Kro Fthollhzdeh,Elhm Mrdneh,Mehmet Cigl,Mohmmd Wqr Ali Asd,*

a Western Australian School of Mines:Minerals,Energy and Chemical Engineering,Curtin University,Kalgoorlie 6430,Australia

b School of Electrical Engineering,Computing and Mathematical Sciences,Curtin University,Perth 6845,Australia

Keywords:Mine planning Grade Engineering Optimization Scheduling Stockpiling Exact methods

ABSTRACT This paper presents the development and implementation of an innovative mixed integer programming based mathematical model for an open pit mining operation with Grade Engineering framework.Grade Engineering comprises a range of coarse-separation based pre-processing techniques that separate the desirable(i.e.high-grade)and undesirable(i.e.low-grade or uneconomic)materials and ensure the delivery of only selected quantity of high quality(or high-grade)material to energy,water,and cost-intensive processing plant.The model maximizes the net present value under a range of operational and processing constraints.Given that the proposed model is computationally complex,the authors employ a data preprocessing procedure and then evaluate the performance of the model at several practical instances using computation time,optimality gap,and the net present value as valid measures.In addition,a comparison of the proposed and traditional (without Grade Engineering) models reflects that the proposed model outperforms the traditional formulation.

1.Introduction

Open pit mine production scheduling (OPMPS) is a strategic decision-making problem that seeks to define the optimal sequence of material extraction and the flow of these materials within the mineral value chain(i.e.processing streams,stockpiles,and waste dumps)of an operation(Fig.1).An OPMPS model maximizes the net present value (NPV) and satisfies reserve,mining precedence,production (mine and processing plant) capacity,grade control,and stockpile handling constraints [1].Therefore,the model accounts for the following inputs:(1) geological or orebody block model that is a collection of identical-sized mining blocks with quality (grade) and quantity (tonnes) attributes allocated on a block-by-block basis,(2) economic (commodity price,operating and fixed costs,and discount rate,etc.),(3) technical(slope requirements,metallurgical recoveries,etc.),and (4) operational (mine,processing plant,and stockpiling) capacities.

Within the framework in Fig.1,the reserve constraint ensures that the production from a mining block may not exceed the reserves available in this block.Slope or precedence constraint requires the extraction of a set of overlying blocks before gaining access to blocks located at lower levels (or benches).Accordingly,for a 45° pit slope angle at 360° azimuth,the slope constraint applies an extraction precedence of five overlying blocks for a block located at the next bench.Production capacity constraints in OPMPS reflect the limitations of equipment capacity to extract material from the orebody and then sending valuable mineral to the processing streams during each period or year.More specifically,the mine production capacity enforces limitations on the quantity of waste and valuable material (ore) that should be extracted from the pit and processing capacity enforces limitations on the quantity of ore that should be fed to processing streams during a given year.Stockpiles keep excess valuable mineral until processing streams becomes available or serve as a storage for low-grade ore for future processing as it helps the processing streams running continuously with low extraction rate [2,3].

Since the 1960s,researchers and mine planners have addressed the complex mining industry problems.Accordingly,the OPMPS framework has remained relevant to the industry needs as it evolved over the years from a simple system that captured only mine,processing plant,waste dump,and market [4] to a complex framework with multiple sources and destinations within the system[5–7]presented in Fig.1.Currently,in a boom-and-bust environment,global mining industry looks for innovative methods towards sustainable operations.In this context,apart from responsible waste handling in open pit mines [8],one of the important aspects is to save water and energy costs associated to the processing streams that convert the raw ore from the mines and the stockpiles into the customer-specified metallic resources.Preseparation or pre-concentration is among the strategies implemented to achieve these savings.It separates high and low grades ores(and other undesirable materials) and allows the flow of only deserving materials(i.e.high grade ores)to the expensive,energyconsuming processing streams.

