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Technical Papers

Optimal blast design

using a discrete-event simulation model in a hard-rock mine by S.S. Nageshwaraniyer, K.M. Kim and Y.J. Son

Abstract ■ This paper reports on the development of a simulation-based framework for obtaining optimal blast design parameters for a hard-rock mine. To do this, a conventional medium-sized hard-rock mine was taken as a case study. The mine has two pits containing hard and soft rock types. The proposed framework contains a combination of regression analysis using Excel software by Microsoft and discrete-event simulation using Arena software by Rockwell Automation, applied to construct a model of the material-handling network of the mine. Specifically, blasting parameters are estimated by Excel-based regression analysis, processes including blasting and haulage from pits to crushers are modeled by a Forward Blasting simulation submodel, and material-handling operations including crushing, storage in stockpile, and haulage via conveyors and into SAG and ball mills are modeled by a Crusher to Ball Mill simulation submodel. A Reverse Blasting submodel is then used to obtain the optimum blast design corresponding to a target P80 particle size. Using the proposed framework, an economic analysis was performed to demonstrate the cost savings that could be realized for each rock type as a function of specific explosive energy. Mining Engineering, 2015, Vol. 67, No. 11, pp. 47-53. Official publication of the Society for Mining, Metallurgy & Exploration Inc.

Introduction In a modern hard-rock mine, various processes such as drilling, blasting, crushing, grinding and mineral liberation are optimized to increase productivity, a practice known as mine-to-mill (MTM) optimization. Blasting is imS.S. Nageshwaraniyer is graduate student at the Department of Systems and Industrial Engineering, K.M. Kim is assistant professor at the Department of Mining and Geological Engineering, and Y.J. Son is professor at the Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ, USA; email: [email protected]. Paper number TP-14-047. Original manuscript submitted November 2014. Revised manuscript accepted for publication May 2015. Discussion of this peer-reviewed and approved paper is invited and must be submitted to SME Publications by Feb. 29, 2016. www.miningengineeringmagazine.com

portant in MTM optimization as it is the first comminution process and has an impact on all downstream processes (Julius Kruttschnitt Mineral Research Centre, 1996; Nielsen and Lownds, 1997; Nielsen and Malvik, 1999; Chung and Katsabanis, 2000; Kim and Kemeny, 2011a; Kim and Kemeny, 2011b). Two main research topics have emerged in the area of blasting design: (1) identifying the optimal target blast fragmentation at each shot and (2) controlling the blast fragmentation for that target. As part of an MTM optimization study, Kim and Kemeny (2011a; 2011b) developed a site-specific blast fragmentation prediction model that can be used to identify the target blast fragmentation and can be incorporated into MTM optimization. They also reported methodology for assigning model parameters easily and with minimal human bias. However, applying the site-specific blast fragmentation model for MTM at a hard-rock mine has practical limitations because of the complexity of the comminution processes involved and

shortage of engineering resources. An “automated” system would be more useful. This paper describes such a model: specifically, a discrete-event simulation model that identifies the target P80 particle size (the representative particle size in a cumulative distribution of muckpile rock fragmentation). It considers comminution energy changes in a crusher and mill by rock type and predicts the optimum blast design parameters of burden and spacing to achieve the target fragmentation at each shot. This automated simulation model, which was constructed using Arena software from Rockwell Automation, uses a trial-and-error process to estimate the optimum blast at a specific mine.

Background of site-specific blast fragmentation model and economic analysis Traditional models for predicting blast fragmentation such as the KuzRam model can be difficult to apply in certain field applications. They require reliable data for pre-blast block size Mınıng engıneerıng 

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Technical Papers Resumen ■ Este estudio informa acerca de un marco de simulación para la obtención de parámetros de diseño de voladura óptimo para una mina de roca dura. Para lograr esto, se tomó como caso de estudio una mina convencional de tamaño medio de roca dura. La mina tenía dos socavones con tipos de roca dura y suave. El marco propuesto contiene una combinación de análisis de regresión usando el programa Excel de Microsoft y simulación de evento discreto con el programa Arena de Rockwell Automation, aplicado a la construcción de un modelo del material manejado por la red de la mina. En específico, los parámetros de voladura son estimados por el análisis de regresión de Excel; los procesos, que incluyen las voladuras y la extracción de los socavones a los molinos son modelados por un sub-modelo de simulación Forward Blasting, y las operaciones de manupulación de material incluyendo molienda, almacenamiento en pila y transporte mediante fajas transportadoras a los molinos SAG y de bola son modelados por un sub-modelo de simulación Crusher to Ball Mill. Luego se usa un submodelo de Voladura en Reverso a fin de obtener el mejor diseño de voladura para lograr una partícula de tamaño P80. Con el marco de trabajo propuesto se realizó un análisis económico para demostrar el ahorro en costos que podría obtenerse para cada tipo de roca como una función de energía explosiva específica.

