1 Introduction

Blasting is still practiced for fragmenting rocks in surface and underground mining projects. A huge amount of energy is generated during the blasting process and only a small portion of this energy is effectively used to fragment and displace the rock mass and the rest of the energy is wasted in the form of undesirable events, such as air blast, fly rock, ground vibration, etc. [1,2,3,4,5,6,7,8,9]. Therefore, optimizing blast design parameters should be targeted to get the best possible rock fragmentation to be efficient for subsequent operations, including loading, hauling and crushing [10,11,12]. As a matter of fact, there are several influencing uncontrollable (rock mass properties) and controllable (blast geometry) factors affecting fragmentation quality making blast design a process with high complexity [13,14,15].

Investigating 432 blasting events, Mehrdanesh et al. attempted to evaluate the effect of rock mass properties on fragmentation. They concluded that in comparison of controllable parameters, uncontrollable parameters are more effective on rock fragmentation. Their study results showed that, from the rock mass properties group, point load index, uniaxial compressive strength, Poisson’s ratio, cohesion and rock quality designation, respectively, are the most important parameters on rock fragmentation and from the blast geometry group, stemming, spacing and hole diameter are the least important parameters on the quality of rock fragmentation [13]. Numerous empirical Formulas have been introduced to model rock fragmentation due to blasting. However, due to the complex nature of the fragmentation and limitation of effective variables in conventional models, these formulas are not adequately accurate. Consequently, they will not be capable to predict rock fragmentation suitably. It seems that more precise techniques are needed to predict the rock fragmentation [16].

Nowadays, artificial intelligence (AI) is being applied in a range of geo-engineering projects and AI is a fruitful approach to cope with such types of problems [17,18,19,20,21]. In this regard, a number of research studies have been carried out to utilize various AI tools to improve blast design parameters obtained from conventional and empirical methods [13, 22,23,24]. Table 1 briefly summarizes some researchers’ work in rock fragmentation, where they have used different AI tools and techniques. In this paper, for which four different mines were adopted as case studies, various techniques including regression analysis, classification and regression tree, support vector regression and artificial neural network were applied to predict rock fragmentation in the open pits blasting operation.

Table 1 Summary of researches regarding rock fragmentation prediction
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2 Artificial neural network

Artificial neural network is a branch of artificial intelligence [36,37,38]. It is made of a multilayer topology in which the layers are connected to each other. The first layer is considered for placing inputs, whereas the last one is for output(s). In addition to the mentioned layers, there are one or more layers known as hidden (transitional) layers which are placed in between the first and last layer. In fact, the hidden layers’ components known as neurons are responsible for the required computations. Number of the neurons in each hidden layer is determined by a try and error mechanism. When facing very low correlation ANN would be the best possible solution as compared to the available conventional alternatives [12, 13]. Amongst various advantages of ANN modeling, function approximation and feature selection can be considered as a specific capability [39,40,41].

To start working with ANN, a reasonable number of data sets (a set of inputs and their respective outputs) should be collected and used for training various network architectures from which the best combination would be selected. Artificial neural network (ANN) is increasingly being used to solve various nonlinear complex problems, such as rock fragmentation. However, it is not clear that what appropriate sample size should be there when using ANN in this context. The amount of data required for ANN learning depends on many factors, such as the complexity of the problem or the complexity of the learning algorithm. Till now, it is not clear that how much sample data should be there in a predictive modeling problem. However, there are some empirically established rule-of-thumb are there to estimate sample size requirements when using ANN. For example, one rule-of-thumb is that the sample size needs to be at least a factor of ten times the number of features. During this process, first, the connections between the neurons should be assigned a random weight, thereafter the initial given weights would be updated in each modeling run to gain the best possible efficient network. The next important item which should be thought of is adopting a proper method of training such as a back propagation algorithm with many advantages as compared to the other existing approaches [42,43,44,45].

A trained network can be examined by comparison of the model outputs with that of the measured outputs. To do this four statistical indices including determination coefficient (R2), mean absolute error (MAE), root mean square of errors (RMSE), and variance account for (VAF) can be calculated [46,47,48,49,50]. The following formulae are the mathematical expressions of the aforesaid indices:

$${R}^{2}=1-\frac{{\sum_{i=1}^{N}(O-{O}^{^{\prime}})}^{2}}{{\sum_{i=1}^{N}(O-\tilde{O })}^{2}}$$
(1)
$$\mathrm{VAF}=\left[1-\frac{\mathrm{VAR}(O-{O}^{^{\prime}})}{\mathrm{VAR}(O)}\right]\times 100$$
(2)
$$\mathrm{RMSE}=\sqrt{\frac{1}{N}{\sum }_{i=1}^{N}{(O-{O}^{^{\prime}})}^{2}}$$
(3)
$$\mathrm{MAE}=\frac{1}{N}{\sum }_{i=1}^{N}\left|(O-{O}^{^{\prime}})\right|,$$
(4)

where \(O\), \({O}^{^{\prime}}\) and \(\tilde{O }\) are the measured, predicted and mean of the O (output) values, respectively, and N is the total number of data.

