditions of these pieces under stress fields. These conditions are controlled by the shape and condition of the joint surfaces that separate these blocks. Based on this, the GSI classifies the rock mass according to two parameters: the rock structure, which indicates the degree of blockiness and interlocking of the rock pieces, and the condition of the discontinuity surfaces. According to\u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e, the GSI can be calculated and expressed based on these parameters using Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eBased on the above procedures, 150 valid series of RMR and GSI measurements were collected along the 64 km length of the Beheshtabad tunnel with different rock types. Accordingly, a powerful database was prepared for the desired meta-heuristics modeling in this research. The statistical characteristics and symbols of the modeling variables are given in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Also, the boxplot of statistical information of used data is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \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\u003eStatistical characteristics and symbols of the parameters.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\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=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\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\u003eMin\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMax\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMean\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eStd. Dev.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRMR\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e42.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12.66\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGSI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e46.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e13.29\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\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Used methods\u003c/h2\u003e \u003cp\u003eIn this section of the present study, three metaheuristic algorithms, including SA, PSO, and GWO, have been employed to develop an optimal relationship between the GSI and RMR parameters. The aim of this section is to introduce these algorithms and explain their applications in this research. The rationale for selecting these algorithms lies in their high speed, extensive search capability within the solution space, and ability to escape local optima \u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e,\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e,\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e,\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e, making them suitable choices for modeling the relationship between GSI and RMR. In the following, each algorithm and its application in the present study are explained in detail.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1. PSO\u003c/h2\u003e \u003cp\u003eThe PSO algorithm is inspired by the social behavior of groups of organisms such as birds or fish. This algorithm, widely used for solving optimization problems, is considered one of the most popular metaheuristic algorithms. In PSO, a group of particles moves within the search space and updates their positions based on two key factors: personal experience (the best local solution) and collective experience (the best global solution). Each particle has two main characteristics: velocity and position, which are improved using motion equations. Simplicity in implementation, low requirement for parameter tuning, and high convergence speed are the main advantages of the PSO algorithm. These features have made the application of PSO successful in various engineering problems.\u003c/p\u003e \u003cp\u003eIn the present study, the PSO algorithm is used to model the relationship between the RMR and GSI parameters. Specifically, this algorithm searches the solution space to provide optimal coefficients for the proposed model. PSO demonstrates a strong ability to explore different regions of the search space and avoids being trapped in local optima. Moreover, this algorithm performs effectively when dealing with complex variables \u003csup\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e such as RMR and GSI. Considering these attributes, the use of the PSO algorithm in the present study is deemed appropriate. The flowchart of the PSO algorithm is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. Based on this figure, the particles are initially initialized with random values. Their performance is then evaluated using model performance evaluation criteria such as R\u0026sup2;, RMSE, MARE, MAPE, MAE and ASE. Subsequently, the best individual and collective solutions are identified, and the velocity of the particles is adjusted to improve the coefficients. This process continues until the optimization criteria are satisfied. Finally, the equation with the optimized coefficients and the model performance evaluation metrics are presented as the output.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2. SA\u003c/h2\u003e \u003cp\u003eThe SA is another metaheuristic algorithm commonly used to solve optimization problems. This algorithm is inspired by the annealing process of metals, which encompasses progressively reducing the temperature to reach a low-energy equilibrium state. In SA, the search space is explored randomly, and with a certain probability, worse solutions are also accepted. This feature enables SA to discover a wide range of possible solutions. The SA algorithm is simple and flexible in parameter tuning, making it highly adaptable for various optimization tasks \u003csup\u003e\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eIn the present study, SA has been employed to optimize the relationship between the RMR and GSI parameters. This algorithm can provide the best coefficients for the proposed model. The optimization is achieved by gradually reducing the temperature and effectively exploring the search space\u003csup\u003e\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e. By accepting worse solutions in the initial stages, SA can establish a highly accurate and stable relationship between the RMR and GSI parameters. This aspect is particularly important in scenarios where the data exhibits significant variability. In Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e, the flowchart of the SA algorithm for optimizing the coefficients of the relationship between RMR and GSI parameters is presented. According to this figure, the initial coefficients are generated, and the objective function is designed. In each iteration, new coefficients are randomly generated, and the objective function is recalculated. It is then evaluated whether the new coefficients improve the objective function. If they do, the coefficients are accepted. Otherwise, they are accepted with a certain probability based on the Metropolis rule. This probability depends on two factors: the change in the objective function and the present temperature. The temperature is gradually reduced after each iteration to decrease the probability of accepting mediocre solutions. This process continues until one of the stopping conditions, such as accomplishment a predefined number of iterations or minimal changes in the objective function, is met. Finally, the coefficients optimized by SA, which best represent the relationship between RMR and GSI parameters, are displayed as the output.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e2.2.3. GWO\u003c/h2\u003e \u003cp\u003eThe GWO introduced an inventive metaheuristic algorithm in 2014 by Mirjalili and colleagues\u003csup\u003e\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e. This algorithm inducements stimulus from the social hierarchy and cooperative hunting strategies of grey wolves in the wild. In the GWO framework, wolves are divided into four distinct groups: Alpha wolves, which signify the finest answers; Beta and Delta wolves, which assist in guiding the search process; Omega wolves, representing the rest of the population. The positions of the wolves are iteratively updated, guided by the Alpha, Beta, and Delta wolves. This hierarchical approach allows the algorithm to effectively balance exploration searching for new solutions and exploitation refining the finest answers found so far. Such a balance helps the algorithm avoid becoming trapped in local optima. a, A, and C are three key parameters in GWO that help optimize the positions. The parameter a can assists of the adjustment of the search range and gradually decreases from its initial value to zero. Parameters A and C are coefficient vectors used to calculate the distance and direction of the wolves' movement towards the best positions. These parameters adjust the direction and movement of the wolves in the search space to help the algorithm find better solutions \u003csup\u003e\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eIn this study, the GWO algorithm has been applied to optimize the coefficients of the relationship between RMR and GSI parameters. The development commences with the generation of an original population of random positions for the wolves. For each position, the objective function is evaluated, and the three best positions are selected as the Alpha, Beta, and Delta wolves, completing the initialization phase. During the position update phase, the positions of the Omega wolves are adjusted based on the positions of the Alpha, Beta, and Delta wolves. This adjustment is governed by equations that mimic the direction and distance of the wolves' movements toward their prey, which corresponds to the optimal solution. This iterative procedure endures pending a predefined stopping standard is met, such as attainment of the peak number of repetitions or observing negligible improvements in the objective function. Due to its adaptability and its ability to avoid local optima, the GWO algorithm is highly suitable for deriving accurate coefficients for the relationship between RMR and GSI parameters. The flowchart in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e illustrates the steps of the GWO algorithm as applied to optimize the coefficients of the relationship between RMR and GSI parameters, providing a visual representation of the procedure described above.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3. GSI-RMR modeling","content":"\u003cp\u003eHoek and colleagues announced a classification system identified as the GSI. The GSI specifically addresses the characteristics of rock mass discontinuities that affect their strength and deformability \u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e. The characteristics of the rock mass depend on the sliding and rotational characteristics of the blocks under stress fields. These conditions are controlled by the shape and surface characteristics of the blocks, which differentiate them from one another. The GSI classification establishes a relationship based on two parameters of the rock mass: the rock structure, which indicates the degree of blockiness and interconnecting of rock fragments, and the surface conditions of discontinuities \u003csup\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e. Based on this, Hoek and colleagues suggested that the GSI can be calculated and expressed using a diagram (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e) derived from these two parameters. The use of this diagram to determine the GSI value requires specialized expertise with a high level of geological knowledge and field observations, which demands considerable time and relatively high costs.\u003c/p\u003e \u003cp\u003eIn contrast to this method, researchers in recent years have developed relationships to calculate GSI from other rock mass classification parameters. As a result, by having one of the rock mass classification parameters and using the developed models, the GSI parameter can be easily calculated \u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e,\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e,\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e. As shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, these relationships are mostly linear. Additionally, the coefficient of determination (R\u0026sup2;) for these relationships indicates the average accuracy of the corresponding relationships in calculating the GSI. Furthermore, the type of rocks examined for estimating GSI using these existing models is very limited. This limitation can restrict the applicability of the existing relationships in different geological conditions, where a wider variety of rock types are present.\u003c/p\u003e \u003cp\u003eTo address the shortcomings of the existing models, this study aims to develop five different relationships between GSI and RMR, including linear, power, exponential, quadratic polynomial, and logarithmic relationships, using three different metaheuristic algorithms: PSO, SA, and GWO. To achieve this goal, five initial equations, including linear, power, exponential, quadratic polynomial, and logarithmic equations, are first defined for the model (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). The model then optimizes the coefficients of each equation and determines the best coefficients to achieve the equation with the highest accuracy for estimating GSI. By substituting these coefficients into the initial equations, the final equations between the RMR and GSI parameters are developed. Each of these five equations, which has the highest R\u0026sup2; value and the lowest MARE, MAE, MAPE, RMSE, and ASE values, is selected as the proposed relationship developed by the algorithm used for its development. Then, the selected equations of each algorithm are compared with the selected equations of the other two algorithms according to the aforementioned performance evaluation indices. Thus, the relationship with the highest accuracy for estimating GSI based on RMR is determined. In addition, this study utilizes 14 different types of rocks to develop the models under discussion. These rocks include shale, slate, phyllite, chert, conglomerate, limestone, marl, dolomite, sandstone, mudstone, andesite, serpentinite, mylonite, and tuff. This can expand the generalizability of the established models in this research.\u003c/p\u003e \u003cp\u003eIn the continuation of this section, the modeling of relationships between the RMR and GSI parameters is carried out based on the metaheuristic algorithms PSO, SA, and GWO.\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\u003eInitial equations for modeling the GSI-RMR relationship.\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\u003eR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEquation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eType\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI=(a*RMR)\u0026thinsp;+\u0026thinsp;b\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eLinear\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;a*RMR\u003csup\u003eb\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePower\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;a*e\u003csup\u003e(b*RMR)\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eExponential\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI=(a*RMR\u003csup\u003e2\u003c/sup\u003e)+(b*RMR)\u0026thinsp;+\u0026thinsp;c\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePolynomial\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;a*ln(RMR)\u0026thinsp;+\u0026thinsp;b\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eLogarithmic\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Modeling using the PSO\u003c/h2\u003e \u003cp\u003eIn this section, the PSO algorithm is employed to model the relationship between the GSI and RMR parameters. The objective of this effort is to develop an optimized relationship for estimating the GSI parameter using the RMR. The accuracy of this relationship is evaluated using performance indices, including R\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e, RMSE, MARE, MAPE, ASE, and MAE. In the current PSO modeling, the number of particles in the initial population was set to 30 to maintain a balance between accuracy and computational cost, as an excessive increase in the number of particles could lead to longer processing times. Additionally, the maximum number of iterations was set to 300 to provide sufficient opportunity for the algorithm to converge while avoiding unnecessary executions. These values were determined based on previous studies and an analysis of the algorithm's convergence behavior.\u003c/p\u003e \u003cp\u003eIn this model, the coefficients of the initial equations referenced in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e were optimized using the PSO algorithm. A list of these coefficients for each of the five models including linear, power, exponential, quadratic polynomial, and logarithmic model is provided in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. By substituting these coefficients into their corresponding equations from Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the final equations describing the relationship between GSI and RMR are obtained. These relationships, along with the performance evaluation indices for each equation, are presented in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. The graphs of these five equations are also plotted in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e. Based on Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, by evaluating and comparing the performance assessment parameters of the models developed using the PSO algorithm, the linear model is selected as the best relationship among the five equations developed by PSO for GSI prediction. This is because, in this model, the R\u0026sup2; value is higher than that of other models, while the values of RMSE, MARE, MAPE, MAE, and ASE are lower than those of other models, indicating a relatively higher accuracy of this model compared to the other four models.