and intact rock strength, which are difficult to obtain for sites with variable geological conditions. To address this challenge, the site-specific blast fragmentation model developed by Kim and Kemeny (2011a) uses innovative approaches to obtain data. This empirical model applies to a specific rock type or area within a mine and relates several variables: specific explosive energy, pre-blast block size, post-blast fragmentation and intact rock strength. Techniques for obtaining data include using image-processing software to determine the pre-blast, or in situ, block size quickly and consistently, with the least human bias and using tensile stress instead of uniaxial compressive strength, since it is the most common failure mode in rock fracturing. The main inputs for this model are intact rock strength To in MPa, pre-blast block size F80 in cm, and post-blast fragmentation P80 in cm. The model’s output is specific explosive energy ESE, which is the explosive or mechanical energy required to fragment a unit mass of rock, in kJ/t:

(1)

where a, b, c and d are fitting coefficients determined by regression analysis using Excel. The following equation for properly spacing drill holes with burden B and spacing S, both in meters, yields the target P80 for a blast design: (2) where Ex. is the average explosive energy per hole in kJ, H is the bench height in meters and S.G. is the specific gravity of the rock. In general, higher explosive energy induces finer blast fragmentation, which, in turn, reduces costs for all downstream processes including crushing and grinding, although the specific conditions at each mine, such as energy saturation, should be considered. Within a certain range, however, further increases in blasting energy increase the total cost. Kim and Kemeny (2011b) used the site-specific blast model to investigate total cost savings versus explosive energy, and demonstrated the impact of blasting — precondi48

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tioning rock by inducing microfracturing — by decreasing the Bond Work Index, following the work of Nielsen and Lownds (1997) and Nielsen and Malvik (1999). Initial simulations were conducted with two rock types: hard and soft. Although the results were preliminary, it showed the optimal blast energy range for both rock types.

Background of discrete-event simulation of material-handling networks in mining Analytical models and computer simulation can be applied to model surface and underground mines in order to obtain various mine system performance measures such as throughput and equipment utilization. Within analytical models, queuing theory and mathematical programming have been very popular. Queueing models of mines have been used to estimate system performance measures such as production (Koenigsberg, 1958; Muduli and Yegulalp, 1996) as well as to evaluate operating conditions such as numbers of trucks and shovels (Trivedi et al., 1999). In addition to throughput as a measure, different alternative operating conditions can be evaluated based on equipment utilization, wait time and other system performance measures to choose the best alternative in a queueing model of a real mine (Trivedi et al., 1999; Ercelebi and Bascetin, 2009). Queueing theory has also been used to mathematically model shovel throughput based on the number of trucks assigned to it (Ta et al., 2010). In the past, computer simulation using discrete-event paradigm had been used as a tool for analysis as well as decision support in manufacturing supply chains. Its applications have appeared in the semiconductor and automotive industries. As crushing, storage and ore haulage processes in many surface mines resemble assembly/processing, inventory and haulage in the manufacturing supply chain, discrete-event simulation has been applied to model them since the 1960s. In the discrete-event simulation paradigm, the changes (events) in the system modeled occur at specific simulation times. Another way to understand this is that only a limited number of events could be happening at any time www.miningengineeringmagazine.com

Technical Papers in the simulation. SPS (Rist, 1961), GPSS (Harvey, 1964) and Fortran (Bucklen et al., 1968) were the first computer simulation languages used in mining. A comprehensive review of software used in mine system simulation in the U.S. was provided by Sturgul (1999). Lately, simulation packages such as Arena, ProModel and Simio have become popular because they offer animation capabilities in addition to easy-to-use graphical user interfaces. The main output from a discreteevent simulation model of a mine is the mean production achieved up to a particular point in time, along with confidence intervals. Monte Carlo technique applied in discreteevent simulation is used to estimate the mean production at any point in the mine at a given simulation time, along with confidence intervals due to uncertainties such as variances in shovel loading times, truck haul and dumping times. Other outputs, such as mean equipment utilization, mean queueing times at shovels or crushers and mean time in stockpile or silos, along with their respective confidence intervals can also be obtained. Detailed status reports such as locations of trucks in mine, tracking of ore on conveyors, and animated visualization of the operations are also possible.