To this end,Cooperative Research Centre for Optimising Resource Extraction (CRC ORE) developed Grade Engineering (GE)that includes three main coarse-separation based pre-processing or pre-concentration techniques,i.e.screening by natural deportment,differential blasting,and bulk sorting.GE suggests that some of the ore types exhibit a natural property that causes valuable minerals to concentrate in fine(or course)size fractions when subjected to blasting,crushing,or screening operations[9].Exploiting this material property,CRC ORE focuses on improving the productivity while minimizing energy and water footprints of mining operations [10].Therefore,aligning with the industry requirements,Fig.2 presents an updated OPMPS framework that introduces these GE techniques into the traditional framework given in Fig.1.As illustrated in Fig.2,GE plant consists of the three GE techniques and the material coming from an individual block in the open pit may be pre-processed in one of the GE techniques for separation into fine and coarse materials.In this case (Fig.2),fine(under-size)material is considered as high-grade and it is sent to the processing plant or the heap leach whereas coarse(oversize)material is considered low-grade that is destined either to the heap leach or the waste dump.The major improvement from GE plant is derived through a reduction in the processing cost by rejecting low-valued(or uneconomic)portion of the ore that was previously(Fig.1)set to be processed at the plant.The rejected portion is then directed to the destinations such as heap leach or waste dump.

CRC ORE employs response rank (RR) system that defines the coarse-separation potential of the material attribute (in a mining block) to a particular GE technique.Accordingly,RRof a material attribute classifies or ranks this material as to which of the GE techniques would be more beneficial.A higherRRsuggests the stronger natural grade by size response to a particular GE technique[9,11].RRand the corresponding under-size(or accept)mass(m) are the geo-metallurgical parameters defined using a series of laboratory scale tests on drill-core or blast-hole samples (for details refer to [12,13]).Given these parameters,Eq.(1) derives response factor (RF) as follows [14].

GivenRF,ifginputrepresents the original grade of a material attribute,then new grade value of the under-size (fine) material,gup,may be derived using Eq.(2).

Finally,Eq.(3) generates the new grade value of the oversize(coarse) material,gdown,as

Table 1 illustrates an example of this GE concept presented in Eqs.(1)to(3).According to this example,a mining block in a large orebody block model constitutes gold (Au:1.5 g per tonne) and copper (Cu:0.23%) as material attributes with correspondingRRto screening along with under-size(or accept)mass(m)applicable to various screen sizes.Given this information,Table 1 demonstrates the calculated valuesRF,gup,andgdownusing Eqs.(1) to(3).Similar calculations are performed for the differential blasting and the bulk sorting as GE techniques.

Based on the calculated up-grade and down-grade values in Table 1,the ore will be sent to the most profitable destination while satisfying the operational and technical constraints.This highlights the importance of the proposed framework in Fig.2.More specifically,in this example,given the response rank associated to the mining block material attributes (RR=120 for Au andRR=80 for Cu) for screening,the total quantity of material within this block is sent to the GE plant(i.e.screening)for pre-processing or separation into under-size (up-grade) and oversize (downgrade) materials.Thus,only a selected (i.e.up-grade or highgrade in this case;using Eqs.(1) and (2) for screen size of 30 mm that achieves 20% accept (under-size) mass and yields Au grade of 3.94 g per tonne and Cu grade of 0.571%) quantity of material will be sent to the processing plant and the remaining (oversize low-grade in this case;using Eqs.(2) and (3) for screen size of 30 mm that achieves 20%accept mass(i.e.retains 80%as oversize)and yields Au grade of 0.89 g per tonne and Cu grade of 0.232%)will be sent to the heap leach,stockpiles,or waste dump (if not economical).Thus,pre-separation or pre-concentration through screening in the GE plant ensures less energy consumption (because only a selected‘‘high-grade”material is processed in the processing plant) as opposed to the traditional framework in Fig.1 that sends the total quantity of material within this block (at Au grade of 1.5 g per tonne and Cu grade of 0.23%) to the processing plant requiring excessive water and energy consumption.

Fig.2.A framework of an open-pit mining operation with GE techniques.Note that HG is the high grade and LG is the low grade.

Table 1 Example of screening by natural deportment.

With this background,the authors present the development and implementation of a new mathematical model that incorporates GE techniques into the production scheduling problem.This adds to the computational complexity of the proposed model and therefore the authors introduce preprocessing of the data to reduce the number of decision variables and possible routes to increase the efficiency of the proposed model.The performance of the new model is demonstrated by its implementation using a realistic case study and then by comparing the results with the traditional OPMPS (without GE).

