Bull Eng Geol Environ DOI 10.1007/s10064-016-0872-8

ORIGINAL PAPER

Development of a new model for predicting flyrock distance in quarry blasting: a genetic programming technique Roohollah Shirani Faradonbeh1 • Danial Jahed Armaghani2 • Masoud Monjezi3

Received: 20 September 2015 / Accepted: 19 March 2016  Springer-Verlag Berlin Heidelberg 2016

Abstract This research was aimed at developing a new model to predict flyrock distance based on a genetic programming (GP) technique. For this purpose, six granite quarry mines in the Johor area of Malaysia were investigated, for which various controllable blasting parameters were recorded. A total of 262 datasets consisting of six variables (i.e., powder factor, stemming length, burden-tospacing ratio, blast-hole diameter, maximum charge per delay, and blast-hole depth) were collected applied to developing the flyrock predictive model. To identify the optimum model, several GP models were developed to predict flyrock. In the same way, using non-linear multiple regression (NLMR) analysis, various models were established to predict flyrock. Finally, to compare the performance of the developed models, regression coefficient (R2), root mean square error (RMSE), variance account for (VAF), and simple ranking methods were computed. According to the results obtained from the test dataset, the best flyrock predictive model was found to be the GP based model, with R2 = 0.908, RMSE = 17.638 and

& Masoud Monjezi [email protected] Roohollah Shirani Faradonbeh [email protected] Danial Jahed Armaghani [email protected] 1

Young Researchers and Elite Club, South Tehran Branch, Islamic Azad University, Tehran, Iran

2

Department of Geotechnics and Transportation, Faculty of Civil Engineering, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor, Malaysia

3

Department of Mining, Tarbiat Modares University, Tehran 14115-143, Iran

VAF = 89.917, while the corresponding values for R2, RMSE and VAF for the NLMR model were 0.816, 26.194, and 81.041, respectively. Keywords Blasting operation  Flyrock distance  Genetic programming  Non-linear multiple regression

Introduction During mining and civil engineering works such as the construction of roads, hydraulic channels, and tunnels, there is a need to remove rock mass. Blasting, i.e., the controlled use of an explosive material, is considered a common technique for rock mass removal (Bhandari 1997). In many blasting works, a large amount of explosive energy is wasted, and there are several environmental issues, e.g., air-overpressure, ground vibration, flyrock and back-break, which can affect surrounding areas (Khandelwal and Singh 2007, 2009; Monjezi et al. 2010; Ghasemi et al. 2012; Hajihassani et al. 2014; Ebrahimi et al. 2015; Hasanipanah et al. 2015; Tonnizam Mohamad et al. 2016). Among the environmental impacts of blasting, flyrock is considered the main cause of human injury, fatalities and structural damage (IME 1997; Khandelwal and Monjezi 2013; Raina et al. 2014). In the flyrock mechanism, there is a strong relationship between the mechanical strength of the rock mass, the explosive energy distribution, and charge confinement (Bajpayee et al. 2004). According to Bajpayee et al. (2004), any mismatch between these factors can create a flyrock phenomenon. When this happens, a large amount of the explosive energy is utilized to move rock rather than produce rock fragmentation (Roy 2005). As shown in Fig. 1, there are three mechanisms for generation of flyrock, namely, cratering, rifling and face

123

R. S. Faradonbeh et al. Fig. 1 Three mechanisms for flyrock generation

bursting, which are described briefly in the following (Little and Blair 2010; Ghasemi et al. 2012). When the ratio of stemming length to diameter of blast-hole is too small, cratering will occur. If the stemming material utilized is inefficient or absent, rifling will occur. Face bursting happens when an explosive material intersects major geological structures, or is located in close proximity to a weakness plane. In this situation, flyrock can be generated due to highpressure gas jets of the explosives along the weakness planes. According to Little and Blair (2010), in the face bursting mechanism, flyrock can be derived from front row blast-holes where insufficient burden exists due to poor design and drilling deviation towards the free face. Based on previous investigations, flyrock phenomena are known to be influenced by both controllable and uncontrollable factors. The main effective controllable factors affecting flyrock include insufficient burden and spacing, inaccurate drilling, inadequate stemming, improper delay timing, unwarranted powder factor and too much explosive energy (Bhandari 1997; Ghasemi et al. 2012). In addition, the most influential uncontrollable parameters affecting flyrock are poor and uncertain geological and geotechnical conditions related to the rock mass (Fletcher and D’Andrea 1987; Adhikari 1999; Ghasemi et al. 2012). Several equations/models were developed in order to empirically predict flyrock distance arising from blasting (Lundborg et al. 1975; Roth 1979; Chiapetta et al. 1983; Roy 1993). These models considered only one or two of the parameters influencing flyrock. As a result, the performance capacities of these models are poor. This is may be because of the complicated nature of flyrock (Rezaei et al. 2011). However, prediction of flyrock with high accuracy is important in assuring safety in a blast area (Monjezi et al. 2010). Therefore, there is a need for more accurate models/ techniques for predicting flyrock distance. Previously, many attempts have been made to predict flyrock distance using numerous techniques/models, such