3 Case study

In this paper, the required database is obtained from four different open pit mines [13]. All the mines are situated in Iran (Fig. 1) and considered to be the main sources of copper and iron ore in the country. Table 2 gives some descriptions about the mines.

Fig. 1
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Location map of studied mines

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Table 2 Various mines and rock formation of case studies
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4 Collection of data sets

In this research, the database has been collected by performing 353 blasting operations in 4 mines mentioned in chapter 3. Descriptive information of the data sets is given in Table 3. Controllable parameters including burden, spacing, stemming, bench height, hole diameter, powder factor and uncontrollable rock characteristics comprising universal compressive strength (UCS), uniaxial tensile strength (UTS), Is50, density, Young’s modulus, P-wave velocity, Schmidt hardness value, Poisson’s ratio, rock quality designation (RQD), cohesion and friction angle were considered to the inputs.

Table 3 Variables used for developing models
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In this research, image analysis techniques were applied to calculate size distribution using Split-Desktop software. Fragmentation has been calculated on the basis of 50% of passing size (X50). Finally mean-blasted particle size (X50) was selected as output in the modeling process.

5 ANN architecture

In this study, a total number of 353 data sets were used for training and testing groups. Back propagation approach was implemented for the model training. To have an applicable database and to improve efficiency of the training process, the whole data sets were normalized between values of − 1 and 1 [51]. After preprocessing of the data sets, to find out the best possible model with maximum accuracy and minimum error, numerous networks were created by varying pertinent elements, such as number of hidden layers and their respective neurons [52]. MAE, RMSE, VAF and R2 were determined for the various network topologies (Table 4). As it is seen in this table, the best model is a back propagation network with an architecture 18-14-1 and a hyperbolic-tangent transfer function in both the hidden and output layers (no.10). From Fig. 2, an optimum architecture of the ANN model is depicted. The determination coefficient was computed 0.9947, which is adequate to show competency of the developed ANN model.

Table 4 Comparison of different neural network structures
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Fig. 2
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Architecture of the optimum ANN model

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6 Multivariate regression analysis (MRA)

Multivariate regression analysis was used to evaluate the relationship between the inputs and output. MRA is considered as a conventional method of trend analysis in scientific tasks [53,54,55]. Using Statistica 12.0 software [56,57,58], regression analysis was performed to develop a mathematical function for predicting mean size of the fragment size (X50) (Eq. 5). As it is deduced from this equation, burden, spacing mean in-situ block size, uniaxial compressive strength, Schmidt hardness value, cohesion, Young’s modulus and density have a direct relevance with X50, whereas bench height, hole diameter, stemming, powder factor, Poisson’s ratio, UTS, Is50, friction angle, P-wave velocity and RQD are indirectly effective in the X50 magnitude. The determination coefficient and RMSE were computed 0.8863 and 0.026, respectively, which indicates the relatively lower performance of the developed MRA model compared to the ANN model:

$${X}_{50} = 0.01 \left(B\right) + 0.009 \left(S\right)-0.003 \left(H\right)-0.0005\left(D\right)-0.001\left(\mathrm{ST}\right)- 0.33 \left(\mathrm{PF}\right) - 0.001 {(\mathrm{Is}}_{50}) + 0.002 (\mathrm{UCS}) - 0.005 (\mathrm{UTS})+ 0.022 (\rho )+0.002 (E)-0.1 ({V}_{\mathrm{p}})+0.007 (\mathrm{SHV})-0.524 (\vartheta )- 0.001 (\mathrm{RQD})+ 0.515 (C)- 0.004 (\varphi )+ 0.4 \left({X}_{\mathrm{B}}\right) + 0.233$$
(5)

7 Classification and regression tree

Decision tree (DT) is fundamentally a branch of hierarchical approach which is used worldwide due to its capability to cope with classification-based problems. Structure of a tree contains different parts including, root, branches, leaves and nodes. DT is an ascending way of solution in which the root is placed at the topmost of the tree. In this technique, solution process is started with selecting a random node as a potential root for the tree. Each node represents a variable of the problem in hand and is divided into two branches. Division of the nodes is done with help of one of the independent variables. It is noted that a range has to be selected during the division process using a try and error mechanism. The selected range should be such a way that model performance indices such as root mean square error (RMSE) be minimized for each and every node [59, 60].

This method is also employed for regression analysis [61,62,63,64,65]. Due to various merits of classification and regression tree (CART) over other decision tree algorithms, it is normally preferred to be applied by many researchers [66,67,68]. In this paper, Matlab software was used to predict rock fragmentation incorporating the CART method. Developed decision tree for predicting X50 is shown in Fig. 3.