\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\u003eThe optimized coefficients calculated using the PSO algorithm.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\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=\"left\" 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=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003eCoefficient\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEquation type\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ea\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eb\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ec\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.0703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.6760\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLinear\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e26.2852\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLogarithmic\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e22.2040\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.2057\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePower\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e19.6419\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.0197\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eExponential\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0143\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.2615\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e30.3932\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePolynomial\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\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\u003eThe developed equations between RMR and GSI using the PSO algorithm.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\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=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" 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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEquation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"6\" nameend=\"c8\" namest=\"c3\"\u003e \u003cp\u003eModel performance evaluation parameters\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eR\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMARE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMAPE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eASE\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;1.0703 * RMR\u0026thinsp;+\u0026thinsp;1.6760\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8785\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.6563\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0814\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e8.9980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.8298\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e21.6808\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;26.2852 * log(RMR) \u0026minus;\u0026thinsp;50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.7813\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.2486\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.1053\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12.3518\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e4.9544\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e39.0444\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;22.2040 * RMR\u003csup\u003e0.2057\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.3782\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10.5356\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0864\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e8.9497\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e8.8907\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e110.9993\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;19.6419 * exp(0.0197 * RMR)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8522\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.1362\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0623\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.2941\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e4.1747\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e26.3811\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;0.0143 * RMR\u003csup\u003e2\u003c/sup\u003e \u0026minus;\u0026thinsp;0.2615*RMR\u0026thinsp;+\u0026thinsp;30.3932\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8326\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.6563\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0814\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e8.9980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.8298\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e21.6808\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\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Modeling using the SA\u003c/h2\u003e \u003cp\u003eThe SA algorithm was applied to establish an optimized relationship between the GSI and RMR parameters. This modeling effort aimed to accurately estimate GSI based on RMR values by enhancing the predictive capability of the proposed equations. The accurateness of the established models was assessed using several performance metrics, including the coefficient of determination (R\u0026sup2;), root mean square error (RMSE), mean absolute relative error (MARE), mean absolute percentage error (MAPE), average squared error (ASE), and mean absolute error (MAE). To ensure effective convergence and reliable outcomes, the maximum number of function evaluations for the SA algorithm was set to 2000, and the convergence tolerance was defined as 1e-4. These parameters were selected through iterative testing to minimize error and enhance model performance. In this study, the SA algorithm was employed to optimize the coefficients of the five fundamental equation types presented in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, including linear, power, exponential, quadratic polynomial, and logarithmic forms. The resulting optimized coefficients are listed in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. By incorporating these coefficients into their respective equations, the final predictive models for GSI estimation were formulated. These models, along with their corresponding performance indices, are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. Additionally, graphical representations of the five equations are illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e. According to the comparative evaluation presented in Table\u0026nbsp;\u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, the linear model outperforms the other equations developed using the SA algorithm. This conclusion is supported by its higher R\u0026sup2; value and lower RMSE, MARE, MAPE, MAE, and ASE values, indicating superior accuracy and robustness in predicting GSI from RMR.\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\u003eThe optimized coefficients calculated using the SA algorithm.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\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=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003eCoefficient\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEquation type\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ea\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eb\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ec\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.0296\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.3090\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLinear\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e36.2266\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-86.9522\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLogarithmic\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.6820\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8903\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePower\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e8.6264\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.0352\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eExponential\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.0086\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.5968\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-2.7388\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePolynomial\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\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\u003eThe developed equations between RMR and GSI using the SA algorithm.