123

as Monte Carlo simulation (Little and Blair 2010), artificial neural networks (ANN) (Monjezi et al. 2010, 2011), multiple regression analyses (Trivedi et al. 2014), fuzzy inference system (FIS) (Rezaei et al. 2011; Ghasemi et al. 2014), rock engineering systems (RES) (Faramarzi et al. 2014), adaptive neuro-fuzzy inference system (ANFIS) (Jahed Armaghani et al. 2015a, b; Trivedi et al. 2015), support vector machines (SVM) (Khandelwal and Monjezi 2013), and hybrid ANN models (Monjezi et al. 2012; Marto et al. 2014). Despite all these available techniques, it seems there is still an undeniable need to develop new predictive models for flyrock prediction. A possible means of controlling and preventing flyrock accidents is predicting the range of flyrock distance. This enables contractors and mining engineers to determine blast safety area and to select the best location for equipment. The importance of this issue is highlighted when residential areas are close to the location of blasting operations. To overcome such problems, nature-inspired techniques like genetic programming (GP) can be performed. Although GP is considered as a black box technique (Asadi et al. 2011), it can produce a non-linear equation between predictors or inputs and output parameters. This is an advantage of GP compared to ANN-based models. Although intelligent systems have been developed and are widely utilized and to predict flyrock distance in previous investigations, there has been no conducted research introducing GP techniques in this field. In the present study, a series of analyses was performed in order to demonstrate the capability of GP in flyrock prediction using blasting parameters operated in six quarry sites in Malaysia. For the sake of comparison, a non-linear multiple regression (NLMR) model was also applied to predict flyrock. The performance prediction of the proposed models are compared and discussed.

Development of a new model for predicting flyrock distance in quarry blasting: a genetic…

Flyrock prediction methods Over the past few decades, several studies have been carried out to develop empirical models/equations for flyrock prediction. Lundborg et al. (1975) developed an empirical equation based on hole and rock diameters as follows: Flyrock ¼ 260  D2=3

ð1Þ

Tb ¼ 0:1  D2=3

ð2Þ

where D is the hole diameter in inches, and Tb is the size of the fragmented rock in meters. In order to determine the horizontal (FSH) and vertical (FSV) flyrock safety factors, Raina et al. (2006) carried out research based on collected parameters of rock mass and blast design. In another study, McKenzie (2009) used hole diameter, density, confinement state, and explosive density to propose two empirical equations for prediction of flyrock and particle (rock) size. Trivedi et al. (2014) developed a new empirical model to estimate flyrock using data collected from 95 blasting operations in four opencast limestone mines. The proposed equation is described as follows: Flyrock ¼

105:1 q0:51 q0:14 I 0:93 0:64 0:75 B St rc RQD0:93

ð3Þ

where q is specific charge (kg/t), qI is linear charge concentration (kg/m), rc is unconfined compressive strength (MPa), B is burden (m), St is stemming length (m), and RQD is rock quality designation (%). Utilizing dimensional analysis, Ghasemi et al. (2012) suggested an empirical equation for flyrock prediction. The equation is formulated as follows: h i Flyrock ¼ 6946:547 B0:796 S0:783 St1:994 H 1:649 D1:766 ðP=QÞ1:465 ð4Þ where S is spacing (m), B is burden (m), St is stemming (m), H is hole length (m), P is powder factor (kg/m3) D is hole diameter (m), and Q is mean charge per blast-hole (kg). In Eqs. (1)– (4), flyrock can be calculated in meters. Two power empirical equations were introduced in the study carried out by Marto et al. (2014) for flyrock estimation using results of powder factor and charge per delay obtained from 113 blasting operations in Johor, Malaysia. Moreover, Jahed Armaghani et al. (2015a) suggested an empirical graph for predicting flyrock. This graph was proposed for various amounts of maximum charge per delay in the range of 75–550 kg, and also for different powder factors in the range 0.5–1.1 kg/m3. Aside from empirical models, several models have also been proposed to estimate flyrock using soft computing techniques. Ghasemi et al. (2014) examined two intelligent