Automated simulation model in Arena We developed an automated simulation model using Arena. This model incorporates the site-specific blast model defined by Eqs. (1) and (2) along with Excel-based regression analysis to achieve two goals: (1) calculate the fitting coefficients a, b, c and d for different rock types and locations within the mine and (2) determine the optimum blast design, including the drill-hole burden and spacing given other blasting parameters, to obtain the target P80 after blasting for

each rock type, using the calculated fitting coefficients. One of the main contributions of this paper is to specify the mean cost savings, with confidence interval, achieved as a function of specific explosive energy. To this end, all operations from blasting and excavation to grinding at ball mill were modeled using a combination of regression and discrete-event simulation. Specifically, the relationship between the blast design and the resulting particle sizes was modeled using regression; a combination of real data for blasting parameters such as B, S, S.G. and Ex., rock properties and particle size distribution was used to form a blasting regression model; and Excel’s linear regression tool was used to estimate the coefficients of the blasting regression model. The remaining operations such as excavation, truck and conveyor transportation of ore, crushing, and milling at SAG and ball mills were modeled by discrete-event simulation in Arena. Details of the simulation model. Three simulation submodels were developed to determine the effects of mine material-handling operations on the rock type and volume prediction: (1) Forward Blasting, (2) Reverse Blasting and (3) Crusher to Ball Mill. Blasting is one of the key processes of this work. Hence, it is necessary to obtain burden and spacing parameters at a target fragmentation for each rock type before proceeding to simulate comminution processes. The purpose of the Forward Blasting submodel is to obtain the coefficients in the blasting regression model when only limited data for burden and spacing are available. This submodel is useful in real cases where these shot parameters were either not measured or not recorded every time a shot was made. These coefficients will help determine the burden

Figure 1 Overview of simulation model of blasting and comminution processes.

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Technical Papers Table 1 Various inputs for the regression model. Explosive energy (kJ)

B (m)

S (m)

To (MPa)

F80 (cm)

P80 (cm)

Fixed H (m) S.G. (density, t/m3)

1,443,480

7.25

8.70

7.86

33.02

25.14

12

Table 2

2.65

Table 3

ESE, F80 and P80 inputs for each scenario.

Data set for scenario No. 6.

Scenario no.

ESE increase F80 increase P80 increase (%) (%) (%)

ESE (kJ/t)

F80 (cm)

P80 (cm)

734

34.34

22.63

1

1.00

1.00

−2.00

748

35.66

22.63

2

2.00

1.00

−2.00

763

36.98

22.63

3

3.00

1.00

−2.00

777

38.30

22.63

4

2.00

2.00

−2.00

792

39.62

22.63

5

2.00

3.00

−2.00

806

40.94

22.12

6

2.00

4.00

−2.00

820

40.94

21.61

835

40.94

21.13

849

40.94

20.62

864

40.94

20.11

and spacing parameters for an optimum or target fragmentation, as deemed by the mine operators, without having to test these parameters in a real mine. The submodel has four parts, as shown in Fig. 1: Error, Blast, Monitor and Write. The model parameters are numbers of pits and trucks. In Error, an estimated error in blasting parameters such as burden and spacing, induced by drill error and the mismatch between drill pattern and blast direction, as given by the blasting engineer, is also entered into the model. For demonstration purposes, in our model this error is assumed to be uniformly distributed. For example, if the error in burden is 10 percent, then the actual burden in a blast can vary between −10 and +10 percent of its mean value. In Blast, two types of ore, hard and soft rock, are considered and a shot is simulated each day at their respective pits. Based on the inputs of number of trucks assigned to each pit and their cycle times, the corresponding number of truckloads is generated. In Monitor, it is assumed that several images of rock fragments will be captured as each truck is dumping into the crusher. The objective of taking these images is to determine P80 of the particle size distribution of each truckload. These cameras can be positioned at the crusher in a real mine. In Arena’s Input Analyzer add-on, a whole shot’s worth of data from a real mine was used to construct a particle-size distribution. In Write, for each pit the values of burden, spacing, bench height, tensile strength, F80 and P80 for each truckload are recorded into a Microsoft Access database. Excel-based regression model for blasting. The simulation run length has to be varied based on the quality of real data available. This means that the input data should be able to capture some of the intuitive relationships between ESE, F80 and P80. The blasting model and its logarithmic form used in regression analysis are given in Eqs. (1) and (3), where 50

