Research On Practical Model To Calculate The Cai Based On The Gene Expression Programming (Gep) Approach
The Cerchar abrasivity test is very popular for determination of rock abrasivity. Accurate estimation of Cerchar abrasivity index (CAI) is useful for excavation operation costs. This paper presents a practical model to calculate the CAI based on the gene expression programming (GEP) approach. The model is trained and tested based on a database collected from the experimental results available in the literature. Proposed GEP model predicts CAI based on two basic geomechanical properties of rocks, i. e. rock abrasivity index (RAI) and Brazilian tensile strength (BTS). Root mean square error (RMSE), mean absolute error (MAE), Nash-Sutcliffe efficiency (NSE) and coefficient of determination (R2) are used for measuring the model’s performance. The results show that GEP is a strong technique for prediction of the CAI.
Introduction
In mining and civil projects, abrasiveness of rocks plays a crucial role in the wear of cutting tools in any rock excavation operation including drilling-blasting and mechanical excavation. This feature can lead to increase in costs, decrease of efficiency, and increase in the life of projects. To estimate rock abrasiveness, several tools and techniques have been developed by various researchers as well as international standards. Among them, Cerchar Abrasivity Index (CAI) test is the most commonly used method for laboratory assessment of rock abrasivity due to its simple, fast test procedure and economic merits. This test has been introduced in the 1970s by the Centre d’Etudes et Recherches des Charbonages (CERCHAR) de France and standardized by French standard AFNOR (NF904-430-1), ASTM (D7625-10) and ISRM [2]. In the laboratory, CAI is determined while a rock sample is fixed on the sliding platform; over which a scratching pin of Rockwall Hardness 54-56 is fixed with a loading arrangement. A static load of 70 N is applied on the fresh surface of the sample and the sample is displaced at a rate of 1 mm/s over a length of 10 mm. The wear of the pin is determined through high precision microscope. To eliminate the error, the test is repeated five times and the arithmetic mean is reported in the result.
Rostami et al. reported that Cerchar testing is influenced by many parameters. One of the main parameters on the results of cerchar test is petrographical and geomechanical properties of rock such as compressive strength, shear strength, Young’s modulus, quartz contact, grain size, etc. During last decades, many researchers have studied the effect of petrographical and geomechanical properties on CAI. For example, Suana and Peters, West and Yarali et al. mentioned that quartz content of the rock is a main influencing parameter on the CAI. Plinninger et al. showed that the combination of Young’s modulus and the equivalent quartz content (EQC) has a fair correlation with CAI. Lassnig et al. studied the impact of grains size on CAI. Al-Ameen and Waller, Alber, kahraman et al. , Dipova and Deliormanli investigated the relationship between rock strength and CAI. Plinninger developed an empirical correlation between CAI and rock abrasivity index (RAI) based on 60 types of igneous, metamorphic and sedimentary rocks. Khandelwal and Ranjith proposed an empirical correlation between CAI and P wave velocity. Gharahbagh et al. and Rostami et al. examined the behavior of rough and smooth rock surfaces on CAI values and proposed some correlations. Tripathy et al. investigated the correlation of CAI values of different rock types with uniaxial compressive strength (UCS), point load index, P wave velocity and Young’s modulus using multivariate regression analysis and artificial neural networking.
Majeed and Abu Bakar evaluated CAI measurement methods and their dependence on petrographic and mechanical properties of 64 rock units in Pakistan. Er and Tugrul [19, 20] developed empirical relationships between geological and physico-mechanical properties and CAI of 12 different granitic rock samples using simple regression analysis. Moradizadeh et al. investigated correlations between CAI and EQC, point load index, slake durability index and percentage of water absorption of 36 samples of igneous, metamorphic and sedimentary rock using simple and multivariate regression. Abu Bakar et al. [2]investigated the influence of water saturation on CAI values based on laboratory testing of 33 sedimentary rock units. Ko et al. evaluated the correlation between CAI and geomechanical properties of rocks (including QC, UCS, BTS, and brittleness index) using statistical analysis.