systems, including ANN and fuzzy inference system (FIS) to predict flyrock distance. They concluded that, although both intelligent systems are applicable for flyrock prediction, the FIS technique provides higher performance capacity in comparison with the ANN model. In another study of flyrock prediction, Monjezi et al. (2012) established a new model based on a combination of genetic algorithm and ANN. Support vector machine (SVM) and statistical models were utilized by Khandelwal and Monjezi (2013) to estimate flyrock distance, and these authors presented SVM as a superior model for close flyrock prediction. Marto et al. (2014) developed a hybrid imperialist competitive algorithm (ICA)-ANN predictive model for predicting flyrock. Another hybrid system, with name genetic algorithm (GA)-ANN, was utilized as a new method in the field of flyrock prediction by Monjezi et al. (2012). Finally, the latter authors noted that a hybrid model can be introduced as a reliable model for predicting flyrock. The RES technique was employed and developed in a study carried out by Faramarzi et al. (2014) to analyze the assessment of flyrock risk and also predict flyrock distance. An ANFIS predictive model was constructed and established to estimate flyrock induced by quarry blasting in the study conducted by Jahed Armaghani et al. (2015a). Several recently developed models for estimating flyrock distance and their performance capacity based on regression coefficient (R2) are shown in Table 1. Genetic programming Genetic programming (GP), which was developed by Koza (1992), is based on natural selection and the biological model of evolution. Genetic algorithm (GA) techniques can be applied to solve many non-linear problems. GP is based on Darwin’s theory of ‘survival of the fittest’ and is categorized as circulating algorithms (Faradonbeh et al. 2015). There is a great similarity between GP and GAs; however, it is noticeable that the former acts on tree structures whereas the latter act on binary strings. The flexible structure of trees allows them to show mathematical relationships or logical expressions appropriately. Leaves of trees that are selected from a pre-defined terminal set specify typically constants or variables. However, other nodes, which are selected from a pre-defined function set, specify generally functions or operators. The tree displayed in Fig. 2 demonstrates the mathematical expression (a - b) ? (c 9 d). a, b, c, d and ?, -, 9 are employed as variables and operators, respectively. The tree does not use the division operator or variable e. The sets that have been pre-defined in Fig. 2 are as follow: Function set ¼ fþ; ; ; =g Terminal set ¼ fa; b; c; d; eg

123

R. S. Faradonbeh et al. Table 1 Recently-developed models for estimating flyrock distance References

Method

Input

Dataset no.

R2

Monjezi et al. (2010)

ANN

SD, RD, HD, ST, BS, PF, N, C

250

R2 = 0.98

Rezaei et al. (2011)

FIS

HD, S, B, SD, ST, PF, RD, C

490

R2 = 0.98

Monjezi et al. (2011)

ANN

ST, HD, BS, PF, C, D, SD, B

192

R2 = 0.97

Monjezi et al. (2012)

GA-ANN

PF, HD, B, S, ST, SD, D, C, RMR

195

R2 = 0.89

Amini et al. (2012)

ANN, SVM

PF, HL, SD, S, D, B, ST

245

R2ANN = 0.92 R2SVM = 0.97

44

R2 = 0.94

HD, S, B, D, C

310

R2 = 0.98

SVM

HL, S, B, ST, PF, SD

187

R2 = 0.95

Tonnizam Mohamad et al. (2013)

ANN

RD, BS, HD, ST, PF, C, N, D, SD

39

R2 = 0.97

Ghasemi et al. (2014)

FIS, ANN

PF, S, HL, ST, B, C

230

R2ANN = 0.94 R2FIS = 0.96

Marto et al. (2014)

ICA-ANN

HD, RD, PF, BS, ST, C, Rn

113

R2 = 0.98

Trivedi et al. (2014)

ANN

B, ST, qI, q, rc , RQD

95

R2 = 0.98

Jahed Armaghani et al. (2015a)

ANFIS, ANN

BS, C, PF, ST

Jahed Armaghani et al. (2015c)

PSO-ANN

PF, B, RD, HD, S, N, ST, C, D, SD

Monjezi et al. (2013)

ANN

Khandelwal and Monjezi (2013)

166

R2ANN = 0.83 R2ANFIS = 0.98

Jahed Armaghani et al. (2015b)

