1 Introduction

Rock excavation is one of the most important steps in surface mines, tunneling and civil works. In this regard, blasting operation is an economical and a common and a widely utilized method. Nevertheless, a huge amount of blasting energy is wasted to produce several undesirable phenomena, such as ground vibration, flyrock, back break and air overpressure [1–11]. Among the mentioned environmental issues, ground vibration is considered as the most unwanted effect produced by blasting which can cause severe damage to the surrounding structures. Therefore, accurately approximating the blast-induced ground vibration is very significant. The several descriptors, including peak particle velocity (PPV), displacement, acceleration and frequency can be applied to evaluate the level of ground vibration. Among them, PPV is the most well-known descriptor for vibration measurement and is widely used in many studies [12–20]. The effective parameters on the PPV may be divided into two main groups including blast design parameters and related parameters to rock mass properties [21–28]. Blast design parameters or controllable parameters such as stemming, spacing, burden, sub-drilling, maximum charge weight used per delay (W), blast-hole diameter, blast-hole depth and specific charge can be changed by the explosive engineers. Whereas, the related parameters to rock mass properties or uncontrollable parameters, such as compressive and tensile strength, cannot be changed by the explosive engineers. Based on the literature [29–31], W and distance between blasting site and monitoring stations (D) are the most influential parameters on the blast-produced PPV. A number of researchers and institutes have been proposed different empirical models for the prediction of blast-induced PPV [32–41]. According to some studies (e.g., [28, 30]), the performance estimation of the empirical models are not good enough, while the suitable prediction of PPV is necessary. As an example, several empirical models were used for the estimation of PPV in the study conducted by Hasanipanah et al. [42]. Their results indicate that the accuracy of the empirical models is not good enough. Recently, application of such models like artificial intelligent (AI) in approximating problems in science and engineering areas, especially in the field of mining and civil [43–50], has been highlighted by various scholars. For predicting PPV, many researchers used AI techniques as their main methodology. Monjezi et al. [30] employed several empirical models and artificial neural network (ANN) for the prediction of blast-produced PPV at Shur river dam, in Iran. As a result, it was found that the ANN indicates a better prediction capability than empirical models. Support vector machine (SVM) and empirical models were proposed for the prediction of PPV in the study conducted by Khandelwal [51]. He used 150 datasets which were collected from Sinagreni Collieries Company Limited (SCCL), India. He concluded the SVM has superior fitting specification for PPV prediction compared to the used empirical models. A comprehensive study for the prediction of blast-produced PPV by fuzzy logic (FL) and empirical models were presented by Ghasemi et al. [28]. Their datasets were obtained from Sarcheshmeh copper mine, in Iran. They demonstrated that FL provided more accurate predictions in comparison with empirical models. In the other study of the AI methods, Armaghani et al. [52] investigated the results of blast-produced PPV at ISB granite quarry, in Malaysia. They developed ANN, empirical models and adaptive neuro-fuzzy inference system (ANFIS) for approximating PPV. Finally, it was revealed that ANFIS can estimate PPV better than ANN and empirical models. Recently, Amiri et al. [31] proposed a hybrid model of ANN and K-nearest neighbors (KNN) for forecasting of the PPV. In their studies, ANN and the United States Bureau of Mines (USBM) models were also applied. They showed that ANN-KNN model is more suitable for forecasting of the PPV compared to the ANN and USBM models. In this study, imperialist competitive algorithm (ICA) is proposed to get a precise and reliable equation for the predicting the PPV in three quarry sites, namely Ulu Tiram, Pengerang and Masai, Malaysia. For comparison purposes, the most well-known empirical models were also developed. The rest of the present research is organized as follows. Detail of the studied cases is given in Sect. 2. In Sect. 3, the ICA is shortly described. Then, developments of empirical techniques as well as ICA in estimating PPV will be discussed in Sects. 4 and 5, respectively. Evaluation of the developed models (Sect. 6) will be given before conclusions of the study (Sect. 7).