Capik and Yilmaz developed new prediction models for CAI based on some rock properties using simple and multiple regression analysis. Moreover, they modeled drill bit lifetime based on CAI. Balani et al. investigated the effect of rock parameters on Cerchar abrasivity index using PFC3D modeling. Javier Torrijo et al. studied the relation between CAI and chemical compounds and petrographical properties of andesitic rocks from the central area of Ecuador. In this paper, the correlation between CAI and geomechanical properties of rocks including rock abrasivity index (RAI) and Brazilian tensile strength (BTS) is investigated using Gene Expression Programming (GEP) technique. GEP is a new soft computing technique, first invented by Ferreira. The main advantage of GEP approach is the capability to generate prediction equations that can be easily manipulated in practical circumstances. An increase in the application of GEP technique for solving many mining and rock mechanics problems has been observed in recent years. For example, GEP has been successfully applied for prediction of tunneling-induced settlement, TBM and roadheader performance, rock properties such as uniaxial compressive strength, tensile strength, modulus of elasticity, side-effects of blasting operation such as ground vibration and flyrock, rockburst hazard, etc.
All researchers pointed out that GEP has the ability to solve complex problems. The literature surveys show that there is no study about the application of GEP in the field of rock abrasiveness prediction. Hence, an effort has been made, in this paper, to make use of GEP for predicting the CAI based on geomechanical properties of rocks (RAI and BTS). Data collectionIn this paper, to develop and assess the performance of GEP model, a database including 106 rock units has been employed. This database has been compiled from published literature in this field. The examined rock types were from sedimentary, metamorphic and igneous origins. The database contains two inputs (RAI and BTS) and one output parameter (CAI). RAI is calculated using equation by Plinninger et al. as follows: RAI=(EQC×UCS)/100 (1)where EQC= equivalent quartz content (%) and UCS= uniaxial compressive strength (MPa). The RAI ranged between 0. 07 and 190. 38; the BTS ranged between 0. 48 and 22. 67 MPa; and the CAI varies between 0. 19 and 4. 88. It should be mentioned that among 106 data, 90 sets (85% of data) were randomly chosen as training sets for the GEP modeling and 16 sets (15% of data) were used as testing the generalization capacity of proposed model. The testing sets have not been utilized in training of corresponding model.
Gene Expression Programming (GEP)
Gene Expression Programming (GEP) is a new evolutionary algorithm first invented by Ferreira [25] based on genetic algorithm (GA) and genetic programming (GP). GEP incorporates both the idea of a simple, linear chromosome of fixed length used in Genetic Algorithms (GAs) and the tree structure of different sizes and shapes used in Genetic Programming (GP). According to Ferreira the primary difference between GEP and its predecessors, genetic algorithms (GAs) and genetic programming (GP), stems from the nature of the individuals: in GAs, the individuals are linear strings of fixed length (chromosomes). In GP, the individuals are nonlinear entities of different sizes and shapes (parse trees). In GEP, the individuals are encoded as linear strings of fixed length (chromosomes) that are expressed as nonlinear entities of different sizes and shapes.
. To develop a GEP model, five components; the function set, terminal set, fitness function, control parameters and stop condition are required. After the problem has been encoded for candidate solution and the fitness function has been specified, the algorithm randomly creates an initial population of viable individuals (chromosomes) and then converts each chromosome into an expression tree corresponding to a mathematical expression. Afterwards the predicted target is compared with the actual one and the fitness score for each chromosome is determined. If it is sufficiently good, the algorithm stops. Otherwise, some of the chromosomes are selected using roulette wheel sampling and then mutated to obtain the new generations. This closed loop is continued until desired fitness score is achieved and then the chromosomes are decoded for the best solution of problem. Readers can refer to Ferreira for more details about GEP.