ANFIS, ANN

C, PF

232

R2ANN = 0.92 R2ANFIS = 0.98

ANN Artificial neural network, FIS fuzzy inference system, GA-ANN genetic algorithm-ANN, SVM support vector machines, PSO-ANN particle swarm optimization-ANN, ICA-ANN imperialist competitive algorithm-ANN, ANFIS adaptive neuro-fuzzy inference system, HL hole length, S spacing, B burden, ST stemming, PF powder factor, SD specific drilling, SVM support vector machine, C charge per delay, D hole diameter, HD hole depth, RD rock density, BS burden to spacing, N number of row, B blastability index, GA genetic algorithm, RMR rock mass rating, PSO particle swarm optimization, SD subdrilling, ICA imperialist competitive algorithm, qI linear charge concentration, q specific charge, rc unconfined compressive strength

GP are the model inputs of the system. As a result, three control models, i.e., full, grow, and ramped half-and-half, are provided by the system (Abhishek et al. 2014). Subsequent to generating the population, using the fitness function, each individual is evaluated. The fitness function shows the capability of a program in solving the problem. Genetic operation for forming the new generation

Fig. 2 Tree structure of the expression (a - b) ? (c 9 d)

The GP procedure can be explained as follows (Abhishek et al. 2014). Generation of initial population Normally, GP gets started with an initial random population of individuals (models or computer programs), which consist of terminals and functions that can be appropriately applied to the problem. Using non-linear functions (tan, cos, sin), standard arithmetic operators, or Boolean operators, the functions can be presented. In fact, terminals in

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The population that is generated in the first stage is improved gradually by means of three genetic operators, namely crossover, mutation, and reproduction (Abhishek et al. 2014; Faradonbeh et al. 2015). During the reproduction operation, through the duplication of a fraction of the parents in the current generation, children are produced to be used in the next generation. Based on the fitness value, two individuals are chosen as parents, and they are reproduced (copied) for the next generation. During the crossover operation, new offspring program(s) are created through randomly recombining the selected parts from two already-chosen programs (parents) (Baykasoglu et al. 2008). In general, there are two crossovers, i.e., sub-tree crossovers and node crossovers. The former have been shown to be more practical compared to the latter

Development of a new model for predicting flyrock distance in quarry blasting: a genetic…

(Abhishek et al. 2014); thus, sub-tree crossovers were employed for the purposes of the present study. Termination of algorithm Fitness is measured iteratively over numerous generations until a termination criterion is met. The number of runs that are needed for a satisfactory solution depends highly upon the complexity of the problem (Abhishek et al. 2014). After reaching the termination criterion, the best program in the population that is generated during the run (the best-so-far individual) is taken into account as the result of the run. A simple flowchart of the GP implementation is shown in Fig. 3.

Studied quarry sites and data collection To develop predictive models for flyrock prediction, six granite quarry sites, i.e., Ulu Tiram, Taman Bestari, Trans Crete, Putri Wangsa, Ulu Choh and Masai, were investigated in Johor state, Malaysia. Blasting operations are conducted in these quarries in order to produce aggregate ranging from 8000 to 240,000 tons per month. In these

sites, 6–15 blasts are operated monthly depending on the weather conditions. Mass weathering zones ranging from slightly weathered to completely weathered were observed at these sites. Note that no completely weathered zone was observed in Taman Bestari and Ulu Choh quarry sites. The rock type in the studied sites is granite, and the sites are near to each other. Generally, in terms of geological conditions and rock mass properties, there is a high level of similarity between the studied sites. Hence, we decided to collect and use blasting parameters from all six quarry sites in order to develop flyrock predictive models. In overall, rock quality designation (RQD) values in these sites were measured in the range of 25–55 %. Moreover, a range of 17–42 was observed for Schmidt hammer rebound (Rn) values. It is worth mentioning that all field observations and tests are based on methods suggested by ISRM (2007). Some descriptions of the studied sites, together with their rock quality designation RQD and Rn ranges, are presented in Table 2. Since the blasting works in the aforementioned sites are carried out close to the location of existing facilities, equipment and residential areas, flyrock is a serious problem in these quarries. A total of 262 blasting works in six granite quarries were investigated in this study. During data collection, all controllable parameters of blasting, i.e., burden, spacing, stemming length, powder factor, blasthole diameter, blast-hole depth, and maximum charge per delay were obtained from the study sites. Furthermore, maximum flyrock distance as a favorable parameter was measured in each blasting operation. Burden, stemming length, spacing, blast-hole depth and blast-hole diameter were recorded by a measuring tape. In these operations, blast-hole diameters of 75, 89, 115 and 150 mm were utilized. A range of 0.44–1.14 kg/m3 was used for powder factor values. Additionally, maximum and maximum values of stemming length were obtained as 1.4 and 4.5 m, respectively. Moreover, ammonium nitrate and fuel oil (ANFO) was set as explosive, whereas dynamite was utilized as initiation. Furthermore, fine gravels were used as stemming material for the blast-holes.