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\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=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" 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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEquation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"6\" nameend=\"c8\" namest=\"c3\"\u003e \u003cp\u003eModel performance evaluation parameters\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eR\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMARE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMAPE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eASE\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;1.0296 * RMR\u0026thinsp;+\u0026thinsp;3.3090\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9542\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.8201\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0483\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.4549\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.2672\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e8.0663\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;36.2266 * log(RMR) -86.9522\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9244\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.6426\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0645\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.0924\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.0229\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e13.2682\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;1.6820 * RMR\u003csup\u003e0.8903\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9533\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.8615\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0492\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.5966\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.3086\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e8.1879\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;8.6264 * exp(0.0352 * RMR)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5511\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.8755\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.1673\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e18.1669\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e7.8457\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e78.7748\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI = -0.0086 * RMR\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;+\u0026thinsp;1.5968 * RMR \u0026minus;\u0026thinsp;2.7388\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9139\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.8870\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0673\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.7855\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.1562\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e15.1085\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\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.3. Modeling using the GWO\u003c/h2\u003e \u003cp\u003eIn a similar approach, the GWO method was applied to model the relationship between the GSI and RMR parameters. The aim of this modeling was to develop a more accurate estimation of GSI using RMR. The estimation capability of the established models was measured using th standard evaluation metrics including R\u0026sup2;, RMSE, MAE, MAPE, MARE, and ASE. Based on trial-and-error procedure, the number of wolves was set to 20 and the number of iterations was set to 500 in order to maintain a balance between computational efficiency and solution accuracy. The coefficients of five models, namely linear, power, exponential, quadratic polynomial, and logarithmic, introduced earlier in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, were optimized using this method. The updated coefficients are presented in Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. By substituting these values into their respective equations, the final relationships between GSI and RMR were obtained. These equations and their corresponding performance metrics are provided in Table\u0026nbsp;\u003cspan refid=\"Tab9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. Graphical representations of the five models are illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e. According to the results, both the linear and power models demonstrated the highest R\u0026sup2; values, and their error indices showed only minor differences. However, given the simplicity and practical applicability of the linear model, it is proposed as the most suitable equation among the five models developed using the GWO method for GSI estimation.\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\u003eThe optimized coefficients calculated using the GWO algorithm.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\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 \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003eCoefficient\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEquation type\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ea\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eb\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ec\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.0254\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.4845\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLinear\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLogarithmic\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.5524\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.9110\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePower\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e18.6041\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.0210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eExponential\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-0.0013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.1384\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.2749\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePolynomial\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\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\u003eThe developed equations between RMR and GSI using the GWO algorithm.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\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=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" 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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEquation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"6\" nameend=\"c8\" namest=\"c3\"\u003e \u003cp\u003eModel Performance Evaluation Parameters\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eR\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMARE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMAPE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eASE\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;1.0254 * RMR\u0026thinsp;+\u0026thinsp;3.4845\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9541\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.8396\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0484\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.4644\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.2676\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e8.8396\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;10 * log(RMR)\u0026thinsp;+\u0026thinsp;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4071\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10.2010\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.1869\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e21.4527\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e8.7637\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e104.0597\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;1.5524 * RMR\u003csup\u003e0.9110\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9541\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.8371\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0484\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.4958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.2789\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e8.0491\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI\u0026thinsp;=\u0026thinsp;18.6041 * exp(0.0210 * RMR)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9329\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.4309\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0592\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.7951\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.7751\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e11.7708\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSI = -0.0013 * RMR\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;+\u0026thinsp;1.1384 * RMR\u0026thinsp;+\u0026thinsp;1.2749\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9543\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.8329\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0484\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.4431\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.2618\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e8.0255\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\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Assessment of the suggested models","content":"\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e4.1. Assessment of predictive accuracy using standard performance metrics.