2 Field investigation

In this section, the studied cases and collection of datasets are shortly explained. Three granite quarry sites, i.e., Ulu Tiram, Pengerang and Masai near, in Malaysia, were considered in the present research work (see Fig. 1). The Ulu Tiram, Pengerang and Masai sites are located 18, 62 and 25 km of Johor City, at 103°49′20″E, 104°7′58″E and 103°52′27.79″E Longitudes as well as at 1°36′41″N, 1°22′58″N and 1.5°29′42.16″N Latitudes, in order. In these sites, drilling and blasting methods is used in order to fragment the rock mass. In the drilling process, Wagon Drill Machine is mainly used, and the blast-hole diameters are 75, 115 and 150 mm. Also, in the blasting process, ANFO was utilized as the main explosive material. Moreover, fine gravels are used to stem the blast-holes and the benches height in the The Ulu Tiram, Pengerang and Masai sites are 10–15, 10–30 and 5–30 m, respectively. Ground vibration is the most undesirable phenomena produced by blasting operations in the mentioned sites. Therefore, accurate prediction of blast-produced ground vibration is very significant. For this aim, 73 blasting events were monitored, and the values of W, D and PPV in kg, m and mm/s, respectively, were measured. In the other words, W and D were considered as independent parameters, and PPV was considered as dependent or target parameter. Table 1 and Figs. 2, 3 and 4 show the range and the frequency distributions of the W, D and PPV in this research, respectively.

Fig. 1
figure 1

Studied cases in this research

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Table 1 Used parameters and their range
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Fig. 2
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Frequency histogram of the measured D

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Fig. 3
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Frequency histogram of the measured W

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Fig. 4
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Frequency histogram of the measured PPV

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3 Imperialist competitive algorithm

The imperialism competitive algorithm (ICA) which was developed by Atashpaz-Gargari and Lucas [53] is considered as one of the global search population-based algorithms to optimize science and engineering problems. Similar to some other optimization algorithms like genetic algorithm (GA) and particle swarm optimization (PSO), ICA starts with a random initial population called countries. After creating N countries (also named as N country), some of them with the lowest costs or objective functions, for instance, root mean square error (RMSE), are chosen as the imperialists (N imp). Therefore, colonies or N col are defined as the remaining countries. All colonies are distributed among the empires in accordance with the initial power of them. So, it is clear that the more powerful imperialists (lowest RMSE) can attract more colonies. In ICA, there are three algorithm operators, i.e., assimilation, revolution and competition [53]. Colonies attract to the imperialists through assimilation operator. With the help of revolution operators, some sudden changes in the locations of the countries will happen. Through these two operators, for a colony, there is a possibility to reach a state better than its imperialist state.

In the last ICA operator or competition, imperialists try to adopt more number of colonies. During competition, there is a competition among all empires to obtain the colonies of other empires. All of the empires have at least a chance to adopt one colony of the weakest empire based on their power. After a few decades, the weak empires will be gradually collapsed, and on the other hand, more powerful empires increase their power. The mentioned procedures will be continued until all of the empires (except the most powerful ones) collapse or a termination criterion is met. Here, termination criteria can be defined as root men square error (RMSE) or maximum number of decade. It should be noted that in ICA, the maximum number of decades (N decade) works like the number of iteration (N ite) in PSO algorithm. Figure 5 illustrates a flowchart of ICA algorithm. Nevertheless, the focus of this paper is not to cover ICA mathematical formulation. More details in this regard can be found in the previous studies (e.g., [7, 54, 55]).

Fig. 5
figure 5

ICA algorithm

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4 Prediction of PPV by empirical models

In the present paper, the most conventional empirical models are utilized for forecasting of the blast-produced PPV, as listed in Table 2. In this Table, PPV, W and D are in terms of mm/s, kg and m, respectively. Also, K, n, a, b are site constants and can be determined from the multiple regression analysis.

Table 2 Used empirical models in this research
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The total datasets were considered, and then, the site constants were calculated using SPSS version 16 (SPSS 2007). Table 3 summarizes the values of site constants for the used empirical models in this paper. The performance of the empirical models is evaluated in Sect. 6.