Table 2 Some information regarding the studied areas together with their rock quality designation (RQD) and Schmidt hammer rebound (Rn) ranges

Fig. 3 A simple flowchart of genetic programming (GP) implementation (Liong et al. 2002)

Quarry name

Latitude

Ulu Tiram

1360 4100 N 0

Longitude

00

Rn

RQD (%)

103490 2000 E

20–38

40–51

0

00

Taman Bestari Trans Crete

160 41 N 1310 2100 N

10378 32 E 103520 6000 E

22–37 17–30

35–52 27–41

Putri Wangsa

1350 3200 N

103480 400 E

20–42

31–50

0

00

Ulu Choh

131 48 N

103320 4100 E

25–39

33–55

Masai

1290 4200 N

103520 2800 E

18–32

25–39

123

R. S. Faradonbeh et al.

To measure the flyrock, a work bench was colored and two video cameras were placed to monitor the flyrock projections. After each blasting, relevant videos were reviewed to locate the maximum rock projections. The maximum horizontal distance between the free face and landed fragments was considered as flyrock distance. A view of Ulu Tiram quarry is shown in Fig. 4. According to Table 1, and considering some other studies (e.g., Bhandri 1997; Little and Blair 2010; Ghasemi et al. 2012), burden, spacing, maximum charge per delay, powder factor, blast-hole diameter, stemming length and blast-hole depth, which are considered as controllable parameters, are the most important factors impacting flyrock. Hence, in this research, stemming length, burden to spacing ratio, powder factor, the maximum charge per delay, blast-hole diameter and blast-hole depth were selected and set as input parameters to predict flyrock. A summary of input and output data utilized in the modelling analysis of this study is presented in Table 3. Moreover, Figs. 5, 6, 7, 8, 9, 10 and 11 illustrate the frequency distributions of the measured blast-hole diameter, blast-hole depth, burden to spacing ratio, stemming length, maximum charge per delay, powder factor and flyrock distance, respectively. Prediction of flyrock distance In order to develop new equations for predicting flyrock, two predictive models, i.e., GP and NLMR were utilized. As discussed above, D, HD, BS, ST, MC and PF were set as model inputs to estimate flyrock. The following sections describe modeling procedures of the GP and NLMR approaches.

divided randomly into training and testing datasets. In this regard, Swingler (1996) and Looney (1996) suggested 20 % and 25 % of whole dataset for testing purpose, respectively. Furthermore, Nelson and Illingworth (1990) introduced a value between 20 % and 30 % of whole data in order to evaluate the performance prediction of model development. Using these suggestions, 210 datasets (80 % of the total of 262 datasets) were utilized randomly for flyrock model development, whereas the remaining 52 datasets were chosen to test the model performance capacity. Then, five different datasets were selected randomly to train and test for developing new models to estimate flyrock as recommended by Zorlu et al. (2008), Yagiz et al. (2009) and Jahed Armaghani et al. (2015a). The main objective of using GP in this study is to develop an equation for estimating flyrock distance using tree structure. In order to construct GP models, the software of Gene Xpro Tools 4.0 was utilized. To propose GP models, each randomly selected dataset was presented separately to the software. In the modelling procedure, some mathematical ffi 3 , ^2, ^3, Exp, Ln) non-linear operators (?, -, *, /, H, p functions ðsin; cos; tan; A tanÞ, and Boolean operators were applied to obtain the minimum error and maximum regression coefficient of the system. In GP, the choice of operators was based on the type of problem that needed to be solved. In fact, these operators and functions are the choices in the hands of the designers. Using a trial-and-error procedure, as a criteria for fitness function, mean absolute error (MAE) was selected for sets nos. 1, 2, 4 and 5, while relative absolute error (RAE) was applied to set no. 3. The equations for MAE and RAE are expressed as follows: n   1X Xipred  Ximes  n I¼1  Pn   I¼1 Xipred  Ximes RAE ¼ Pn  jXimes  X j

MAE ¼

Flyrock model development using GP This section presents the GP modelling procedure for developing a flyrock predictive model. All 262 datasets were

ð5Þ

ð6Þ

I¼1

where Ximes and Xipred are actual and predicted values by GP respectively. In addition, X can be calculated using the following equation: 1 X ¼ n

