1 Introduction

Fully grouted rock and cable bolting systems are the most common retaining systems widely employed in various civil and mining engineering aspects. Applications of rock bolting systems dates back to the late 1940s, which had been utilized as mining underground supports. However, the first use of cable bolts as a secondary support system in Australian underground coal mining was in the 1970s. Although rock bolting systems differ fundamentally and structurally from other types of reinforcement elements, demands utilizing these types of retaining systems are widespread and increased worldwide due to the simplicity, availability of materials, and ease of installation process in the field [96, 97].

Although many parameters can affect the performance of the fully grouted rock and cable bolting systems, these retaining systems' success heavily depends on some influential parameters of grout, such as the quality, mechanical behaviours, and type of the grouts. In fact, grout acts as a stable interface between the retaining system, and surrounding rock mass with their loading transfer abilities [1]. In other words, grout plays a vital role as a medium to transfer the initiated stresses from bolt to stable rock mass and also to transfer the in situ stress (lateral confining stress) from surrounding rock to bolt–grout interface. Since the failure of the fully grouted rock bolting system occurs commonly at the bolt–grout interface [2]. As a result, some researchers focused on investigating the strength properties of grouts and their sample preparations for cable and rock bolting systems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. For instance, Aziz et al. [9] suggested a general standard for the preparation and testing procedures of various grouts and resins. Several crucial parameters, such as resin sample shape, size, height-to-width or diameter ratio (H/D), resin type, resin age, and curing time (CT), were investigated carefully. Also, some mechanical properties of resin, including the uniaxial compressive strength (UCS), elastic modulus (E), shear strength (τ) , and creep, were examined. Li et al. [10] presented an analytical model for investigating a fully grouted cable bolt's shear behaviour, including shear strength and displacement. Mirza et al. [1] examined some mechanical parameters of two common grouts used in Australia, Jennmar Bottom-Up 100 (BU-100) and Orica Stratabinder HS. For this, the samples were cast on 50 mm cube moulds. The prepared samples were subsequently cured for 1 to 28 days. Then, the UCS, E, and rheological properties were measured. The results revealed that both grouts were suitable for applying in the cable bolting systems. Ma et al. [16] characterized the bond of fully grouted rock bolting systems based on the installation procedure. Mirzaghorbanali et al. [17] investigated the mechanical behaviours of Stratabinder HS by casting and then curing small- and large-scale samples in cube and cylindrical moulds. The UCS values of cured samples, ranging from 1 to 28 days, were measured. Along with measuring UCS values, the bending resistance values of grouts were also determined by conducting four points bending test. In another research, Mirzaghorbanali et al. [18] determined UCS values of grouts by considering a mixture of fly ash and Stratabinder HS, for different CTs. Indeed, this research was one of the early steps towards utilizing mine waste materials as a part of grout in the fully rock bolting systems. They found that the UCS and tensile strength of grouts had been increased in fly ash mixed samples. In another research, Entezam et al. [12] suggested that the UCS values of grout and the axial bearing capacities of fully grouted rock  bolting system resulting from pull-out tests, increased by replacing small amount of fly ash contents in the grout mixture.

Since the direct UCS experiments are extensively destructive, time- and cost-consuming, over the past two decades, several identified approaches, including empirical and statistical methods and intelligent machine learning techniques, have been proposed and developed as popular alternative methods for conventional direct UCS test, for predicting UCS values in different media. In addition to empirical techniques, numerous studies have explored the development of statistical models and formulas to estimate the UCS values (Table 1). However, addressing highly nonlinear issues, through purely statistical methods can be considered as a challenging task. As a result, the application of artificial intelligence (AI) and soft computing (SC) becomes effective and particularly relevant when addressing such a nonlinear relationship in different scopes of geosciences [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. Some of the most significant research works to predict UCS values applying AI techniques in different medium were given as follows and in Table 2. For instance, Meulenkamp and Grima [34] applied artificial neural networks for predicting UCS values by considering physical properties such as hardness, porosity, density, and rock type information from hardness tests on rock samples. Sonmez et al. [35] predicted E and UCS values of Ankara Agglomerate by using regression and fuzzy logic methods. Tiryaki [36] applied regression and ANNs methods for predicting UCS and E values of the intact rock materials. Jahed Armaghani et al. [37] estimated UCS values by applying three AI techniques, including adaptive neuro-fuzzy inference system (ANFIS), ANNs, and nonlinear multiple regression (NLMR). A comparison of the results revealed that ANFIS was more reliable than the other methods. In another research, Jahed Armaghani et al. [38] presented a relationship for predicting the UCS of sandstone by considering some parameters, such as dry density, slake durability index, and Brazilian tensile strength. Moussas and Diamantis [39] predicted the UCS of serpentine by using ANNs. For this, they considered serpentinization percentage and physical, dynamic, and mechanical characteristics of serpentinites as input data, while UCS values were the outputs. In another research, Moussas and Diamantis [40] applied ANN technique to estimate UCS values of peridotites. Cao et al. [41] presented a new hybrid AI method called the XGBoost-FA model, which was a combined model of the extreme gradient boosting machine (XGBoost) with the firefly algorithm (FA). The method was employed for predicting UCS and E values. Gowida et al. [42] utilized three AI methods, including ANNs, ANFIS, and support vector machine (SVM), to estimate UCS's downhole formation by considering drilling mechanical properties such as rate of penetration (ROP), gallon per minute (GPM), standpipe pressure (SPP), rotating speed in revolution per minute (RPM), torque (T), and weight on bit (WOB).