In the mining literature,there is no published research that reveals mathematical formulation of OPMPS with GE (OPMPS+GE).However,the GE framework in Fig.2 relates to the recent studies in the context of simultaneous optimisation of mining complexes with multiple mines,material attributes,stockpiles,processing streams,and products [15–22].With the exception of these selected studies,other researches addressed the traditional OPMPS (i.e.without multiple processing streams) and this body of research can be classified into two main categories:the OPMPS with and without stockpile option.As the research herein is about OPMPS+GE with stockpiling policy,the authors describe the relevant literature on mining complexes as well as the traditional OPMPS with stockpile options.For those readers who are interested in the OPMPS without stockpile options,the authors recommend Caccetta and Hill [4],Boland et al.[23],Askari-Nasab et al.[24],and Jélvez et al.[25].In addition,Osanloo et al.[26],Newman et al.[27],and Lamghari [28] present a review of such OPMPS models.

In the context of mining complexes models that have relevance to GE,Montiel and Dimitrakopoulos[15,16,19]share mathematical models and their applications that account for multiple processing streams,geological uncertainty,and geo-metallurgical properties of the diverse material attributes within the system.More specifically,the geo-metallurgical properties in these models define the suitability of material attributes to multiple processing(or operating) and transportation alternatives.However,their models ignored a variation in metal recoveries,energy,and steel consumption applicable to diverse material types moving from the mine to different stages of the processing alternatives.Redwood and Scott[9] shared an implementation of a commercial mining software[29] to develop a production schedule considering GE techniques.While the report proves the worth of considering GE,it does not present the intricacies of the formulation and its implementation.As opposed to a mixed-integer programming (MIP) based OPMPS model,Espejel et al.[11] relied on Lane’s model [30,31] for GE framework.Thus,this application ignored the block-by-block delineation of the orebody because Lane’s model takes gradetonnage distribution as the geological input.More recently,Saliba and Dimitrakopoulos [20] proposed a mathematical model for a mining complex (multiple mines,processing options,stockpiles,and waste dumps) in the presence of grade and price uncertainty.The authors employed a combination of themeta-heuristic algorithms (simulated annealing and particle swarm optimization) to implement the proposed model at a gold mine complex that included two mines,three processing destinations,a waste dump,and a set of stockpiles for various materials.