\u003c/h2\u003e \u003cp\u003eThe prediction capability of the PSO, GWO, and SA models is further assessed through the standard metrics, including R\u0026sup2;, RMSE, MAE, MAPE, ASE, and MARE. Ideally, a model with an R\u0026sup2; value of 1 and zero values for the remaining error indices is considered perfect. The equations used to calculate these metrics are provided in Equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) to (\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e). Table\u0026nbsp;\u003cspan refid=\"Tab10\" class=\"InternalRef\"\u003e10\u003c/span\u003e presents the calculated values of these performance indices, reflecting the predictive capability of each model in estimating GSI from RMR. As shown, the PSO model performs less accurately than GWO and SA across all six metrics and can thus be considered the least effective among the three. The SA model, on the other hand, demonstrates superior performance in five out of the six criteria. Although the difference between SA and GWO in the MARE value is minimal, SA generally shows the best overall accuracy in this study. While SA is identified as the most accurate model, it is noteworthy that all three methods exhibit acceptable predictive capability, as supported by the presented evaluation indices.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${R^2}=1 - \\frac{{\\sum\\limits_{{i=1}}^{n} {{{(yi - \\hat {y}i)}^2}} }}{{\\sum\\limits_{{i=1}}^{n} {{{(yi - \\bar {y}i)}^2}} }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$MAPE=\\frac{{100\\% }}{n} - \\sum\\limits_{{i=1}}^{n} {\\left| {\\frac{{yi - \\hat {y}i}}{{yi}}} \\right|}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$RMSE=\\sqrt {\\frac{1}{n}\\sum\\limits_{{i=1}}^{n} {(yi - \\hat {y}} {)^2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$MARE=\\frac{1}{n}\\sum\\limits_{{i=1}}^{n} {\\frac{{\\left| {yi - \\hat {y}} \\right|}}{{\\left| {yi} \\right|}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$MAE=\\frac{1}{n}\\sum\\limits_{{i=1}}^{n} {{{\\left| {yi - \\hat {y}} \\right|}^{}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$ASE=\\frac{1}{n}\\sum\\limits_{{i=1}}^{n} {{{(yi - \\hat {y}i)}^2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the above equations, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e signifies the \u003cem\u003ei\u003c/em\u003eth amount of actual data (observed), \u003cem\u003eŷ\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e donates the \u003cem\u003ei\u003c/em\u003eth amount of estimated data (predicted), \u003cem\u003en\u003c/em\u003e is the datasets number, ȳ is the average amount of the actual data (observed).\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\u003eThe obtained values of performance evaluation metrics for the proposed models.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\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=\"char\" char=\".\" 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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eModel\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"6\" nameend=\"c7\" namest=\"c2\"\u003e \u003cp\u003eIndex\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eR\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMAPE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eASE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMARE\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8785\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.6563\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.0814\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8.9980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3.8298\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e21.6808\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9542\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.8201\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.0483\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5.4549\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.2672\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e8.0663\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGWO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9541\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.8371\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.0484\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5.4958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.2789\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e8.0491\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/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e4.2. Scatter plot analysis of predicted vs. actual GSI values\u003c/h2\u003e \u003cp\u003eTo further assess the predictive capabilities of the proposed models, scatter plots comparing the actual GSI values with the predicted ones are illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e. In this figure, each sub-plot demonstrates the performance of one of the three models: PSO, SA, and GWO. The diagonal line (1:1 line) represents the ideal case where predicted values perfectly match the actual observations. According to the plotted results and performance metrics summarized earlier, the SA model yields the highest prediction accuracy with R\u0026sup2; = 0.9542 and RMSE\u0026thinsp;=\u0026thinsp;2.8201. Its scatter points are densely clustered along the 1:1 line, indicating close agreement with the actual data. The GWO model also shows a similar level of accuracy (R\u0026sup2; = 0.9541 and RMSE\u0026thinsp;=\u0026thinsp;2.8371), with only minor deviations from the ideal line. In contrast, the PSO model performs comparatively worse, with a lower coefficient of determination (R\u0026sup2; = 0.8785) and a higher RMSE (4.6563), as its scatter points display greater dispersion from the diagonal. Despite these differences, the scatter plots reveal that all three models demonstrate reasonable accuracy in estimating GSI values from the RMR input, supporting their applicability in geomechanical predictive modeling.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e4.3. Sensitivity analysis\u003c/h2\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e illustrates the error distribution of the SA, GWO, and PSO models under two different conditions: utilizing the entire dataset and after randomly removing 20% of the data. Sensitivity analysis is crucial in evaluating the robustness of predictive models, as it helps determine how changes in the input data affect model performance. In this case, the removal of a portion of the data allows us to assess whether the models remain stable and reliable when faced with incomplete datasets. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e, the overall error distribution has remained nearly unchanged after data reduction, maintaining an approximately normal shape. This suggests that the examined models exhibit high stability, as their performance does not deteriorate significantly with reduced data availability. In other words, removing 20% of the data has not caused a sudden increase in prediction error or introduced substantial variations in model behavior. Among the three models, PSO appears to have the least deviation in its error distribution, indicating greater robustness against data reduction.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e4.4. Error distribution analysis\u003c/h2\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig17\" class=\"InternalRef\"\u003e17\u003c/span\u003e illustrates the error distributions of the GWO, SA, and PSO models. The histograms represent the frequency of error values, while the red curves depict the fitted normal distribution for each model. The overall shape of the distributions indicates that the prediction errors are symmetrically centered around zero, suggesting that the models exhibit unbiased performance. Additionally, the error distributions closely resemble a normal pattern, which implies that the residuals are well-behaved and do not show significant skewness or irregularities. This characteristic is particularly important in predictive modeling, as it suggests that the models do not exhibit systematic bias and that their errors follow a stable and predictable pattern. Such stability enhances confidence in the reliability of the employed optimization algorithms.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e4.5. Model assessment through REC criterion\u003c/h2\u003e \u003cp\u003eTo further assessment of the proposed models accurateness, the Regression Error Characteristic (REC) curves were employed. Figure\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e illustrates the REC curves for the SA, GWO, and PSO models based on the absolute error between the predicted and actual GSI values. The area under the curve (AUC) was also computed for each model, resulting in values of 5.3618 for SA, 5.3530 for GWO, and 5.3168 for PSO. A higher AUC value reflects a better predictive performance across a range of error thresholds \u003csup\u003e\u003cspan additionalcitationids=\"CR29 CR30\" citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e, the SA model achieved the highest AUC, indicating the most accurate performance in estimating GSI. The GWO and PSO models ranked second and third, respectively. However, the relatively close AUC values among all three models suggest that each algorithm provides an acceptable level of accuracy in GSI prediction.