Table 1 Various techniques for UCS prediction
Full size table
Table 2 Range of effective parameters
Full size table

As mentioned, the AI and SC techniques are being applied to predict the UCS values. It is noteworthy that some of these AI optimization techniques, such as ANN, FIS, are available as the first options where complicated simulations are needed. However, training and testing procedures might be intensively time-consuming process. Moreover, the predictions would be more accurate than the other methods with lower errors if ensemble models were selected. This is because of combining outputs resulting from multiple techniques. Due to the importance of exact UCS values, this research introduced novel ensemble modelling approach for predicting the UCS values of grout in fully grouted rock bolting systems. Along with applying ensemble modelling, jellyfish search (JS) algorithm, which is innovative and distinguish, has been used to create weighted averaging ensemble predictions. Indeed, the results from other SC techniques, including multi-layer perceptron (MLP), Bayesian regularized (BR) neural networks, generalized feed-forward (GFF) neural networks, classification and regression tree (CART), and RF, will be combined to achieve more reliable and robust predictive system.

The literature reviews and Table 2 revealed that although many researchers applied soft computing techniques for predicting the UCS values of rocks, these methods have yet to be used for estimating the UCS values of grouts in bolt or cable systems. It is because of the difference in the material, complex composition, and mechanical behaviour. For instance, in comparison with the rocks that come with monolithic minerals, some of the other components, such as fly ash, cement, or water, might be found in the grout. Moreover, SC models suitable for the rocks might not consider the time-dependent behaviour of grout or the influence of environmental factors like temperature and humidity. The lack of comprehensive datasets is one of the other main reasons that should be mentioned. Indeed, building a robust soft computing model requires a substantial amount of data for training and validation. While datasets might be available for predicting the UCS of rocks, similar datasets for grouts in bolt or cable systems may need to be expanded or more present. As a result, as mentioned in the paper, one of the aims of this research is to build a dataset and showcase the performance of SC methods as a reliable tool instead of applying time- and cost-consuming conventional laboratory tests in predicting UCS values of grout in rock and cable bolting system, a valuable contribution to both academia and industry. Therefore, the main objective of this paper is (a) to build a dataset by considering  CTs, water-to-grout (W/G) ratios, and the type of cementitious grouts (2) to predict the exact UCS values of grouts used in either rock or cable systems with suggesting ensemble methods.

2 Experimental analysis

2.1 UCS tests

Seventy-three tests have been conducted to investigate the UCS values of different grouts for fully grouted rock bolting systems. For this, two common Australian grouts, including Stratabinder HS and BU-100, were chosen for further investigations. Six different amount of fly ash contents, including 5%, 10%, 15%, 20%, 25%, and 30%, were used in the grouts to investigate the effect of them on mechanical behaviour of grouts specifically UCS values. Also, to cover common various mixtures of grouts, eight different W/G ratios of grouts including, 22%, 25%, 30%, 32%, 35%, 36%, 38%, and 40%, have been considered in the required tests. The grout samples were then prepared and cast into the cube moulds with 50 mm × 50 mm × 50 mm dimensions in a line with ASTM standard C579 [60]. Subsequently, the samples were cured at five different times, 1, 7, 14, 21, and 28 days. Grout preparation and a view of moulds in one of the conducted tests are shown in Fig. 1.

Fig. 1
figure 1

a Grout preparation and b a view of UCS samples in the curing room

Full size image

After the curing procedure, the UCS values have been measured by a compression testing machine made by an impact test equipment, under a displacement of 1 (mm/min) at the University of Southern Queensland. Figure 2 shows a sample before and after the test.

Fig. 2
figure 2

A sample before and after the UCS test

Full size image

3 Material and methods

3.1 Classification and regression tree (CART)

Classification and regression tree (CART) is one of the decision tree algorithms categorized in the data mining techniques by Li et al. [61]. This technique belongs to the rules-based technique that can provide binary trees for recursively dividing a prediction space into subsets. Unlike other machine learning methods, the CART is a “white box” procedure that simplifies the relationships between the inputs and output(s) [62]. Instead of considering the relationships between the variables, the CART algorithm dataset is divided into uniform parts based on yes/no responses concerning the predictor values with the CART algorithm. This process leads to the creation of a binary tree. If the dependent variable is quantitative, the structured tree will be representative of a regression tree. However, if it is qualitative, it is representative of a classification tree. In the recursive partition process, at the starting point exists a root node, which includes the whole dataset and represents an attribute or input variable. Whereas branches surround the sides of each root, each shows a range of values. Figure 3 shows a general structure of a simple CART tree. The nodes are comprised of a till and leaf that predicts model output at the end of each step. This structure will continue till the termination criteria are met. It is worth noting that defining the optimum value for the number of intervals and the maximum tree depth will control the overgrowth of produced trees and overfitting problems.