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ln(a) is the intercept and b, c and d are the coefficients in the regression expression: (3) For a demonstration of some of these relationships, we considered six scenarios, each corresponding to a fresh data set that could be used to estimate the regression model coefficients. For instance, each scenario could be thought of as a data set from a different mine. In each of these scenarios, neither ESE and F80 nor ESE and P80 were allowed to be constant throughout the whole data set. The value of d was fixed at 1 to simplify our demonstration. Intuitively, for increasing values of ESE and constant F80, P80 should be decreasing. Alternatively, for increasing values of ESE and constant P80, F80 should be increasing. Although d was fixed in our demonstrative tests, it can be varied as well by increasing ESE corresponding to increasing To and keeping F80 and P80 constant in some of the data. The mentioned relationships were assumed for synthesizing data so as to demonstrate the combined use of simulation and regression to construct the sitespecific blasting model for different mines, with valid regression coefficients of a, b, c and d. During our demonstrative tests, the synthesized data not necessarily following these relationships (such as P80 or F80 being constant throughout a data set) resulted in invalid values for regression coefficients (for example, power of F80—b having negative values and power of P80—c having positive values). During field application of the proposed framework, the relationships will be www.miningengineeringmagazine.com

Technical Papers Table 4 Regression coefficients determined using Arena software and Excel’s regression analysis tool. Scenario no.

R2

ln (a)

b

c

Mean ESE (kJ/t)

Mean F80 (cm)

Mean P80 (cm)

1

1

3.01

0.88

−0.34

759

34.52

21.87

2

1

1.51

1.71

−0.63

799

34.52

21.87

3

1

0.03

2.50

−0.88

838

34.52

21.87

4

1

3.63

0.89

−0.63

799

35.99

21.87

5

1

4.35

0.61

−0.64

799

37.49

21.87

6

1

4.71

0.47

−0.64

799

38.96

21.87

figured out based on the actual data so as to obtain valid regression coefficients. We began with the same initial values for each type of input into the blasting regression model to synthesize the data sets for different scenarios, as shown in Table 1. Each of these values corresponds to one shot for a pit in a day, and they were taken from a real medium-sized hard-rock mine. Table 2 shows the variations in ESE, F80 and P80 that were specified for each scenario. These variations defined the behavior of the data set for that scenario. An example of the whole data set for scenario No. 6 based on the specified variations in Table 2 is shown in Table 3. The data set of each scenario is composed of 10 rows (or observations). Starting from the first row, ESE was incremented by 2 percent of the initial value until the 10th row. For half of the observations, either F80 or P80 was kept constant but ESE was varied throughout. This was done so that valid regression coefficients could be obtained since monotonically increasing or decreasing F80 or P80 in any data set also led to invalid regression coefficients. The regression analysis tool in Excel was used to determine the blasting regression coefficients. The resultant coefficients are shown in Table 4. ln(a) implies it is the natural logarithm of coefficient a that corresponds to the constant

in the blasting model. The R2 value expresses the proportion of the variation in ESE that is explained by variation in To, P80 and F80. Values of R2 above 0.9 represent good fit between the response (ESE) and predictor variables (To, P80 and F80) in the regression. In other words, it leads us to conclude that the ESE values can be explained by our sample data. ln(a), b and c are the final coefficient values of the blasting as determined by the Excel regression analysis tool. Finally, Mean ESE, Mean F80 and Mean P80 values are the sample means of ESE, F80 and P80, respectively, over the entire 10 observations in each scenario. These regression coefficients can be entered into the Reverse Blasting submodel to determine the burden and spacing for a desired P80 at a pit, given its F80, height of shot, mean explosive energy, mean tensile strength and mean ore specific gravity. The Reverse Blasting model has two parts: (1) Error and (2) Record Burden, Spacing. Error is used to introduce variation using uniform distribution in explosive energy, tensile strength and specific gravity. This distribution can be other than uniform, such as gamma, exponential or normal. A number of replications of simulation can be run wherein for each replication the product of burden and spacing is recorded. If there is a specific relationship between burden

Figure 2 Arena simulation of cost savings versus specific explosive energy for two rock types.

