Accurate estimation of equipment costs is a key factor in feasibility study and evaluation of design alternatives of mining projects. In this paper, capital and operation costs of backhoe loaders are estimated using multiple linear regression (MLR), based on principle component analysis (PCA). These cost functions are consisted of five independent variables; bucket size, digging depth, dump height, weight and horse power. The MLR is conducted in two steps. At the first correlation between independent variables is omitted using PCA technique. Thereafter, MLR functions are established using selected significant PCs and total cost functions are constituted as functions of initial variables. At the end, accuracy of functions are evaluated using mean absolute error rate method

Lashgari, A

Oraee, B

Sayadi, AR

English

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 SME Annual Meeting  Feb. 27-Mar. 02, 2011, Denver, CO   1 Copyright © 2011 by SME   Preprint 11-040    ESTIMATION OF CAPITAL AND OPERATION COSTS OF BACKHOE LOADERS  B. Oraee, Univ. College London, London, England, UK A. Lashgari, Tarbiat Modares Univ., Tehran, Iran A. R. Sayadi, Tarbiat Modares Univ., Tehran, Iran   ABSTRACT Accurate estimation of equipment costs is a key factor in feasibility study and evaluation of design alternatives of mining projects. In this paper, capital and operation costs of backhoe loaders are estimated using multiple linear regression (MLR), based on principle component analysis (PCA). These cost functions are consisted of five independent variables; bucket size, digging depth, dump height, weight and horse power. The MLR is conducted in two steps. At the first correlation between independent variables is omitted using PCA technique. Thereafter, MLR functions are established using selected significant PCs and total cost functions are constituted as functions of initial variables. At the end, accuracy of functions are evaluated using mean absolute error rate method. INTRODUCTION Over the past decades, mining equipment complexity and size has been increased. On the other hand, improving the productivity of the mines required large and complex equipment. In this term, the mining industry continues to adhere to a “bigger is better” approach, because in addition of higher production and productivity, the operating cost of mined material of larger equipment is low. But, there are some indications that this policy may not always be a realistic approach. Some difficulties such as dilution, lack of flexibility and complexity have accompanied with large equipment which may limit the advantage of larger size of mine machines. Loading operation take a great deal of expenses in surface mines and selecting the most appropriate equipment wich both minimizes the costs and meets production needs is one of the main concerns about designing and planning. Capital and operating cost are of extreme important elements for scaling the mining equipment and the most precise possible estimation of costs is one of the most important issues for prefeasibility and feasibility studies, to decrease the financial risks of the mining projects. The costs are often estimated in one of two ways; qualitative and quantitative (Stewart and Wyskida, 1987). In qualitative approach heuristic norms and expert judgments are the base of estimations. Quantitative approach is classified to three methods; generative-analytical models, analogous models and statistical models (Asiedu et al., 2000). Analogous methods are based on comparison of costs information of similar projects and adjusting these data for differences between the products (Layer et al., 2002). Generative-analytical methods aim to describe product formation method and estimate costs.  Regression analysis, artificial neural network are some kinds of statistical methods. The most commonly used methods for cost model development effort have been based on linear-least-squares regression, because of their ability to classify, summarize and extrapolate collections of data (Bode, 2000). Application of different cost estimation tools has showed that professional cost estimators usually use regression to build their cost models (Mason and Smith, 1995). So far, a number of cost models using the Single Regression Analysis (SRA), for the estimation of mineral industry costs, have been published (O’Hara, 1980, 82, 87; Mular, 1982, 1998; CANMET, 1986; USBM, 1987; Lanz and Noakes, 1993; O’Hara and Suboleski, 1992; Camm, 1994; Rudenno, 1998). The existing methodologies to evaluate mining costs do not allow estimating them with acceptable margines of error throughout the evaluation of the mining projects. These models are often in univariate form and other effective independent variables has simplified been ignored. Moreover these are old enough and updating them result in high errors. In this study, a multiple linear regression based on principal component analysis model is presented to make quick estimates of the capital and operating costs of backhoe loaders which are of the most prevalent machines in loading operation of open-pit mining. Figure.1 shows a normal rock loading operation by using bachkoe loader and dump track (loading machine: Komatsu PC450-7, PC450-7E0 and dump truck: Komatsu HD255-5).  Figure 1.  