Fig. 3
figure 3

Structure of simple CART tree

Full size image

3.2 RF

RF algorithm, which is generally recognized as a robust nonparametric statistical technique for classification and regression issues, was first proposed by Breiman [63]. Compared to other SC techniques, RF was developed as an ensemble approach relying on the outcomes from several trees to obtain prediction accuracy. Indeed, the RF integrates the estimated values from structured trees in the forests to provide the optimal result for each generated observation. Each tree in the forest serves as a vote for the RF's ultimate decision [63]. Figure 4 illustrates the general scheme of the RF algorithm.

Fig. 4
figure 4

General scheme of the RF algorithm

Full size image

3.3 ANN

In 1949, ANN was first constructed using the biological neural network as its primary source of inspiration. ANNs are noticeably superior, which may be attributed to the fact that nonlinear mapping can be carried out across a dataset when utilizing it. The ANN splits the available data into three primary datasets: training, testing, and validating data. The performance of ANN is controlled by three following factors, including transfer functions, the number of epochs, and learning roles. The ANN generally consists of three main layers, including the input, the output, and the hidden layers [64].

3.3.1 Multi-layer perceptron (MLP)

The MLP is one of the most widely used ANNs that are being applied in different scopes. The MLP neural network should first be trained to predict targets, which this process is commonly performed by utilizing the feed-forward back-propagation (FFBP) learning algorithm [65]. This algorithm minimizes the system errors between the predicted and actual values. In the MLP, a more compatible output may be generated by the system’s parameters of after its iterations have been mastered. The number of hidden layers is generally determined based on a trial and error and also depends on the complexity of the problem [66]. The general scheme of MLP structure is shown in Fig. 5. The output values of the MLP will be determined using the following equation [67]:

$$y = f_{i} \left( {\sum\limits_{i = 1}^{n} {w_{ij} x_{j} + b_{i} } } \right)$$
(1)
Fig. 5
figure 5

General architecture of the MLP model

Full size image

In which, x and y are, respectively, the value of inputs and output(s), w stands the weight vector, b shows the bias connected to the layers, and f denotes the transfer function (tansig, logsig, purelin, radbas, etc.).

3.3.2 Bayesian regularized (BR) neural network

The Bayesian regularized algorithm was first introduced by MacKay in 1992 as a solution to problems like determining the optimum hidden neurons while constructing ANN topology. He applied Bayes’ theorem to the regularization process [68]. The BR neural network is a variety of propagation neural networks that combine the traditional sum of the least-squares error function (Fig. 6) [69]:

$$E_{D} = \sum\limits_{i = 1}^{n} {\left( {y_{i} - t_{i} } \right)^{2} = } \sum\limits_{i = 1}^{n} {\left( {e_{i} } \right)^{2} }$$
(2)
$$S\left( w \right) = \beta E_{D} + \alpha E_{W}$$
(3)
$$E_{W} = \sum\limits_{i = 1}^{m} {w_{i}^{2} }$$
(4)

whereas n and t denote the number of training data and the target values, respectively, α and β are hyperparameters (regularization parameters), Ew suggests penalty phrase (large penalizer values of the weights), m indicates the number of weights, and S(w) signifies the performance function of the network [70].

Fig. 6
figure 6

Topology of the BR neural network model

Full size image

3.3.3 Generalized feed-forward (GFF) neural networks

GFF neural networks are generalizations of MLPNNs that allow for connecting one or more layers. Although the GFF neural network works theoretically similarly to MLPNN, it eliminates the complexity of the systems rather than MLPNN does. Besides that, MLPNN has often been trained hundreds of times, adding more learning epochs to externally elucidate the problem. However, GAFFNN uses only a few numbers of training epochs (Fig. 7) [71].

Fig. 7
figure 7

Structure of GFF neural network

Full size image

3.4 Ensemble model

As mentioned before, ensemble machine learning (EML) techniques have been used for predicting UCS values in this article. Indeed, the EML model integrates several base models called sub-models. The integration process includes four following methods: simple averaging, weighted averaging, integrated stacking, and separate stacking ensemble models. Also, bagging and boosting methods can be applied for implementing super learner ensembles.

3.4.1 Sub-models

Sub-models can be provided by employing one of the AI methods. Indeed, the average of these basic models, which have different structures, is required. For instance, an ANN model has n basic MLP, which has different hidden layers, transfer functions, and optimizers.

3.4.1.1 Weighted averaging ensemble

Due to the equal weights allocated for each sub-model, the predicted values were improved in the SAE technique. Moreover, the weighted averaging ensemble (WAE) combined results by averaging the outputs for all basic models (Fig. 8). It is noteworthy that the weight of the sub-model is determined by applying optimizing algorithms.