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e4.6. Model Assessment through score analysis\u003c/h2\u003e \u003cp\u003eIn this study, in addition to the REC curve, score analysis was also employed to more precisely evaluate the performance of the GWO, SA, and PSO models. While the REC curve presents model performance in a continuous manner based on the percentage of accurate predictions across different error thresholds, score analysis enables a discrete comparison of the models based on six statistical indices: R\u0026sup2;, RMSE, MAE, MAPE, ASE, and MARE. In this approach, for each of the six mentioned indices, a score ranging from 1 (for the weakest model) to 3 (for the best model) was assigned to the models under evaluation. The total score of each model was then visualized using a stacked bar chart, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig19\" class=\"InternalRef\"\u003e19\u003c/span\u003e, to facilitate easier comparison. In this figure, the height of each column represents the overall score of each model, while the colored segments reflect the contribution of each individual index \u003csup\u003e\u003cspan additionalcitationids=\"CR29 CR30\" citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e. As illustrated, the SA model, with a total score of 17, achieved the best overall prediction capability, followed by the GWO and PSO models with scores of 13 and 6, respectively. It is worth noting that the results obtained from this score-based comparison are in full agreement with those derived from the REC curve, further confirming the superior accuracy of the SA model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e4.7. Assessment of models performance using A20, IOA, and IOS metrics\u003c/h2\u003e \u003cp\u003eTo complete the evaluation of the proposed models, three additional indices including Acceptable 20% Accuracy Index (A20), Index of Agreement (IOA), and Index of Symmetric Agreement (IOS) were calculated and presented in Table\u0026nbsp;\u003cspan refid=\"Tab11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. The a20 represents the percentage of data points with a relative error below 20%. This metric is determined using Eq.\u0026nbsp;(7) \u003csup\u003e28,32\u003c/sup\u003e. The IOA assesses how closely the predicted values match the observed ones, with scores ranging from 0 to 1. A value nearer to 1 signifies stronger agreement. It is calculated using Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e8\u003c/span\u003e). The IOS is an enhanced version of IOA that additionally accounts for the average of predicted values. It is computed using Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e9\u003c/span\u003e). In Equations (7) to (\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e9\u003c/span\u003e), Pi, Qi, and N refer to the estimated value, the real value, and the total data points number, respectively\u003csup\u003e\u003cspan additionalcitationids=\"CR33\" citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e\u003c/sup\u003e. According to Table\u0026nbsp;\u003cspan refid=\"Tab11\" class=\"InternalRef\"\u003e11\u003c/span\u003e, the PSO technique concluded the maximum relative accuracy in the low-error range with an A20 value of 0.985612. On the other hand, the GWO model outperformed the others in terms of IOA and IOS, with values of 0.987845 and 0.797099, respectively. The SA model yielded A20, IOA, and IOS values of 0.978417, 0.987726, and 0.796304, respectively, indicating that although its performance is comparable to the other two models, it ranks slightly below the GWO model. These results are graphically depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig20\" class=\"InternalRef\"\u003e20\u003c/span\u003e. While the SA model demonstrated superior performance over the GWO and PSO models in both the REC curve and score analysis, the findings based on these three complementary indices suggest that the GWO model holds a slight advantage in certain aspects of predictive accuracy. This observation highlights the importance of utilizing a diverse set of evaluation criteria to gain a more comprehensive and precise understanding of model performance.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(a20=\\frac{1}{N}\\sum\\limits_{{i=1}}^{N} {\\delta i}\\)\u003c/span\u003e \u003c/span\u003e Where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta i=1if\\left| {\\frac{{Pi - Qi}}{{Qi}}} \\right| \\leqslant 0.2\\)\u003c/span\u003e\u003c/span\u003e, otherwise\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta i=0\\)\u003c/span\u003e\u003c/span\u003e (7)\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$IOA=1 - \\frac{{\\sum\\limits_{{i=1}}^{N} {{{(Pi - Qi)}^2}} }}{{\\sum\\limits_{{i=1}}^{N} {{{(\\left| {Pi - \\overline {Q} } \\right|+\\left| {Qi - \\overline {Q} } \\right|)}^2}} }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$IOS=1 - \\frac{{\\sum\\limits_{{i=1}}^{N} {{{(Pi - Qi)}^2}} }}{{\\sum\\limits_{{i=1}}^{N} {{{(\\left| {Pi - \\overline {P} } \\right|+\\left| {Qi - \\overline {Q} } \\right|)}^2}} }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\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\u003eThe obtained amounts of performance assessment metrics for the proposed models.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\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=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eModel\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003eIndex\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eA20\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eIOA\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eIOS\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.978417\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.987726\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.986304\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGWO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.978417\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.987845\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.797099\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.985612\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.987815\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.793174\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\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec23\" class=\"Section2\"\u003e \u003ch2\u003e4.8. Taylor diagram analysis\u003c/h2\u003e \u003cp\u003eTo provide a comprehensive comparison of model performance, a Taylor diagram was employed, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig21\" class=\"InternalRef\"\u003e21\u003c/span\u003e. This graphical representation simultaneously visualizes the correlation coefficient, standard deviation, and centered root mean square error (RMSE) of the PSO, GWO, and SA models in predicting GSI values from RMR input data. As depicted in the figure, all three models are located in close proximity to the reference point, which represents perfect agreement with the observed data. This close clustering indicates that the proposed metaheuristic models exhibit strong predictive performance, with high correlation and minimal deviation from actual GSI values. Moreover, the convergence of the three models in the Taylor space reinforces the findings obtained from other evaluation metrics discussed in this study, including the REC curve, score-based analysis, A20 index, IOA, and IOS. These consistent results confirm that all three algorithms, despite slight variations in individual indices, are highly reliable for estimating GSI. The differences between their performances are considered statistically negligible for practical applications.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5. Comparative analysis with similar researches","content":"\u003cp\u003eIn this section of the research, to further evaluate the proposed relationships between RMR and GSI parameters developed by the SA, GWO, and PSO algorithms, the relationships obtained from these algorithms are compared with the results of models from previous studies based on various quantitative and qualitative criteria. To achieve this goal, first, the R\u0026sup2; index obtained from the proposed equations of each of the three models used in this research is compared with the R\u0026sup2; values of the proposed equations of existing relationships developed by other researchers. Additionally, the types of rocks studied are examined as a basis for a qualitative comparison of the proposed relationships of this research and existing relationships. Based on the comparison of R\u0026sup2; values of existing relationships between RMR and GSI parameters (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) and the proposed models in this study (Table\u0026nbsp;\u003cspan refid=\"Tab10\" class=\"InternalRef\"\u003e10\u003c/span\u003e), it can be concluded that the R\u0026sup2; values obtained from the proposed relationships in this study are higher than the R\u0026sup2; values of existing models, which indicates the relatively better performance of the proposed models in this study compared to existing models. Furthermore, In Fig.