Fig. 4 A view of Ulu Tiram quarry

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n X

Ximes

ð7Þ

I¼1

A model is excellent if Ximes ¼ Xipred and MAE ¼ RAE ¼ 0. Range of MAE and RAE is between zero and 1 where a value of zero is the best for MAE as well as RAE. It is worth mentioning that MAE and RAE cannot be applied directly as fitness functions. In order to select an appropriate fitness, the value of the fitness function should increase with efficiency and can be presented as follows:

Development of a new model for predicting flyrock distance in quarry blasting: a genetic… Table 3 Summary of input and output data utilized in the modelling analysis of this study. SD Standard deviation

Parameter

Unit

Symbol

Category

Min

Max

Mean

Blast-hole diameter

mm

D

Input

75

150

115.81

28.15

Blast-hole depth

m

HD

Input

10

29

19.76

5.64

Burden to spacing ratio

–

BS

Input

0.40

0.95

0.70

0.14

Stemming length

m

ST

Input

1.40

4.50

2.98

0.77

Maximum charge per delay

kg

MC

Input

48

594

257.63

152.60

3

Powder factor

kg/m

PF

Input

0.45

1.14

0.78

0.16

Flyrock distance

m

Flyrock

Output

67

354

204.58

64.16

Fig. 5 Frequency-histogram of blast-hole diameter

Fig. 7 Frequency-histogram of burden to spacing ratio

Fig. 6 Frequency-histogram of blast-hole depth

Fig. 8 Frequency-histogram of stemming length

fI ¼ 1000 

1 1 þ MAE

SD

ð8Þ

where the MAE range is between 0 and 1000. In the equation presented, a value of 1000 indicates the best performance in terms of unique distribution. Since GP is influenced by several initial factors, a series of analyses were conducted to obtain them using a trial-and-error procedure. Table 4 presents the obtained GP parameters for five sets in predicting flyrock. Moreover, considering the five different sets, the developed GP equations and obtained regression coefficient for both training and testing are given in Table 5. The R2 values were found to range

from 0.798 to 0.837 for training and from 0.801 to 0.908 for testing of the GP equations. Figure 12 displays tree structure of the best GP equation for prediction of flyrock. More discussion regarding selection of the best GP model is given later. Flyrock model development using NLMR Regression analysis is a statistical tool that is used to recognize relationships between variables. Typically, researchers attempt to ascertain the effect of one variable on another. To explore such issues, researchers assemble the data on the underlying variables and make use of

123

R. S. Faradonbeh et al.

Fig. 9 Frequency-histogram of maximum charge per delay

Since GP is considered as a non-linear model, in order to have a fair comparison between obtained results, NLMR technique was selected and performed. Therefore, based on simple relationships between inputs and output, non-linear equations were developed to estimate flyrock. NLMR equations were built using the statistical software package SPSS version 16 (SPSS 2007). The developed NLMR equations to predict flyrock are presented in Table 6. In these equations, results of blast-hole diameter, blast-hole depth, burden to spacing ratio, stemming length, maximum charge per delay, and powder factor were utilized as model inputs. The relevant results from five constructed NLMR equations are discussed and evaluated according to their accuracy and applicability in the next section. Results of models performances

Fig. 10 Frequency-histogram of powder factor

Fig. 11 Frequency-histogram of flyrock

regression to estimate the quantitative effects of variables on the variable that they influence. In regression techniques, relationship between an independent (predictor) variable and a dependent (output) variable is determined systematically in the form of a function (regression function) (Bahrami et al. 2011). There are two types of multiple regression; linear (LMR) and non-linear (NLMR). Multiple regression techniques have been used extensively to develop multiple equations for flyrock prediction (e.g., Rezaei et al. 2011; Marto et al. 2014).

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This section aims to check the capability of the proposed models to predict flyrock distance. Both GP and NLMR equations were developed in order to solve the flyrock problem. In these equations, results of D, HD, BS, ST, MC and PF were used to predict flyrock. Some performance indices, including R2, root mean square error (RMSE) and variance account for (VAF), were computed to check the performance prediction of developed equations: PN ð y  y0 Þ 2 2 R ¼ 1  PI¼1 ð9Þ N ~Þ2 I¼1 ðy  y   varðy  y0 Þ VAF ¼ 1   100 ð10Þ var ð yÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X ð11Þ ðy  y0 Þ2 RMSE ¼ t N I¼1 where y, y0 and y~ are the measured, predicted and mean of the y values, respectively, and N is the total number of data. In theory, a predictive model will be considered as perfect, if (R2 = 1), (VAF = 100) and (RMSE = zero). Obtained performance indices for the developed GP and NLMR equation in predicting flyrock are presented in Table 7. In Table 7, looking at the developed models, the results from the performance indices are very similar; hence, selecting the best models among them is not easy. To overcome the difficulty of selecting the best GP and NLMR equations, Zorlu et al. (2008) suggested a simple ranking procedure. The developed models were also analyzed according to the total ranking index: each performance index was ordered in its class, and the best performance index was assigned the highest rating. For example, R2 values of the test datasets for the NLMR technique were obtained as 0.751, 0.737, 0.816, 0.679 and 0.821, respectively. So, the rating values of 3, 2, 4, 1 and 5