Fig. 8
figure 8

Diagram of the WAE-DE hybrid algorithm

Full size image

4 Jellyfish search (JS) algorithm

In general, metaheuristics algorithms can be divided in two categories, including nature inspired and human inspired. The nature-inspired metaheuristic algorithms can be categorized in three sub-classes as evolutionary, physical-based, and swarm-based algorithms [72]. In this research, JSO was chosen instead of applying other evolutionary algorithms such as genetic algorithms (GA) [73], physical-based algorithm such as gravitational search algorithm (GSA) [74], or swarm-based algorithm such as particle swarm optimization (PSO) [75]. It is because of its unique biological inspiration, diversity, adaptability, scalability, convergence speed, parallelism, and ability to strike an effective balance between exploration and exploitation [67]. In 2020, the jellyfish search (JS) algorithm was firstly proposed by Chou and Truong [72] by identifying and monitoring jellyfish food-searching behaviour in oceans. Since the proposal of the jellyfish search (JS) algorithm, several research works have applied it in diverse fields [76,77,78,79,80,81,82,83]. For instance, Abdel-Basset et al. [76] applied JS algorithm to optimize photovoltaic (PV) system in solar/PV generating units. Gouda et al. [78] could solve the identifications problems of polymer exchange membrane fuel cells (PEMFCs) model. Durmus et al. [84] applied the JS with other swarm-based algorithms, such as PSO, artificial bee colony (ABC), and mayfly algorithm (MA), to determine the optimal design of linear antenna arrays in wireless communication. Throung and Chou [79] presented a new model of fuzzy adaptive jellyfish search-optimized stacking system (FAJS-SS) for solving and optimizing engineering planning and designs. Ansari et al. [82] used JS algorithm to estimate state of health (SOH) of lithium-ion batteries. JS algorithm was proposed by identifying and monitoring jellyfish food-searching behaviour in oceans. Jellyfish foods include phytoplankton, small fish, and fish eggs, which are small oceanic animals [85]. Jellyfish bloom refers to a large mass of jellyfish that can swarm [86]. The ocean current and each jellyfish's personal search inside the jellyfish bloom are the two primary searching mechanisms of jellyfish for food that govern how the jellyfish's mechanism for looking for food moves. A giant jellyfish bloom might develop under these circumstances due to abundant nutrients in the ocean [87]. Each jellyfish is around its current area simultaneously to locate a spot with more food. The global and local search capabilities of the JS algorithm are shown by the mobility of jellyfish based on ocean currents and the search of each jellyfish in the jellyfish bloom, respectively. An ocean current often starts a jellyfish bloom, but if the environment changes (for instance, due to wind or temperature), the jellyfish travel to another ocean current [67]. The JS algorithm follows six primary phases: (1) jellyfish in the ocean, (2) ocean current, (3) jellyfish swarm, (4) passive motions, (4) active motions, and (6) jellyfish bloom [88]. Equation 5 is used in the JS method to determine the ocean current [72].

$$\overrightarrow {{{\text{trend}}}} = X^{*} - \left( {\beta \times {\text{rand}} \times \overline{X}} \right)$$
(5)

In which X* denotes the position of the jellyfish with a maximum food source representing the best solution, β stands as a distributions factor, rand signifies values in the range [0, 1] that are randomly determined, and \(\overline{X}\) indicates the mean food source in the jellyfish bloom. Eventually, each jellyfish's location was given by Eq. 6 [72].

$$X_{i}^{t + 1} = X_{i}^{t} + \left( {{\text{rand}} \times \overrightarrow {{{\text{trend}}}} } \right)$$
(6)

In other words, Eq. 6 determines how the JS algorithm's exploration step (global searches) operates. The jellyfish bloom's local search step is carried out by two different kinds of motion: passive and active motion. The passive motion involves each JS rotating around its location, comparing the amount of food in the new locations to its current location, and moving towards the new location if there is much more food available. Each jellyfish (Xi) in the active movement compares its location to another one (Xj), and if the amount of food at Xj is greater than that at Xi, the jellyfish moves towards Xj. Otherwise, it will veer away from the Xj jellyfish. JS algorithm uses Eq. 30 to perform the passive movement [89].

$$X_{i}^{t + 1} = X_{i}^{t} + \left( {\gamma \times {\text{rand}} \times \left( {U^{b} - L^{b} } \right)} \right)$$
(7)

where Ub and Lb denote the upper and lower bounds of the search space, respectively, and γ is the motion parameter.

To accomplish its active motion, each jellyfish uses Eq. 8 [72].

$$X_{i}^{t + 1} = \left\{ {\begin{array}{*{20}c} {{\text{if}}\,\,F_{f} \left( {X_{j} } \right) \le F_{f} \left( {X_{i}^{t} } \right)} & {X_{i}^{t} + \left( {{\text{rand}} \times \left( {X_{j} - X_{i}^{t} } \right)} \right)} \\ {{\text{if}}\,\,F_{f} \left( {X_{j} } \right) > F_{f} \left( {X_{i}^{t} } \right)} & {X_{i}^{t} + \left( {{\text{rand}} \times \left( {X_{i}^{t} - X_{j} } \right)} \right)} \\ \end{array} } \right.$$
(8)

In which Ff is the fitness function. Equation 9 suggests a time management process for the JS algorithm to balance the exploration and exploitation mechanism. The random value c(t) fluctuates between 0 and 1 [72].