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Technical Papers and spacing, then the mean burden as well as mean spacing can be obtained individually based on their recorded values. Truckload P80 values are inputs from the Forward Blasting submodel to the Crusher to Ball Mill submodel that serve to simulate truck haulage, crusher, stockpile and grinder operations. With F80, power settings (with uncertainties) and corresponding Bond’s Work Indices, the Crusher to Ball Mill submodel can be used to estimate P80 at crusher and grinders. In summary, Forward Blasting is used to predict blast fragmentation, Reverse Blasting can be used to determine burden and spacing for a target blast fragmentation using coefficients determined through Forward Blasting and regression, and Crusher to Ball Mill is used to estimate P80 as well as production at each comminution process.

Results An experiment was run to demonstrate how cost savings could be achieved for hard and soft rocks by obtaining their optimum ESE values. To do this, the developed simulation model was used, and the errors earlier included in blasting parameters such as burden and spacing were considered here as well. In addition to that, a percentage error of uniform distribution — UNIF (4.8, 5.2) — was considered for each comminution power setting. Ten replications were performed to determine the mean power consumption and confidence interval at different comminution stages.

Table 5 Sensitivity analysis of reductions in percentage cost savings with increases in mean power consumptions for soft rock at different ESE values. ESE (kJ/t)

At crusher

At SAG mill

At ball mill

3%

5%

3%

5%

3%

5%

711

0.49

0.82

0.77

1.29

1.48

2.46

837

0.47

0.79

0.72

1.19

1.42

2.37

1,046

0.45

0.75

0.69

1.15

1.43

2.38

1,423

0.43

0.71

0.68

1.13

1.47

2.46

Table 6

Conclusions and future work

Sensitivity analysis of reductions in percentage cost savings with increases in mean power consumptions for hard rock at different ESE values. ESE (kJ/t)

At crusher

At SAG mill

At ball mill

3%

5%

3%

5%

3%

5%

711

0.49

0.81

0.77

1.29

1.48

2.46

837

0.46

0.77

0.71

1.18

1.41

2.36

1,046

0.44

0.73

0.68

1.14

1.42

2.37

1,423

0.4

0.67

0.66

1.11

1.46

2.43

1,778

0.38

0.62

0.66

1.1

1.49

2.49

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The initial F80 value at the blast site and other blasting parameters were borrowed from a real mine. Thereafter, as detailed P80 data by rock type at each stage were unavailable, they were calculated for an assumed reduction ratio at each stage. The reduction ratio at a particular stage was defined in this work as the ratio of that stage’s F80 to its P80. The potential cost savings as a function of blast energy is shown in Fig. 2, where the cost savings were obtained from comparison with the cost incurred in the base ESE scenario of 669 kJ/t. The curves for hard and soft rocks, respectively, show the ranges of optimum ESE. It is noticed that the percentage cost savings for hard rocks are larger and last for longer ESE than those for soft rocks. This result suggests that tracking power consumptions by rock type in a real mine will help mine operators choose the optimum ESE for each rock type in order to achieve maximum cost savings. We also analyzed the sensitivity of percentage cost savings to increases in mean power consumption at the crusher, SAG mill and ball mill. In real mines, these fluctuations could be due to reduced efficiencies of crushers or mills as they age so that, effectively, for the same reduction ratio they could be consuming more power. The reductions in percentage cost savings for hard and soft rocks are provided in Tables 5 and 6, respectively, for the extreme values of ESE in Fig. 2 and some ESE values in between the extremes. This analysis was performed for the crusher, SAG mill and ball mill, one at a time. It is observed that the mean percentage cost savings are more sensitive to changes in mean power consumptions at the SAG and ball mills than at the crusher for both rock types. Also, at the crusher and SAG mill, for increasing values of ESE, it is noted that the reductions were decreasing for the same percentage increase in mean power consumption. In contrast, at the ball mill, the extreme ESE values had greater reductions in percentage cost savings than the in-between ESE values. This could be due to the fact that at the extreme ESE values, the ball mill had relatively greater share of total cost incurred than at the in-between ESE values, unlike the crusher and SAG mill. If a target P80 is unavailable, the optimum ESE values can be entered into the Reverse Blasting model to determine the target P80 values in addition to determining the optimum burden and spacing, given other blasting parameters, by playing with different combinations of these design parameters that satisfy the relationship in Eq. (2).