Table 3 Calculated values of site constants
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5 Prediction of PPV by ICA

The present section describes model development procedure of ICA in estimating PPV, considering two model inputs, i.e., D and W. In this regard, two ICA models, i.e., power and quadratic are proposed for forecasting of the blast-produced PPV. As the first stage, in order to develop a model with lower errors, it is well established to normalize the prepared database before modeling. Therefore, all 73 datasets should be normalized considering the following equation which its range is between 0 and 1:

$$X_{n} = \frac{{X - X_{ \hbox{min} } }}{{X_{ \hbox{max} } - X_{ \hbox{min} } }}$$
(1)

where \(X_{n}\) and \(X\) are the normalized and measured values, respectively. Also, the minimum and maximum of measured dataset are \(X_{ \hbox{min} }\) and \(X_{ \hbox{max} }\), respectively. As mentioned before, two functions of power and quadratic were applied for forecasting of the blast-produced PPV and then using ICA, their coefficients were found. The general forms of power and quadratic functions are presented in Eqs. (2) and (3), respectively:

$$y = ax^{b}$$
(2)
$$y = ax^{2} + bx + c$$
(3)

where a, b and c are constants. Note that, b in power function is any nonzero real number. By this introduction, power function type of the problem of this study can be shown as follows:

$${\text{PPV}}_{P} = x_{1} + x_{2} D^{{x_{3} }} + x_{4} W^{{x_{5} }}$$
(4)

In addition, quadratic function of the PPV problem can be formed based on the following equation:

$${\text{PPV}}_{P} = x_{1} + x_{2} D^{2} + x_{4} W^{2} + x_{5} DW$$
(5)

In the above equations, W and D are set as independent parameter of the system, while PPV is considered as dependent parameter.

Here, the role of ICA in determining the optimal values of coefficients will be highlighted. In fact, based on the lowest values of fitness function, ICA attempts to find the fittest model or models to the available data. Of course, this is possible through minimizing errors between the measured values of PPV and the predicted ones. Hence, the used fitness function for solving problem of this study can be defined as follows:

$$f\left( x \right) = \mathop \sum \limits_{i = 1}^{m} ({\text{PPV}}_{\text{measured}} - {\text{PPV}}_{\text{predicted}} )^{2}$$
(6)

where \(f\left( x \right)\) is defined as the fitness function, \({\text{PPV}}_{\text{measured}}\) and \({\text{PPV}}_{\text{predicted}}\) are measured PPVs and predicted ones by the system, respectively. It should be mentioned that the minimum value of the above equation in the solution of this study. An ICA code was written by the authors in MATLAB environmental software, and all modeling stages were conducted using MATLAB version 7.14.0.739 [56]. In order to achieve the aim of this study, the optimum ICA parameters should be designed and subsequently utilized. To do this, some of the previous investigations have been reviewed to identify the range of ICA parameters.

According to the original study of ICA [53], parameters like β, θ, ζ, number of country (N country), number of decade (N decade) and number of imperialism (N imp) are the most effective ones on ICA performance. Review of the literature (e.g., [53, 57, 58]) showed that the best values of β, θ and ζ are considered as 2, π/4 and 0.02, respectively. Therefore, the mentioned values were used in constructing all ICA models. In case of N country, many values such as 40, 56, 135 and 250 were recommended in the studies carried out by Ahmadi et al. [57], Marto et al. [59], Hajihassani et al. [55] and Armaghani et al. [60], respectively, for solving engineering problems. N decade of 800, 500 and 400 were suggested by Armaghani et al. [7, 60, 61], respectively. Moreover, Hajihassani et al. [55], Marto et al. [59] and Armaghani et al. [60] recommended values of 11, 9 and 25, respectively, as N imp for solving geotechnical problems. It seems that there is a need to design these parameters through trial-and-error procedure. Hence, many ICA models were built using different values of N country, N decade and N imp. The optimal values of N country, N decade and N imp for both power and quadratic ICA models are shown in Table 4. As presented in this Table, there is no much difference between the obtained ICA results of power and quadratic functions. Considering the obtained ICA parameters in this study, the proposed power and quadratic equations for approximating PPV values resulting from blasting operations in Ulu Tiram, Pengerang and Masai sites are shown in Eqs. (7) and (8), respectively.