A normal rock loading operation by using bachkoe loader and dump track DATA A set of technical and economical data of 19 different backhoe loader was gathered. Explanatory parameters of these machinery are consisted of bucket size (BS) in cubic meter, digging depth (DD) in  SME Annual Meeting  Feb. 27-Mar. 02, 2011, Denver, CO   2 Copyright © 2011 by SME meter, dump height (DH) in meter, weight (W) in kilogram and horse power (HP). The data are classified on the basis of cost types (capital or operating). The capital cost (CC) is based on US dollar (2007) while the operation cost (OC) is based on US dollar (2007) per hour. The operating costs items include: overhaul (parts and labor), maintenance (parts and labor), power, lubrication and wear parts. The cost of operator's time is not included here. Descriptions of data related to each of these variables are shown on Table 1. Also, the normal distribution of data has been approved. Table 1.  Data distribution.  BS (cu m) DD (m) DH (m) W (1000kg) HP CC (M$) OC ($/h) Min 0.72 5.64 6.1 12.5 88 0.145 14.7 Max 56.24 14.63 14.02 635.0 3000 6.8 553.3 Mean 17 9.2 9.6 176.8 931.7 2.221 179 St. Dev 17 2.6 2.6 180.9 897.5 2.229 177  Table 2 indicates the correlation matrix. As observed, an intense correlation between independent variables. Thus, it is necessary to take this aspect into consideration while analyzing the multiple variables (Gujarati 2003). Table 2.  Correlations of independent variables.  BS DD DH W HP BS 1.000000 0.752393 0.908046 0.993431 0.977302 DD 0.752393 1.000000 0.889620 0.752696 0.769203 DH 0.908046 0.889620 1.000000 0.887726 0.880153 W 0.993431 0.752696 0.887726 1.000000 0.988021 HP 0.977302 0.769203 0.880153 0.988021 1.000000  METHODOLOGY AND APPLICATION MLR was used to estimate the backhoe costs using as predictors machinery explanatory parameters. The multivariable cost functions have been developed at three stages. Principal component analysis One of the greatest challenges in multiple regressions is multicollinearity phenomena in which predictor variables have a high correlation with each other. The high correlation among predictive variables can lead to unstable regression model so that the variance and covariance of coefficients are very high. With respect to intense correlation between independent variables (Table 2), it is needed to take this correlation away before MLR (Gujarati 2003). For this, PCA technique could be used (Jolliffe 1986 and Liu 2003). Apart from omitting the correlation of independent variables, this method simplifies the complexity of relations between them (Sharma 1996; Jolliffe 2002 and Hardle and Simar 2003). In PCA, the p original variables are transformed into linear combinations called principal components. PCA is a dimension reduction technique in which the goal is to find a few principal components (PCs) that explain a large proportion of the total sample variance of the p variables while the PCs have no correlation with each other. Principal components are often used as input to another analysis, such as regression. If we use PCs as predictor variables, the regression model called PCA regression. It is obvious that in PCA Regression multicollinearity problem fundamentally resolved (Jambu 1991). Performing PCA technique on these five backhoe loader parameters to describe their interrelation pattern as well as onto the costs, the number of PCs usually will equal the number of independent original variables. This new variables are independent linear combinations of original variables and retain the maximum possible variance of the initial set. Table 3 shows the eigenvectors of correlation matrix that represents the matrix of the weights for the PCs, which demonstrates the relative importance of each standardized parameter in the PC calculations. Table 3.  Eigenvectors of correlation matrix.  BS DD DH W HP PC1 -0.458995 -0.409054 -0.450719 -0.45808 -0.45718 PC2 0.322988 -0.776273 -0.284404 0.346781 0.303208 PC3 0.138828 -0.442548 0.788619 -0.143169 -0.37744 PC4 0.52757 0.178511 -0.281215 0.306239 -0.718985PC5 0.622414 0.048576 -0.12233 -0.745403 0.199125  The new variables from the PCA become ideal to use as independent variables in MLR, since they optimize spatial patterns and remove possible complications caused by multicollinearity phenomena. Table 4 indicates eigenvalues of correlation matrix. As can be seen, out of five principle components only one PC has eigenvalue higher than 1, so in order to derive more accurate cost functions, all five PCs were selected as significant for the MLR analysis. These components could explain all of the total variance of the abckhoe loader variables. Table 4.  Eigenvalues of correlation matrix.  Eigenvalue Total Variance % Cumulative Eigenvalue Cumulative %PC1 4.527094 90.54187 4.527094 90.5419 PC2 0.376184 7.52367 4.903277 98.0655 PC3 0.078215 1.56430 4.981492 99.6298 PC4 0.015327 0.30654 4.996819 99.9364 PC5 0.003181 0.06362 5.000000 100.0000  Multiple linear regressions Coefficients obtained by PCA technique were used as independent variables in the MLR. The eigenvalue-one criterion also known as the Kaiser criterion (Kaiser 1960) is the simplest and the most common method used to solve the number of principal component problem. Based on this, the principal components with eigenvalue greater than 1 are selected. The PC with the highest eigenvalue is considered the most significant ones, but in this research all five PCs ware selected. To exclude non-significant PCs and identify the best estimation functions for backhoe loader costs, MLR costs analyses on the PC scores were performed using stepwise variable selection procedures. Table 5 shows coefficients of determination for each of the models. Also table 6 and table 7 show the results of MLR analysis on capital and opetating costs of this equipment, respectively. Table 5.  