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This paper proposes a framework that incorporates regression and discrete-event simulation for obtaining the optimum blast design for a hard-rock mine. A medium-sized hard-rock mine with two rock types was considered as a case study. The rock types were assumed to have different hardnesses, so they were classified into hard and soft rocks. An Arena simulation of the material-handling network, from blasting to ball mill, of the mine was used along with Excel to estimate the coefficients of the blasting regression model. In experiments using the Forward Blasting and Crusher to Ball Mill submodels, optimum ESE was obtained for each rock type. It is noted that these optimum values differ by rock type. They can be entered into the Reverse Blasting model to obtain the optimum blast design parameters. The work in this paper is a preliminary study before the www.miningengineeringmagazine.com

Technical Papers proposed framework is tested in a field application. It will be integrated with real data for all its parameters for the purpose of validation and deployment. Overall mine cost, including drilling costs, loading costs and increased haulage costs due to oversized fragments, was not studied in this paper. Nevertheless, the optimum blasting parameters could be obtained by a trial-and-error procedure using the proposed framework after including oversized fragment costs. Using this framework to study the effects of optimizing blasting on overall mine cost will be a future extension of this paper. In addition, although percentage cost savings were determined for a range of ESE values, some of the values above a threshold may result in energy saturation wherein the extra ESE does not result in size reduction but just imparts projectile velocity to the fragments. The authors believe that during field application of the proposed framework such behavior can be modeled based on sufficient real data. ■

Acknowledgment This work was supported by the National Science Foundation STTR Phase I project (Award#: 1332130).

References

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engineering formulae,” International Journal of Blasting and Fragmentation, Vol. 4, No. 3-4, pp. 198-207. Ercelebi, S.G., and Bascetin, A., 2009, “Optimization of shovel-truck system for surface mining,” SIAMM - Journal of The South African Institute of Mining and Metallurgy, Vol. 109, No. 7, pp. 433-439. Harvey, P.R., 1964, Analysis of Production Capabilities, APCOM. Julius Kruttschnitt Mineral Research Centre, 1996, “Open Pit Blast Design: Analysis and Optimization.” Kim, K.M., and Kemeny, J., 2011a, “Site specific blasting model using Schmidt tensile strength and image analysis,” Proceedings of ISEE 37th Annual Conference on Explosive and Blasting Technique, Feb. 6-9, 2011, San Diego, CA. Kim, K., and Kemeny, J., 2011b, “Site specific blasting model for mine-to-mill optimization,” Transactions of the Society for Mining, Metallurgy and Engineering, Vol. 330. Koenigsberg, E., 1958, “Cyclic queues,” Operational Research Quarterly, Vol. 9, No. 1, pp. 22-35. Muduli, P.K., and Yegulalp, T.M., 1996, “Modeling truck–shovel systems as closed queueing network with multiple job classes,” International Transactions in Operational Research, Vol. 3, No. 1, pp. 89-98. Nielsen, K., and Lownds, C.M., 1997, “Enhancement of taconite crushing and grinding through primary blasting,” International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 3-4, pp. 226e1-226e14. Nielsen, K. and Malvik,T., 1999, “Grindaility enhancement by blast-induced microcracks,” Powder Technology, Vol. 105, No. 1-3, pp 52-56. Rist, K., 1961, ‘’The solution of a transportation problem by use of a Monte Carlo technique,” Proceedings of the 1st International Symposium on Computer Application in Mining, Tucson, Arizona, APCOM. Sturgul, J.R., 1999, “Discrete mine system simulation in the United States,” International Journal of Surface Mining, Reclamation and Environment, Vol. 13, No. 2, pp. 37-41. Ta, C.H., Ingolfsson, A., and Doucette, J., 2010, “Haul Truck Allocation via Queueing Theory,” submitted to European Journal of Operations Research. Trivedi, R., Rai, P., and Nath, R., 1999, “Shovel-truck optimization study in an opencast mine - A queueing approach,” Indian Journal of Engineering and Materials Sciences, Vol. 6, No. 3, pp. 153-157.

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