$$c\left( t \right) = \left| {\left( {1 - \frac{t}{{{\text{Max}}_{{{\text{iteration}}}} }}} \right) \times \left( {2 \times {\text{rand}} - 1} \right)} \right|$$
(9)

where the current iteration number, t, is indicated in Eq. 9. When the values of c(t) exceed 0.5, the jellyfish conduct the global searches; otherwise, they conduct the local searches. The function (1-c(t)) regulates the jellyfish's passive and active motions. When the value of rand exceeds 1-c(t), the jellyfish move passively; otherwise, they move in active situations. According to Eq. 9, c(t) and 1 − c(t) tend to be one and zero as time passes, respectively. This means that the JS algorithm's capability to be exploited increases over time. The chaotic logistic map generates the initial population for the JS algorithm at random. The logistic map method (Eq. 10) makes it possible to produce an initial population with a wide variety of models, which aids in the algorithm's quick convergence rate [72].

$$X^{i + 1} = rX^{i} \left( {1 - X^{i} } \right),\left\{ {\begin{array}{*{20}l} {0 \le X^{0} \le 1\,,\,\,r = 4.0\,} \hfill \\ {X^{0} \notin \left( {0.0,0.25,0.50,0.75,1.0} \right)} \hfill \\ \end{array} } \right.$$
(10)

5 Data analysis and data preparation

5.1 Data presentation

Statistical analysis is needed to show the data and interpretation graphically. For this, five influential parameters were chosen as independent variables for estimating the UCS values of the grout in the fully grouted rock bolting system. The dataset includes seventy-three measurements of CT, W/G, Stratabinder HS (GS), BU-100 (GBU), fly ash (GF) contents, and UCS values. Table 2 provides descriptive statistics of the inputs and output parameters. The CT values range between 1 and 28 days with different values of W/G ratios, ranging from 22 to 42%. Notably, the GS, GB, and GF values were 0–100%, 0–100%, and 0–30%, respectively. Moreover, the measured UCS values ranges were between 22.58 and 90.32 MPa.

The Pearson correlation between effective parameters is depicted in Fig. 9. As can be seen, the correlation between UCS and CT is 0.71, which represents a high positive relation. Whereas a value of − 0.0089 is calculated for the correlation between UCS and GS, where indicating a low negative relation. Figure 10 shows the matrix plot of effective parameters.

Fig. 9
figure 9

Pearson correlation between effective parameters

Full size image
Fig. 10
figure 10

Matrix plot of effective parameters

Full size image

6 Results and discussions

This study addresses the performance and accuracy of several  powerful machine learning (ML) techniques, including MLP, BR, GFF, CART, and RF, methods to predict UCS values of grout in the fully grouted rock bolting system. For this, the measured data were standardized and normalized to allow the model’s generating. For this, available data were normalized by the min–max normalized method, which reduces data ranges to 0 to 1 value (Eq. 11) [90]:

$$X_{n} = \frac{{X_{m} - X_{\min } }}{{X_{\max } - X_{\min } }}$$
(11)

In which xn stands the normalized value of x, xmax represents the maximum value of data, xmin is the minimum value of data, and xm signifies the actual values of the data.

After normalizing the data, the training and testing datasets were subsequently built based on random selections. The training dataset included 54 data (or 75% of the dataset), whereas the testing dataset consisted of 25% of the dataset (or 18 data). The ANN, CART, and RF models were created using the datasets supplied in the normalization stage. As a result, the model parameters are changed at the point of prediction to achieve the maximum accuracy and performance of the models. The ideal structures of employed AI techniques were found after comparing the results with applying the following statistical criteria, including determination coefficient (R-squared), value account for (VAF), and root mean square of errors (RMSE) (Eqs. 12–14) [71, 90,91,92,93,94].

$$R^{2} = 1 - \left( {\frac{{\sum\nolimits_{i = 1}^{n} {(O_{i} - P_{i} )^{2} } }}{{\sum\nolimits_{i = 1}^{n} {(P_{i} - \overline{P}_{i} )^{2} } }}} \right)$$
(12)
$${\text{RMSE}} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {(O_{i} - P_{i} )^{2} } }$$
(13)
$${\text{VAF}} = 100 \cdot \left( {1 - \frac{{{\text{var}} (O_{i} - P_{i} )}}{{{\text{var}} (O_{i} )}}} \right)$$
(14)

where Oi and Pi are measured and predicted values, respectively. \(\overline{P}_{i}\) is mean value of the estimated value. n represents the number of available data.

The next step was to compare and evaluate the level of performance of various developed models using the final rating of the model (FRM) and colour intensity system (CIS). The R2, RMSE, and VAF values were rated during the FRM procedure. The model with the highest R2 and VAF values and the lowest RMSE value was deemed to have the highest rate. It is noteworthy that the highest rate depends on the number of models obtained. For instance, if there are ten models, the top model will have a rating of 10 [95]. Equation (15) is used for formulating the FRM rating system.

$${\text{FRM}} = \sum\limits_{i = 1}^{2} {\left( {r_{i}^{{R^{2} }} + r_{i}^{{{\text{RMSE}}}} + r_{i}^{{{\text{VAF}}}} } \right)}$$
(15)

where ri is the rate of statistical indices, i means 1 for training rates of statistical indices or 2 for testing rates of statistical indices.