Table 4 Obtained values of the ICA parameters
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$${\text{PPV}}_{P} = 0.0049D^{ - 6.35} + 0.00075W^{1.56} - 1.74$$
(7)
$${\text{PPV}}_{P} = 2.68 - \left( {8.2 \times 10^{ - 6} \times D^{2} } \right) + \left( {4.14 \times 10^{ - 5} \times W^{2} } \right) + \left( {6.3 \times 10^{ - 5} DW} \right)$$
(8)

It is worth noting that the obtained coefficients in Eqs. (7) and (8) were computed using the actual datasets, not the normalized ones. More details regarding the performance of the proposed ICA power and ICA quadratic equations for the prediction of PPV will be given in Sect. 6.

6 Analysis and results

In the present study, two ICA models namely power and quadratic as well as several empirical models were presented for predicting of the blast-produced PPV in three site quarries in Malaysia. In this regard, the total datasets (73 data) were considered, and the W and D were set as independent parameters, while PPV was set as dependent parameter. The performance of the developed predicting models is carried out using two statistical functions, i.e., root mean square error (RMSE) and coefficient of multiple determination (R 2).

$${\text{Root}}\,{\text{Mean}}\,{\text{Square}}\,{\text{Error}} = \sqrt {\frac{1}{n} \times \mathop \sum \limits_{i = 1}^{n} \left[ {\left( {x_{i} - x_{p} } \right)^{2} } \right]}$$
(9)
$$R^{2} = \frac{{\left[ {\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{i} - x_{\text{mean}} } \right)^{2} } \right] - \left[ {\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{i} - x_{p} } \right)^{2} } \right]}}{{\left[ {\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{i} - x_{\text{mean}} } \right)^{2} } \right]}}$$
(10)

where n denotes number of datasets and represents the predicted and measured PPV values. For an excellent model, the values of RMSE and R 2 should be close to 0 and 1, respectively. Table 5 summarizes the obtained statistical function values for the predictive models. Furthermore, the relationships between the predicted and measured PPVs are plotted in Figs. 6 and 7. The R 2 were obtained as 0.875, 0.879, 0.894, 0.897, 0.93 and 0.94 for USBM, Langefors-Kihlstrom, Roy, Rai-Singh, ICA power form and ICA quadratic form, respectively. The RMSE was also obtained as 0.56, 0.55, 0.51, 0.5, 0.41 and 0.37 for USBM, Langefors-Kihlstrom, Roy, Rai-Singh, ICA power form and ICA quadratic form, respectively. As a result, the ICA quadratic form possessed superior predictive ability than the ICA power form and empirical models.

Table 5 Statistical functions results for the predictive models
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Fig. 6
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Measured versus predicted PPVs by a USBM, b Langefors-Kihlstrom, c Roy, and d Rai-Singh equations

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Fig. 7
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Measured versus predicted PPVs by a ICA power and b ICA quadratic equations

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7 Conclusions

Ground vibration is the most important environmental effect caused by blasting in Ulu Tiram, Pengerang and Masai quarry sites in Malaysia. Hence, accurate prediction of blast-produced ground vibration is critical in these sites. This research explores the possibility of using the ICA to create a model for predicting the PPV. In this regard, two equations namely power and quadratic optimized by ICA were proposed. For comparison aims, several empirical models, i.e., USBM, Langefors-Kihlstrom, Roy and Rai-Singh were also utilized. In order to achieve the aim of this study, 73 blasting events were considered and the most effective parameters on the PPV were measured. For modeling, W and D were used as the independent variables, while PPV was used as the dependent variables. The RMSE and R 2 were used for evaluating the accuracy and performance of the models. Finally, the results indicated the superiority of quadratic form over the other predictive models in the current study. The R 2 were 0.875, 0.879, 0.894, 0.897, 0.933 and 0.945 for the USBM, Langefors-Kihlstrom, Roy, Rai-Singh, power and quadratic forms, respectively. As a conclusion, ICA can be introduced as an acceptable and reliable algorithm in forecasting the PPV.