\u0026nbsp;\u003cspan refid=\"Fig22\" class=\"InternalRef\"\u003e22\u003c/span\u003e, the proposed equations in this study are compared with the equations presented by other researchers. Based on this figure, the proposed equations in this research demonstrate higher accuracy in predicting GSI compared to the equations developed by various researchers. This figure indicates that the equations developed in this study exhibit a better fit with the utilized data, which can be attributed to the use of metaheuristic algorithms such as SA, GWO, and PSO. These methods possess greater capability compared to traditional optimization techniques in modeling complex relationships between parameters such as RMR and GSI.\u003c/p\u003e \u003cp\u003eReviews show that in studies conducted by other researchers, a more limited variety of rock types has been investigated, which is considered a major weakness because it significantly reduces the generalizability of existing relationships. To address this problem, this study has attempted to increase the generalizability of the proposed relationships as much as possible by examining a wider range of rocks, including: shale, slate, phyllite, chert, conglomerate, limestone, marl, dolomite, sandstone, mudstone, andesite, serpentinite, mylonite, and tuff. These comparisons confirm the qualitative and quantitative superiority of the proposed relationships in this study over existing relationships. However, despite the high capability of the developed models in estimating GSI and its relatively lower error compared to other models, further research on other rock types, especially igneous and metamorphic rocks, is necessary to further increase the generalizability of existing relationships.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"6. Research benefits, restrictions and future directions","content":"\u003cdiv id=\"Sec26\" class=\"Section2\"\u003e \u003ch2\u003e6.1. Benefits\u003c/h2\u003e \u003cp\u003eThis study presents optimized mathematical relationships between RMR and GSI using SA, GWO, and PSO algorithms. By formulating separate equations for each algorithm and assessing their accuracy, the effectiveness of these models has been validated. The results demonstrate that all three models provide high accuracy and efficiency in capturing the relationship between RMR and GSI, confirming their reliability in estimating GSI based on RMR.\u003c/p\u003e \u003cp\u003eMoreover, the inclusion of 14 different rock types in this study enhances the generalizability of the proposed models, making them applicable to a wide range of geological formations. These findings can significantly contribute to geomechanical modeling and have practical implications for rock mass classification, tunnel stability analysis, and drilling optimization.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec27\" class=\"Section2\"\u003e \u003ch2\u003e6.2. Restrictions\u003c/h2\u003e \u003cp\u003eDespite the promising results, this study is limited to sedimentary rocks, which may affect the applicability of the developed models to igneous and metamorphic formations. Another limitation arises from the nature of metaheuristic algorithms, which rely on exploratory search mechanisms and do not guarantee finding the absolute optimal solution. Additionally, hyper parameter tuning and model optimization, especially for large datasets, can be time-consuming and computationally demanding.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec28\" class=\"Section2\"\u003e \u003ch2\u003e6.3. Future Directions\u003c/h2\u003e \u003cp\u003eTo enhance the robustness and applicability of the models, future studies should expand the dataset to include igneous and metamorphic rocks, improving the models' generalizability across diverse geological settings. Exploring hybrid approaches that combine multiple metaheuristic algorithms may lead to improved accuracy and model stability.\u003c/p\u003e \u003cp\u003eA promising avenue for further research is the integration of metaheuristic algorithms with machine learning techniques or neural networks, which could enhance predictive accuracy while reducing dependency on extensive parameter tuning. Furthermore, validating the proposed models with real-world geotechnical data, such as exploratory drilling results or slope stabilization projects, would strengthen their practical applicability.\u003c/p\u003e \u003c/div\u003e"},{"header":"7. Conclusions","content":"\u003cp\u003eThree metaheuristic algorithms, including PSO, SA, and GWO, were employed to evaluate GSI in 14 different rock types and to develop reliable relationships for determining GSI based on RMR. RMR was used as the input for the models to predict GSI, as these two parameters exhibit a high correlation. In total, 150 datasets were measured and analyzed to achieve the study's objective, which is predicting GSI using RMR. The results of this study and the analyses performed led to the following key findings:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eThe performance evaluation parameters R\u0026sup2;, RMSE, MAPE, MAE, ASE, and MARE for the PSO model were calculated as 0.8785, 4.6563, 8.9980, 3.8298, 21.6808, and 0.0814, respectively. These results indicate a reasonable predictive capability of the PSO model in estimating GSI.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eFor the GWO model, the performance evaluation parameters R\u0026sup2;, RMSE, MAPE, MAE, ASE, and MARE were calculated as 0.9541, 2.8371, 5.4958, 2.2789, 8.0491, and 0.0484, respectively. These results indicate that the GWO model outperforms the PSO model in predicting GSI. However, its accuracy is still lower than that of the SA model.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eAmong the three models used for GSI prediction, the SA model exhibited the highest accuracy with R\u0026sup2; = 0.9542, RMSE\u0026thinsp;=\u0026thinsp;2.8201, MAPE\u0026thinsp;=\u0026thinsp;5.4549, MAE\u0026thinsp;=\u0026thinsp;2.2672, ASE\u0026thinsp;=\u0026thinsp;8.0663, and MARE\u0026thinsp;=\u0026thinsp;0.0483. These values indicate that the SA model is the most reliable and the best-performing model among the applied models.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe results of scatter plot analysis and sensitivity analysis, confirm the statistical findings and indicate that the SA model outperforms the GWO and PSO models in predicting GSI based on RMR.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe REC curve analysis showed that the SA model had the highest AUC, confirming its superior prediction performance across different error thresholds. This result was also supported by score analysis, where the SA model achieved the highest total score among the models, highlighting its overall effectiveness.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eComplementary evaluation using A20, IOA, and IOS metrics revealed that while PSO had the best low-error accuracy (A20), the GWO model slightly outperformed others in IOA and IOS. However, the SA model showed balanced and competitive results, confirming its robustness and reliability in different evaluation criteria.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe comparison of the developed models in this research with existing empirical models showed that the proposed models exhibit higher accuracy, greater generalizability, and better alignment with real data obtained from the Beheshtabad water transfer tunnel site. The use of metaheuristic models has significantly improved the accuracy of GSI prediction based on RMR compared to existing empirical models.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eConsidering the obtained results, it can be stated that this research involved several innovations, comprising a high diversity of the studied rock types, application of three new algorithms (SA, GWO, and PSO) for GSI estimation with high accuracy. Also, developing a new GSI prediction model based on the RMR is a main novelty of the current study which can be considered as a non-destructive approach.