6.1 Developing CART model

To develop CART model, the XLSTAT software was employed for predicting UCS values. In order to attain a high-accuracy level and minimal errors, the CART algorithm uses of train datasets to attempt to understand the connection between the inputs and output(s). Also, the model's performance is evaluated using the testing dataset. In this study, two stopping criteria, i.e. the maximum tree depth and the number of intervals, were considered after introducing datasets to the software to prevent the model's complexity.

Since choosing the large values causes overfitting and excessive tree growth, the general range of 1–10 for both maximum tree depth and the number of intervals was selected. Through trial and error, the initial ranges for the maximum tree depth and the number of intervals were lowered to (4–8) and (3–8) to achieve the best possible combination of these two parameters. Subsequently, R2, RMSE, and VAF values related to each model were calculated for the training and testing datasets (Table 3). Following this, seven models were then examined, each with a different value for the stopping criteria. Afterwards, the simple ranking of the FRM technique was used to select the most accurate model. The results of structured CART models with the various maximum tree depth and the number of intervals. The specified scores for them are presented in Table 3. As found in Table 3, the best CART model with high performance is model number 5, shown in bold, with a total rate of 42 of 42. The correlation between measured and predicted UCS values using the CART model for training and testing parts is shown in Fig. 11.

Table 3 Obtained performance indices and their rates for CART models
Full size table
Fig. 11
figure 11

Measured UCS values compared to predicted one by the CART model

Full size image

6.2 Developing RF model

The RF models with the different ntree and mtry values as two main stopping criteria were obtained. These stopping criteria control the system's complexity and reduce the model's running time. Therefore, the range of 50 to 200 was considered for ntree, and mtry was selected as 4, 6, and 8. The RF models were developed applying these settings (Table 4). According to Table 4, the 12 various RF models were constructed for predicting the UCS values, whereas only one was suitable for high-accuracy UCS prediction. Therefore, like the CART process, the Zorlu technique and FRM rating system were used to select a model with the highest score. Model no. 3 was one of the RF models that provided better RMSE than other RF models. However, the rating system and total rate indicated that RF model no. 9 was the superior model to estimate UCS values, with a total rate of 69 out of 72 (Table 4). Hence, it can be concluded that RF model no. 9 with ntree = 200 and mtry = 8 was the most accurate model, shown in the Table 4. The measured and predicted UCS values by RF method in the training and testing phases are depicted in Fig. 12. This figure showed that the RF model's performance was more than that of the CART model. Therefore, the RF model predicts the UCS values better than the CART model.

Table 4 Obtained performance indices and their rates for RF models
Full size table
Fig. 12
figure 12

Measured UCS values compared to predicted one by RF model

Full size image

6.3 Developing ANN model

The UCS values were predicted using three different ANNs: MLP, GFF, and BR. The Levenberg–Markvart (LM) algorithm was used to overcome the problems corresponding with issues of a complex nature in the system as a learning function and also for training the ANN. In general, the number of the hidden layer(s), the number of neurons in the hidden layer(s), the types of transfer (activation) functions, and the learning algorithms are some of the hyperparameters that manage ANNs efficiency and capabilities. Hence, the accuracy and result of an ANN heavily depend on the mentioned parameters. These parameters were adjusted with various values to find the most appropriate network structure with the highest potential performance. This approach was carried out through trial and error and does not adhere to rules. An extensive summary of the characteristics of the established networks to estimate UCS values is given in Table 5. The transfer functions "purelin", "logisg", "tansig", and "radbas" were examined. The total number of hidden nodes was also selected to be between 3 and 24. As a result, fifteen various structures were constructed. The results of the MLP models with different properties are presented in Table 5. Based on the justification provided for the FRM technique, Eq. (38) was used to score each R2, RMSE, and VAF of the training and testing phases. The results revealed that model no. 11, shown in bold, with a structure of 5-11-9-1, a transfer function of "tansig-tansig-logsig", and a total rate of 88 of 90, was the top-ranked MLP model. As shown in Table 5, the MLP model's performance level to predict UCS values was shown by R2 values of 0.904 and 0.859 for the training and testing stages, respectively. The predicted and measured UCS values are compared in Fig. 13.

Table 5 Obtained performance indices and their rates for MLP models
Full size table
Fig. 13
figure 13

Measured UCS compared to predicted one by MLP model

Full size image

The UCS values were also predicted using the GFF neural network approach. The GFF neural networks may solve problems with several complicated correlations between certain issue factors. After normalizing the dataset and dividing it into training and testing datasets, 15 diverse GFF neural network models were created, each with a different learning method, number of hidden nodes, and transfer functions. The topology of the GFF neural network was implemented by using trial and error technique. R2, RMSE, and VAF analysis were used to evaluate the model’s accuracy (Table 6). As found in Table 6, the GFF neural network model no. 13 with the architecture 5-13-9-1, shown in bold, was the best model. Notably, the best model's optimum topology produced the best R2 (train 0.915, test 0.903), VAF (train 90.988, test 85.268), and RMSE (train 4.655, test 2.624) results. As a result, this model received the highest rating of 90 out of 90. The comparison between measured and predicted UCS values using GFF neural network No. 13 is shown in Fig. 14.