Development of a new model for predicting flyrock distance in quarry blasting: a genetic… Table 4 Genetic programming (GP) parameters to estimate building blocks (BB)

GP parameter

Value Set number 1

2

3

4

5

3

Terminal set

D (mm), HD (m), BS, ST (m), MC (kg), PF (kg/m )

Fitness function

MAE

MAE

RAE

MAE

MAE

Number of chromosomes

30

30

30

50

30

Number of genes

3

3

3

5

4

Head size

8

10

9

8

7

Linking function

Addition

Multiplication

Multiplication

Addition

Addition

Mutation rate Inversion rate

0.044 0.1

0.03 0.1

0.044 0.1

0.05 0.1

0.05 0.1

Number of generation

1000

1000

1000

1000

1000

Table 5 Developed GP equations and their regression coefficients in predicting flyrock Model no.

1

2

3

4

5

R2

GP equation

Flyrock ¼ ½ðD þ PFÞ þ ðD  0:1310Þ þ ½LnðDÞ  PF  ðBS þ PFÞ  0:5191  HD h hpffiffiffiffiffiffiffiffi  ii 3 þ ST  MC  BS  COSðDÞ  ð5:3358Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Flyrock ¼ ðExpðBSÞÞ3 þD  ð0:951 þ PFÞ  71:714 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii  3:2538 þ 3 MC þ ½ððHD  8:7219Þ  11:9757Þ  ðST  8:7219Þ hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Flyrock ¼ LnðHDÞ  arctan½PF  arctanðð6:2194  ST þ HDÞ  ðHD þ STÞÞ    BS þ HD þ sinðMCÞ  BS2 þ D þ PF Flyrock ¼ ½ððD þ ðD  BSÞÞ  PFÞ þ sinðST þ DÞ h i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 þ HD  2:3060  BS  cosðPFÞ  1:0002 þ ½arctanðMCÞ  4:8343 þ ð2ST  sinðMCÞÞ "   # HD þ MC 2 Flyrock ¼ cosðDÞ  Ln BS

Train

Test

0.837

0.859

0.798

0.876

0.804

0.860

0.829

0.801

0.819

0.908

þ ½HD þ ðððST  0:5720Þ þ 48:7480Þ  ðST  1:1402ÞÞ h  pffiffiffiffii þ ½BS  ðHD  PFÞ þ arctanðPFÞ  2D  3 D

were assigned for testing datasets, respectively. The mentioned process was repeated for utilized performance indices (i.e., R2, RMSE and VAF) as well as all predictive models. The obtained ratings for each dataset (training and testing) were then summed separately. As an example, rating values of NLMR training dataset 1 were 3 for R2, 4

for RMSE and 3 for VAF, so the performance rating was computed as 10 (see Table 7). In the last step, the total rank values were obtained by summing up the rank value of each dataset as listed in Table 8. According to Table 8, model 3 and 5 exhibited the best result for estimating flyrock for NLMR and GP techniques, respectively. As a

123

R. S. Faradonbeh et al. Fig. 12 GP tree structure of the selected model

Table 6 Developed non-linear multiple regression (NLMR) equations, and their regression coefficients in predicting flyrock

Model no.

1

R2

GP equation

Flyrock ¼ 2:54  D þ 4:374  HD1:06 þ 26:786  BS0:472 0:193

 163:332  ST

Train

Test

0.759

0.751

0.741

0.737

0.777

0.816

0.770

0.679

0.733

0.821

1:504

 0:006  MC

þ 72:679  PF1:704  33:752 2

Flyrock ¼ 2:54  D þ 1:719  HD1:26  6:163  BS

0:822

 0:003  MC 3

 300:854  ST

1:584

0:131

þ 84:74  PF1:55 þ 150:644

Flyrock ¼ 2:22  D þ 4:076  HD1:06 þ 54:617  BS0:422 0:131

 77:838  ST

 2:814  MC

0:584

þ 85:864  PF1:45  98:186 4

Flyrock ¼ 2:58  D þ 4:016  HD1:09 þ 56:836  BS0:483  178:915  ST0:183  0:013  MC1:401 þ 65:216  PF1:724  40:418