GEP model development
The fundamental aim of developing GEP model is to generate a mathematical function for prediction of CAI. In developing phase of GEP model, RAI and BTS are entered as input variables, while CAI value is used as output variable. So, a mathematical function is generated in the form of y=f(RAI, BTS) for CAI based on training data sets. In this study, GeneXpro Tools 5. 0 program [42] was employed to develop model based on GEP. The following steps were done to estimate CAI using GEP; the first step is to select the fitness function which can be based on several functions. In this research, the fitness function of root mean square error (RMSE) was used. The second step in GEP modeling involves determining the mathematical functions that chromosomes are allowed to use in their programs and in the final equation. There is no definitive rule in choosing a mathematical function combination. In this paper, the functions set is comprised of four basic arithmetic operators (×, −, /, +) as well as other more complex mathematical functions, e. g. Power (Pow), square root (Sqrt), exponential (Exp), natural logarithm (Ln), logarithm of base 10 (log), cubic root (3Rt), Sines (sin), Cosine (cos), Tangent (tan), Secant (sec), and Cosecant (csc).
The third step is determining the chromosomal architecture, which involves determining the head length and gene numbers. In the present study, the trial and error method was used to establish these two parameters. The GEP was run for various head length and gene number combinations. The results show that the GEP model with 3 genes and head length of 9 produces the most accurate results in modeling the CAI. In addition, since an initial population in the 30–100 interval leads to acceptable results, an initial population of 65 was considered in the current study.
In the fourth step, the genetic operators including mutation, inversion, transposition and recombination are selected. In fact, the mentioned parameters are borderlines of GEP which can affect the performance of GEP. It must be said that all mentioned parameters are selected by user using trial and error procedure to obtain the optimum structure of GEP. Finally, in the last step of GEP modeling procedure, a proper linking function should be chosen to connect expression trees. The functions of +, −, × and / are the most common functions which are used for this aim. Each of these four functions were examined and multiplication (×) was selected as the best linking function.
The mathematical equations related to each gene can be extracted as Eqs. 2–4. These equations were then linked together via chosen linking function (multiplication, in this study) and the final predictive model for CAI prediction is formulated as Eq. 5. (2) (3) (4) (5).
GEP model assessment
As mentioned before, 16 sets out of 106 data sets were randomly selected for testing GEP model. The testing data are unfamiliar to the model and were not included in its development. In this section, the GEP model’ performance is validated using testing datasets.
our standard statistical performance evaluation indices including the root mean square error (RMSE), mean absolute error (MAE), Nash-Sutcliffe efficiency (NSE), and coefficient of determination (R2) were used to assess the model performance. The lowest values of RMSE and MAE indicate the high performance of model. A R2 value equal to 1 indicates that the regression line perfectly fits the data. NSE of 1 corresponds to a perfect match of estimated values to the observed data; on contrary, NSE of 0 indicates that the model predictions are as accurate as the mean of the observed data. The high values of R2 (0. 895) and NSE (0. 806) and the low values of RMSE (0. 357) and MAE (0. 293) show that the GEP model is suitable and can predict the CAI with acceptable error.
Summary and Conclusions
This study explores the potential of GEP in the prediction of CAI value that have great role in the wear of cutting tools in any rock excavation operation including drilling-blasting and mechanical excavation. This study presents the first application of GEP for CAI prediction of rocks. A database including 106 rock units has been employed to develop and assess the performance of GEP model. This database has been compiled from the experimental results available in the literature. The GEP model was trained on 85% of available data and tested using the remaining 15%. Two basic geomechanical properties of rocks, i. e. rock abrasivity index (RAI) and Brazilian tensile strength (BTS) were entered to the GEP model as input parameters. The performance of model was evaluated using four statistical performance evaluation indices (RMSE, MAE, NSE, and R2). In testing phase of GEP model, the values of RMSE, MAE, NSE, and R2 were obtained as 0. 357, 0. 293, 0. 806, and 0. 895, respectively. These findings revealed that GEP is efficient and useful technique for CAI prediction.
The main advantage of GEP in comparison to commonly used soft computing techniques such as ANN is that it suggests a practical and explicit equation between inputs and output parameters.