Coefficients of determination.  R2 Adjusted R2Capital Cost 0.985 0.9797 Operating Cost 0.994 0.9917  Table 6.  Regression summary for capital cost ($) for backhoe loader.  Beta Std. Error of Beta B Std. Error of B T(13) P-valueIntercept - - 2115231 70767 29.8 0.00 PC1 -0.96 0.03 -997854 34124 -29 0.00 PC2 0.25 0.03 904827 118377 7.6 0.00 PC3 0.09 0.03 701317 259611 2.7 0.02 PC4 0.00 0.03 115133 586465 0.2 0.85 PC5 0.02 0.03 774172 1287302 0.6 0.56   SME Annual Meeting  Feb. 27-Mar. 02, 2011, Denver, CO   3 Copyright © 2011 by SME Table 7.  Regression summary for operating cost ($/h) for backhoe loader.  Beta Std. Error of Beta B Std. Error of B T(13) P-value Intercept - - 178.65 3.62 49.3 0.00 PC1 -0.97 0.02 -79.00 1.75 -45.1 0.00 PC2 0.23 0.02 63.75 6.08 10.5 0.00 PC3 0.006 0.02 3.84 12.51 0.3 0.763 PC4 -0.03 0.02 -40.19 29.25 -1.4 0.19 PC5 0.02 0.02 56.16 64.44 0.9 0.4  As can be seen in R2 column, 98.5% of variation in capital cost of backhoe loaders has explained by this method of independent variable selection. Since, percentage of the R2 is raised with the increase of number of independent variables, it is better to use adjusted R2 for comparing equations with different number of explanatory variables. The adjusted R2 is interpreted as that of the R2 value except it takes into account the number of degrees of freedom. It is adjusted by dividing the error sum and total sums of squares through their respective degrees of freedom (Gujarati 2003). Eq. (1) indicates relations between R2 and adjR2: adjR2 =1- [(Res SS / df) / (Total SS / df)] (1) Where Res SS is the error sums of squares, the Total SS is the total sums of squares and df is their respective degree of freedom. In Tables 6 and 7, regression coefficients of independent variables are highlighted on B Column. Since, the Beta coefficients are the standardized regression coefficients, their advantage (compared to the ones that are not standardized) is that their magnitudes allow comparing the relative contribution of each independent variable in the estimation. As seen in these tables, PC1 is the most important independent variable of the cost functions (with regard to beta coefficient). t- Test was used in assessing significance of the regression coefficients that showed that PC1, PC2 and PC3 in capital cost and PC1 and PC2 in operating cost functions are statistically significant. For instance, the following equation was obtained from MLR analysis on capital cost of backhoe loaders: Capital Cost (CC) = -997854 PC1+ 904827 PC2+ 701317 PC3+ 2115231 (2) Since, the obtained relationships are as the function of PCs, the final cost relationships are established as functions of initial backhoe loader parameters. For this, the coefficients determined from multivariable analyses (B coefficients) are multiplied in eigenvector of correlation matrix. While, at the first stage (applying PCA technique), all variables have standardized and thus, to use functions on row data, it is needed that the variables are returned to their initial positions once again. The final equations coefficients are calculated from Eq. (3): X= (Xdl – Xave) / Sx (3) Where X is final regression coefficient, Xdl indicates B coefficient and Xave and Sx are mean and standard deviation of the variable. With multiplying B coefficients of Table 6 and 7 in eigenvectors of correlation matrix of Table 3, coefficient of each original backhoe loader parameters correlated with capital cost are determined that in these relationships, independent variables have standardized. Using Eq. (3), the variables are returned again to their initial positions. The derived functions for capital and operating costs are as follow: Capital Cost (CC) = 48548.37BS -224187.34DD + 274692.17DH + 3.72 W + 557.44HP -349946 (4) Operating Cost (OC) = 3.345BS -7.281DD + 7.69DH + 0.00032W + 0.061HP +2 (5) With reference to the fact that the obtained functions are prepared on the basis of 2007 cost data, it is necessary to apply Eq. (6) to make it up-to-date. Cx= (Ix .C2007) / I2007 (6) Where C indicates cost and X and I are proposed year and cost index, respectively (I2007= 112.4) (Info Mine, 2007). MODEL VALIDATION In this study, each function’s performance is measured by the Mean Absolute Error Rate (MAER) (Kim et al 2004), which is defined in Eq. (7). MAER = [∑ |(Ce - Ca) / Ca|.100] / n (7) Where, Ce is the estimated backhoe loader costs, Ca is the actual backhoe loader costs, and n is the number of data. The results obtained from the MLR represent MAER of 17.44 and 10% for capital and operating costs respectively (Figure 2).  Figure 2.  The difference between the actual costs and the estimated costs. CONCLUSION In this study, an accurate and up-to-date model was developed for backhoe loaders costs estimation. The MLR was applied using PCA technique. The cost functions were classified on the basis of the cost type (capital and operating costs). These functions could be a useful tool for cost estimations in preliminary and detailed feasibility studies of mining projects. Applying the cost index, these functions could be updated and applied in future. REFERENCES 1. 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US Bureau of mines cost estimating system handbook, Mining and beneficiation of metallic and nonmetallic minerals expected fossil fuels in the United States and Canada, United States Bureau of Mines”, Open file report 10-87,

Estimation of capital and operation costs of backhoe loaders

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