5

Flyrock ¼ 2:63  D þ 7:602  HD0:939 þ 60:342  BS0:53 0:191

 144:334  ST

1:356

 0:019  MC

þ 73:373  PF1:664  109:833

result, the prediction performances of the developed GP equations are significantly higher than those developed models of NLMR. The selected NLMR and GP equations in predicting flyrock distance are presented in Eqs. (12) and (13), respectively:

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Flyrock ¼ 2:22  D þ 4:076  HD1:06 þ 54:617  BS0:422  77:838  ST0:131  2:814  MC0:584 þ 85:864  PF1:45  98:186

ð12Þ

Development of a new model for predicting flyrock distance in quarry blasting: a genetic… Table 7 Results of performance indices for NLMR and GP models in predicting flyrock Method NLMR

GP

Model

R2

RMSE

VAF

Rating for R2

Rating for RMSE

Rating for VAF

Rank value 10

Train 1

0.759

31.600

74.172

3

4

3

Train 2

0.741

32.630

73.180

2

2

2

6

Train 3

0.777

30.540

77.612

5

5

5

15

Train 4

0.770

32.333

76.116

4

3

4

11

Train 5

0.733

33.907

71.885

1

1

1

3

Test 1

0.751

36.685

74.250

3

1

3

7

Test 2

0.737

35.032

66.414

2

2

2

6

Test 3

0.816

26.194

81.041

4

5

5

14

Test 4

0.679

33.927

63.644

1

3

1

5

Test 5

0.821

27.979

75.196

5

4

4

13

Train 1 Train 2

0.837 0.798

25.252 28.811

83.466 79.472

5 1

5 2

5 2

15 5

Train 3

0.804

29.435

79.463

2

1

1

4

Train 4

0.829

28.060

82.109

4

3

4

11

Train 5

0.819

27.951

81.105

3

4

3

10

Test 1

0.859

27.312

85.830

2

1

3

6

Test 2

0.876

21.213

87.494

4

4

4

12

Test 3

0.860

22.827

85.646

3

3

2

8

Test 4

0.801

25.870

79.160

1

2

1

4

Test 5

0.908

17.638

89.917

5

5

5

15

"

 # HD þ MC 2 Flyrock ¼ cosðDÞ  Ln BS 

Table 8 Total rank values for NLMR and GP predictive models in estimating flyrock

Method

Model

Total rank

NLMR

1

17

þ ½HD þ ðððST  0:5720Þ þ 48:7480Þ

2

12

ðST  1:1402ÞÞ þ ½BS  ðHD  PFÞ þ ½arctanðPFÞ  pffiffiffiffii  2D  3 D

3

29

4

16

ð13Þ

The graphs of predicted flyrock distance obtained from the NLMR and GP models against actual flyrock values are displayed in Figs. 13 and 14, respectively. Based on these figures, the best flyrock predictive model is the GP technique with R2 values of 0.819 and 0.908 for training and testing data, respectively, compared to the NLMR model.

Summary and conclusions In the present research, in order to develop new models for prediction of flyrock, six sites in Malaysia were studied and the most effective parameters on flyrock were recorded. In these sites, flyrock is considered as a serious problem due to short distance between blast face and the location of existing facilities, equipment and residential areas. To achieve the aim of this study, 262 datasets comprising of

GP

5

16

1

21

2

17

3

12

4

15

5

25

six inputs (i.e., D, HD, BS, ST, MC and PF) and one output (flyrock) were prepared. To develop the models, the established datasets were divided into training and testing parts as suggested in the literature. Then, five different datasets for training and testing were established randomly to obtain the best models from each modeling technique. In order to predict flyrock, two predictive models, namely NLMR and GP, were considered and developed. Using constructed datasets and considering modelling procedure of the mentioned models, five GP equations and five NLMR equations were developed. After developing the models, various performance indices, i.e., R2, VAF and

123

R. S. Faradonbeh et al. Fig. 13 Flyrock values predicted by NLMR equation, and measured values

Fig. 14 Flyrock values predicted by GP equation, and measured values

RMSE were applied to evaluate the accuracy and efficiency of these models. Considering the training datasets of developed models, the highest regression coefficient (R2 = 0.819) was obtained by performing the GP model, while this value for the best NLMR model was achieved as 0.777. Moreover, it is found that the GP model obtains the highest VAF and RMSE indices values. The performance indices obtained by the GP model reveal the high reliability of the new predictive model to predict flyrock. As mentioned before, GP can make a non-linear equation between inputs and output parameters. This is an advantage of GP compared to some soft computing methods like ANN, which can only suggest a superior model.

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