Table 6 Obtained performance indices and their rates for GFF models
Full size table
Fig. 14
figure 14

Measured UCS compared to predicted one by GFF model

Full size image

The BR neural network predictive model was also used to estimate the UCS values in this step. The number of hidden neurons in the BR neural network model causes the system complexity. Therefore, the BR neural network modelling is regulated by using the number of hidden nodes as a stopping condition. The number of hidden nodes was changed to fall within the range of 1 to 15 to prevent overfitting and learning problems. They created the 15 BR neural network models. Table 7 summarizes the outcomes of BR neural network modelling for forecasting  the UCS values. The best BR neural network model architecture was chosen using the FRM method. With a cumulative rating of 87 out of 95, the BR neural network model number 8, shown in bold, is the one that predicts UCS values with the highest degree of accuracy. The ideal BR neural network model has a 5-8-1 design. It should be noted that these models consider colours that are more vibrant than those in previous BR neural network models. Figure 15 compares ' actual and predicted of UCS values using the best BR neural network model.

Table 7 Obtained performance indices and their rates for BR models
Full size table
Fig. 15
figure 15

Measured UCS compared to predicted one by BR neural network model

Full size image

6.4 Ensemble model based on WAE-JS hybrid algorithm

The WAE model is established on the concept that more competent base models should have a higher effect on the result. This is performed by providing weight to the result of several models. These weights may be found in several ways. The adoption of metaheuristic optimization algorithms is one of these ideas. In this study, the JS optimization algorithm was used to find the optimal weight of base models, i.e. CART, RF, MLP, GFF, and BR. Like other evolutionary computation techniques, the JS algorithm starts with initially generated solutions. The JS algorithm followed the four main steps to determine the base models' weight.

  1. (1)

    The chaotic map procedure was used to create the initial population of the artificial jellyfishes, Xi(i = 1, 2,…, n):

  2. (2)

    The jellyfish represented a model in the search space. The maximum number of iterations in this study and the size of the jellyfish population were set as 1000 and 85, respectively.

  3. (3)

    Beta and gamma had respective values of 5 and 0.19 based on trial and error.

  4. (4)

    Finding the X∗:

In this study, the RMSE values are employed to define fitness function as follows (Eq. 16):

$$F_{f} = \sqrt {\left( {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {X_{i}^{O} - X_{i}^{E} } \right)^{2} } }}{{n_{s} }}} \right)}$$
(16)

In which, \(X_{i}^{O}\), \(X_{i}^{E}\), \(n_{s}\) are the observed UCS, estimated UCS, and number of data, respectively. An artificial jellyfish with the lowest fitness function was given to the X∗ by the algorithm.

  1. (3)

    Up until the maximum number of iterations:

  2. (4)

    Determine the time control function, c(t), using Eq. (9).

  3. (5)

    Run a global or local search for artificial jellyfish.

  4. (6)

    Inspect the generated values and replace them with a new one if they fall outside the specified ranges.

  5. (7)

    Consider the new value, and if its fitness function value was the lowest, add it to X∗.

  6. (8)

    Output the final weights

Herein, weights to assemble base models in the WAE model are determined using the JS algorithm. Figure 16 shows the WAE model that has been proposed. Wi is a weight that can be applied to the result of the ith model in Fig. 16. Therefore, to maintain the following relation, \(\sum\nolimits_{i = 1}^{n} {W_{i} = 1}\), a value should be randomly chosen between [0, 1]. In this case, RMSE opted as the cost function (minimization).

Fig. 16
figure 16

Performance of determining the weights of the base models using the JS algorithm

Full size image

A comparison of statistical criteria revealed that the CART model is the superior model for predicting UCS values with the best performance and accuracy (Table 8). Therefore, the models were ranked based on their accuracy: CART > GFF > RF > BR > MLP.

Table 8 Performance indices for CART, RF, MLP, GFF, and MLP models and their ranks
Full size table

In the next step, the final weight of the base model was found using the JS algorithm, as displayed in Fig. 16. Figure 16 shows the convergence diagram of the RMSE to minimal values. Based on the results, base model weights are determined (Fig. 17). The highest and lowest weights are assigned to the CART and MLP models, respectively.

Fig. 17
figure 17

Final weight of the base models

Full size image

In this step, the ensemble model based on WAE was developed using weighted base models. For this, after developing, evaluating, and comparing the required models, the developed models were subsequently weighted by using the JS algorithm and the final predicted values were obtained using Eq. (17):

$$y_{f} = \sum\limits_{i = 1}^{n} {\left( {w_{i} \times y_{i}^{{{\text{base}}}} } \right)}$$
(17)

where yf is the final output from ensemble model, wi denotes the weight of each model, and \(y_{i}^{{{\text{base}}}}\) indicates the output from each base model (CART, RF, MLP, GFF, and BR).

6.5 Evaluation of the predictive models

As mentioned before, the developed models were evaluated utilizing statistical indicators of R2, RMSE, and VAF, which were used for measuring the performance of the predictive models. The best model had values close to 1, 0, and 100 for R2, RMSE, and VAF, respectively. Therefore, the values of the performance indices were calculated for both training and testing datasets (Fig. 18). As seen, the accuracy and performance level of the WAE model were better than CART, RF, MLP, GFF, and BR. In addition, Fig. 19 demonstrates a comparison between measured and estimated UCS values for the developed model by training and testing datasets. Therefore, the ensemble models presented the most reliable results among each constructed best base model in predicting UCS values in the fully grouted rock bolting system.