Fig. 3
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Developed CART model for predicting X50

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8 Support vector regression

Support vector machine is applicable for solving both the classification and regression problems. In machine learning, support vector machines (SVM), which is well-known to handle structural risk minimization, is widely used in different fields of investigation [69,70,71]. Support vector regression (SVR), a subdivision of SVM, is suitable for dealing with interpolative and extrapolative problems using a specific predictive model. In this SVR technique, Vapnik–Chervonenkis (VC) theory is considered as the base for formulization [72,73,74]. Reasonable generalization reaches when VC dimension is quite low which in turn causes the error probability to be definitely low [75, 76]. Also, in this technique, a “loss function” is applied for regression estimation and function approximation. The function is defined as the difference between predicted value and tube radius (ε). Figure 4 shows the idea of the ε-insensitive loss function. As it is seen in this figure, samples situated out of the ± ε margin, would be considered non-zero slack variables and are kept apart from computations. It is obvious that the amount of loss function would be zero within ε-insensitive tube. It is noted that further details about SVM and SVR can be found out in the literature [77].

Fig. 4
figure 4

Graphic description of the SVR model

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9 Performance evaluation of the models

Model evaluation of the developed MRA, CART, SVR and ANN models was performed with the 70 unused data sets in development process of the aforesaid models. The correlation between predicted and measured X50 for all the four models are shown in Figs. 5, 6, 7, 8, 9, 10, 11 and 12. Table 5 shows the calculated values of validation indexes. According to this table, performance of the ANN model with the highest accuracy and lowest is better as compared to the other employed models. On the contrary, efficiency of the conventional MRA is very low amongst the other utilized models. The MRA is bound to follow some valid statistical relations, whereas ANN is unbiased and can make its own relationship based on the sample data sets and due to that it has been found that ANN gives much better results compared to MRA in complex engineering problems. Rock fragmentation is also a very complex and complicated problem, influenced by several controllable and uncontrollable factors. Furthermore, results showed that facing problems with high complexity and nonlinearity such as fragmentation modeling, non-linear methods with high flexibility such as ANN have higher capabilities compared to classical linear methods such as MRA.

Fig. 5
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Scatter plot of the predicted vs. actual X50 for the MRA model (test)

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Fig. 6
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Comparison of predicted and measured outputs for the MRA model

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Fig. 7
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Scatter plot of the predicted vs. actual X50 for the CART model (test)

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Fig. 8
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Comparison of predicted and measured outputs for the CART model

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Fig. 9
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Scatter plot of the predicted vs. actual X50 for the ANN model (test)

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Fig. 10
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Comparison of predicted and measured outputs for the ANN model

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Fig. 11
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Scatter plot of the predicted vs. actual X50 for the SVR model (test)

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Fig. 12
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Comparison of predicted and measured outputs for the SVR model

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Table 5 Calculated validation indices for the ANN, MRA, SVR and CART models
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10 Sensitivity analysis

Normally, sensitivity analysis is performed to evaluate the effect of input variation on the relevant outputs. There are various methods of sensitivity analysis. One of the most frequently used methods is relevancy factor (RF) which is calculated by Eq. 6 [13, 78]:

$$\mathrm{RF}=\left| \frac{{\sum }_{i=1}^{n}({x}_{l,i}-{\overline{x} }_{l})({y}_{i}-\overline{y })}{\sqrt{{\sum }_{i=1}^{n}{({x}_{l,i}-{\overline{x} }_{l})}^{2}{\sum }_{i=1}^{n}{({y}_{i}-\overline{y })}^{2}}}\right|,$$
(6)

where \({x}_{l,i}\) and \({\overline{x} }_{l}\) are the ith value and the average value of the lth input variable, respectively, \({y}_{i}\) and \(\overline{y }\) are the ith value and the average value of the predicted output, respectively.

As it is seen in Fig. 13, uncontrollable parameters are more effective on fragmentation quality as compared to controllable parameters. From the uncontrollable parameters, rock quality designation, Schmidt hardness value, mean in-situ block size and point load index are more effective on rock fragmentation. Accordingly, from the controllable parameters, hole diameter, burden and spacing are the least effective on the fragmentation quality.

Fig. 13
figure 13

Sensitivity analysis of the input variables on fragmentation

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11 Conclusions

In this paper, artificial neural network, support vector regression, decision tree and regression analysis were implemented to investigate the effect of uncontrollable and controllable parameters on fragmentation quality in blasting operation of open pit mines. For this study, a database was prepared from four mines situated in different parts of Iran. In the first step superiority of the different models was inspected from which competence of the neural network modeling was approved. The values of MAE, RMSE, VAF and R2 for ANN model were 0.007, 0.009, 98.612% and 0.986, respectively. In this regard, MRA modelling with the obtained values of 0.021, 0.026, 87.896% and 0.886 in the validation phase for MAE, RMSE, VAF and R2, respectively, displayed the poorest performance. According to outcomes of the application of the network modeling, as a whole, it was concluded that in fragmentation quality uncontrollable parameters are more influential as compared to controllable parameters. Rock quality designation, Schmidt hardness value, mean in-situ block size and point load index from the former group play a vital role in the fragmentation quality and from the latter one, hole diameter, burden and spacing are the least effective parameters in this regard.