\u003c/p\u003e"},{"header":"Abbreviations","content":"\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"601\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eASE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eAverage squared error\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eASTM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eAmerican society for testing and material\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eGSI\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eGeological strength index\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eGWO\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eGrey Wolf Optimization\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eISRM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eInternational society for rock mechanics\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eMAE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eMean absolute error\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eMAPE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eMean absolute percentage\u0026nbsp;error\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eMARE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003e\u0026nbsp;Mean absolute relative error\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003ePSO\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eParticle Swarm Optimization\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eCoefficient of determination\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eRMR\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eRock mass rating\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eRMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eRoot mean squared error\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eRQD\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eRock quality designation\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eSA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eSimulated Annealing\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eUCS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eUniaxial Compressive Strength\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eREC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eRegression Error Characteristic curve\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eAUC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eArea Under the Curve\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eIOA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eIndex of Agreement\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eIOA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eIndex of Symmetric Agreement\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eA20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eAcceptable 20% Accuracy Index\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003ePi\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003ePredicted value\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eQi\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eObserved value\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 31.4476%;\"\u003e\n \u003cp\u003eN\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68.5524%;\"\u003e\n \u003cp\u003eTotal number of data points\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eP. Koureh Davoodi\u003c/strong\u003e conducted the data collection, wrote the first draft of the manuscript, and participated in the research ideation and analysis of the results as a PhD candidate. \u003cstrong\u003eF. Hajizadeh\u003c/strong\u003e proposed the research idea, supervised all data collection activities, and guided the development of the manuscript as the primary supervisor. In addition, \u003cstrong\u003eM. Rezaei\u003c/strong\u003e proposed the research idea, supervised all data collection activities, guided the development of the manuscript, played a significant role in data analysis and manuscript revision, and manuscript editing as the secondary supervisor.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research did not receive any specific grant from funding agencies in public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ePermission for land study\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that all land experiments and studies were carried out according to authorized rules.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eRanasooriya, J. 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Assessment of short and long-term pozzolanic activity of natural pozzolans using machine learning approaches. in \u003cem\u003eStructures\u003c/em\u003e vol. 68 107159 (Elsevier, 2024).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Geological Strength Index, Rock Mass Rating, Particle Swarm Optimization, Simulated Annealing, Grey Wolf Optimization","lastPublishedDoi":"10.21203/rs.3.rs-6690480/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6690480/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAccurate classification of rock masses is an essential task in earth sciences applications. Among various classification systems, the Rock Mass Rating (RMR) and Geological Strength Index (GSI) are the most frequently utilized ones. Unlike the RMR, which is a quantitative classification, GSI is a qualitative system and needs to be converted into a quantitative one as well due to its multiple applicability in both mining and civil engineering projects. With this objective, GSI quantification directly from RMR can be an attractive issue as it remains a complex task still due to the limited accuracy and generalizability of existing empirical models under varying geological conditions. This study addresses this challenge by analyzing data from eleven different rock types and employing three metaheuristic optimization algorithms, namely Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Grey Wolf Optimization (GWO), to develop predictive models for quantifying GSI based on the RMR. Accordingly, five mathematical GSI-RMR equations including linear, power, exponential, polynomial and logarithmic types were first developed using each algorithm. The resulting equations were assessed using six statistical indicators: R\u0026sup2;, RMSE, MAE, ASE, MAPE, and MARE. According to this evaluation, the best-performing equation from each algorithm was selected as the optimum and further evaluated using both graphical and statistical analyses, including comparisons with conventional empirical relationships. The findings revealed that the derived GSI-RMR equation from the SA algorithm achieved superior performance based on the score analysis and the REC curve. However, complementary evaluation using A20, IOA, and IOS metrics showed that the derived equation GSI-RMR equations from the GWO and PSO algorithms outperformed SA in certain aspects. These results demonstrate the unique strengths of all three proposed GSI-RMR equations and highlight the importance of multi-criteria evaluation. Overall, the proposed models provide a more accurate and generalizable framework for quantifying GSI from RMR, improving upon traditional empirical methods and enhancing the required accuracy compared to the qualitative GSI estimation.\u003c/p\u003e","manuscriptTitle":"Application of the metaheuristic algorithms to quantify the GSI based on the RMR classification","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-06-03 10:00:37","doi":"10.21203/rs.3.rs-6690480/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-06-16T07:59:57+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-06-14T16:21:19+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-06-11T08:05:38+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-05-31T15:49:13+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"146315146706630725322738103971308288736","date":"2025-05-31T03:03:34+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"31771985608758761154628142894939500352","date":"2025-05-30T14:14:17+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"276871222946362179074282417143428257222","date":"2025-05-30T12:24:03+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-05-30T12:12:50+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-05-30T12:06:56+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-05-30T10:54:58+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-05-30T05:29:21+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-05-18T08:01:37+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a292dac3-5b24-4c1a-a705-2dac4db7f0eb","owner":[],"postedDate":"June 3rd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":49365209,"name":"Physical sciences/Engineering"},{"id":49365210,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2025-08-11T16:03:02+00:00","versionOfRecord":{"articleIdentity":"rs-6690480","link":"https://doi.org/10.1038/s41598-025-14332-1","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-08-07 15:57:49","publishedOnDateReadable":"August 7th, 2025"},"versionCreatedAt":"2025-06-03 10:00:37","video":"","vorDoi":"10.1038/s41598-025-14332-1","vorDoiUrl":"https://doi.org/10.1038/s41598-025-14332-1","workflowStages":[]},"version":"v1","identity":"rs-6690480","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6690480","identity":"rs-6690480","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}