Fig. 18
figure 18

Measured UCS compared to predicted one by WAE model

Full size image
Fig. 19
figure 19

Comparison of statistical indices for CART, RF, MLP, GFF, BR, and WAE models

Full size image

Table 9 presents the performance of various ML models for predicting the grout’s UCS values in the fully grouted rock bolting system. The CART, RF, GFF, and BR models showed strong performances with high R2 values on both the training and testing datasets. The RMSE values are relatively low, indicating good predictive accuracy. However, a decrease in RMSE on the testing datasets was seen in all models. Also, the VAF values are relatively high, indicating a good level of VAF.

Table 9 Comparison of developed models based on the evaluation metrics
Full size table

Moreover, the WAE-JS model was the clear top performer, exhibiting the highest R2 values, the lowest RMSE values, and the highest VAF values on both training and testing datasets. It outperformed the other models and demonstrated minimal signs of overfitting. They generalized effectively to the testing data. Indeed, the WAE-JS model is the standout choice for predicting grout's UCS values in fully grouted rock bolting systems due to its superior performance across all evaluation metrics.

6.6 Sensitivity analyses

To determine the effectiveness of parameters and suggest the most effective parameters on variation UCS values, a sensitivity analysis was carried out for all the input parameters using the cosine amplitude (CA) method. In the CA technique, all data pairs are arranged into a data array (Eq. 18):

$$X = \left\{ {x_{1} ,x_{2} ,x_{3} , \ldots ,x_{i} , \ldots ,x_{n} } \right\}$$
(18)

In which, xi stands a vector with the length of m as Eq. (19):

$$x = \left\{ {x_{i1} ,x_{i2} ,x_{i3} , \ldots ,x_{ii} , \ldots ,x_{im} } \right\}$$
(19)

Therefore, the sensitivity of each input can be calculated as following formula that establishes the relationships between xi and xj:

$$r_{ij} = \frac{{\sum\nolimits_{k = 1}^{m} {\left( {x_{ik} \times x_{jk} } \right)} }}{{\sqrt {\left( {\sum\nolimits_{k = 1}^{m} {x_{ik}^{2} } } \right)} \times \sqrt {\left( {\sum\nolimits_{k = 1}^{m} {x_{jk}^{2} } } \right)} }}$$
(20)

where xik and xjk are the input and output parameters, m denotes the number of datasets.

The sensitivity analysis results revealed that W/G had the most impact on the grout’s UCS values among all the inputs (Fig. 20). Based on the calculated effectiveness value (rij), input parameters can be sorted in descending order: CT (0.871), GS (0.834), GF (0.592), and GBU (0.405).

Fig. 20
figure 20

Influence degree of effective parameters on UCS

Full size image

7 Conclusions

Mechanical parameters of grouts such as UCS play critical roles in applying ground-controlling methods such as fully grouted rock bolting systems in civil, mining, and geotechnical projects. This research focused on developing a model ensemble system to predict grouts’s UCS of strata reinforcements’ system. The main novelty of this study was the integration of a new algorithm of JS with the WAE technique, called WAE-JS, to develop an ensemble model. For this purpose, two commercial grouts in the Australian industry, including Stratabinder HS and BU-100, were considered for further investigation. Then, comprehensive experimental UCS tests were conducted considering some influential parameters on grout values, such as different curing time ranges, various W/G ratios, and several fly ash contents. After building the dataset, the MLP, BR, GFF, CART, and RF models were used to generate the outputs, while the JS algorithm was employed for weighting obtained outputs. The WAE-JS model consistently outperformed other ML algorithms, as evidenced by high R-squared values, low RMSE values, and exceptional VAF scores for both the training and testing sets. These results proved the reliability and accuracy of this algorithm in predicting grout UCS values. By replacing traditional, time-consuming, and costly conventional measurement methods with ML techniques, the proposed algorithm can significantly accelerate the UCS prediction process, which is particularly valuable in project planning and decision-making. Also, the WAE-JS model demonstrated remarkable generalization capabilities, as indicated by its minimal overfitting on the testing set. This robustness allows for the application of the model in many practical scenarios. While the presented algorithm exhibited outstanding performance, there are several areas for further investigation and improvement. Enhancing the interpretability of the WAE-JS model could provide valuable insights into the underlying factors influencing grout’s UCS values. This could aid in better understanding the relationship between input parameters and UCS values. The quality and quantity of data used to train and test the algorithm are critical to its performance. Continual efforts to collect and curate comprehensive datasets are essential. Indeed, the dataset can be developed by considering the other grout types, additives, and parameters. The development of the WAE-JS ensemble model represented a significant step forward in predicting grout UCS values in the fully grouted rock bolting system, offering exceptional accuracy, efficiency, and generalization capabilities. While the algorithm presented remarkable advantages, addressing challenges related to interpretability and real-world validation will further enhance its utility in ground control methods in construction, mining, and geotechnical projects.