Similarly,within the domain of OPMPS with stockpiling,Bley et al.[32]introduced the OPMPS with basic non-linear formulation for stockpiles.The authors also applied warehouse structure into stockpile formulation and then branch-and-bound method was employed to solve four linear programming relaxations.Topal and Ramazan[33]proposed a linear programming model that considered stockpiles,processing streams and shipping facilities within mineral value chain.The proposed model used continuous variables by employing grade-tonnage distribution of the mineral resource as it offered efficiency in solving the model through an exact approach.Ramazan and Dimitrakopoulos [34] presented stochastic model of OPMPS with stockpiling of the overproduced ore under geological and economic value uncertainties.Similarly,Koushavand et al.[35] proposed a mixed-integer linear programming(MILP)model considering grade uncertainty with stockpiling option.In this study,multi-objective function was employed to maximize the NPV and minimize the cost of uncertainty,respectively.The authors presented clustering technique to reduce number of decision variables and assumed that the stockpile initially has its grade set to mitigate uncertainty,i.e.overproduction can be stored for the next period.Koushavand et al.[35] also pointed out that ideally average grade of the material calculation in stockpiles leads to non-linearity and defined multiple stockpile bins to avoid this non-linearity of the stockpile formulation.Kumar and Chatterjee [36] employed a branch and cut algorithm to solve the OPMPS with stockpile option that considered blending regime in an open pit coal mining operation.The planning strategy was based on a year-wise formulation approach that derived heuristic solution,i.e.it solved the problem sequentially on a year-by-year basis,allowing an update in the formulation through a deduction of the relevant inputs from the previous years.Moreno et al.[37]proposed four linear approximations of the basic and modified warehouse formulations in Bley et al.[38].First,an upper bound model on the warehouse formulation addressed non-linearity by ignoring blending of the stockpiled material,i.e.it assumed retention of the original grade from the mine to the stockpile to the processing plant;however,this strategy offered infeasible solution to the basic and warehouse models.Second,a lower bound model addressed non-linearity by assuming a predefined average grade of the material flowing from the stockpile to the processing plant.This strategy maintained the average grade of the outgoing material as the minimum grade for the material incoming to stockpiles,a compromise that undervalued some of the material going to the processing plant.Third,a grade bin model resolved non-linearity by managing multiple grade bins within the stockpiles.This strategy exercised grade control through an average grade calculated over the minimum and the maximum grade of the materials flowing from each stockpile bin to the processing plant.Fourth,the average lower bound model addressed non-linearity in Bley et al.[38] formulation by mandating a minimum average grade for material flowing from the mine to the stockpile.However,numerical results proved the worth of these linear approximations as the difference in objective function values remained insignificant as compared to the values from the non-linear formulation.Fu et al.[39] proposed a linear stockpile formulation that offered simultaneous solution to both pit and waste dump scheduling.This model minimized the material haulage cost by respecting mining,production capacity,waste dump precedence,waste dump capacity as well as waste dump environmental constraints.The structure of the stockpiles in this model aligns with the third linear approximation in Moreno et al.[37].An application of this model reflected better results as compared to the traditional waste dump scheduling strategies.More recently,Rezakhah et al.[2] presented an extension in Moreno et al.[37]that accounts for the strict contamination limits as part of the blending for the processing streams.The model facilitated an objective function value-based comparison with respect to(1)stockpiling without contamination considerations,and (2) contamination considerations without stockpiling.An implementation of this MILP model revealed significant differences in the optimal production schedule with and without contamination considerations.In addition,Rezakhah and Newman [40] investigated the impact of materials degradation(due to a long-term exposure to the environment)in the stockpiles that compromises mineral recoveries from the processing streams.An implementation of this complex but realistic model demonstrated a reduction in NPV due to degradation of the stockpiled materials.

With this background that confirms limitations of the previous studies in terms of GE framework (GE plant and GE stockpiles;as given in Fig.2),this study presents a novel mathematical model for the first time to simulate the integration of GE with the traditional OPMPS.The authors organize the remainder of this paper as follows.Section 2 presents the proposed MIP model for the GE framework in Fig.2.In Section 3,the authors present the data pre-processing rules to improve the efficiency of the proposed model and Section 4 demonstrates the computational efforts to solve and validate the proposed model.Section 5 is the authors’reflection through concluding remarks and a direction for future research.

2.Mathematical model

In Section 2,the new MIP model considers linear stockpile structure that aligns with previous studies of Moreno et al.[37],Rezakhah and Newman [40],and Fu et al.[39].More specifically,the proposed MIP model maximizes the NPV over the life of operation and it satisfies all operational and technical constraints such as maximum and minimum mining production and processing capacity,average grade constraints to control the feed grades within mineral value chain,recovery constraints that reflect on the response of the valuable mineral to GE techniques,slope constraints,stockpile constraints (both mining and GE),power and steel consumption and fixed-cost constraints.In addition,the structure of the proposed model utilizes the dynamic cut-off grade concept and considers maximum power and steel consumption limitations for the first time in the literature.

However,the proposed model has two important limitations.First,as opposed to the OPMPS models for mining complexes that consider the value of products generated,a complete mineral value chain (multiple mines,multiple processing streams,and multiple products delivered to the diverse customers),the proposed model is limited to the value of raw materials from a single pit(or mine)and ignores the value chain beyond (i.e.product shipping modes,ports,etc.) the open pit mining operation.Second,it is restricted to the deterministic version of OPMPS and ignores the uncertainties around economic (commodity or metal price) and geological(mining block grades) inputs.Therefore,it foregoes the benefits that may be derived through risk-quantified production plans generated using stochastic models [17,18,28].

Given the GE framework(see Fig.2),the following indices,sets,parameters,and variables contribute to the development of MIP formulation for the proposed OPMPS+GE model.Due to a large number of indices,parameters,and decision variables,the authors classify them according to various components of the value chain,as shown below.

The objective function(Eq.(4))aims to maximize the NPV of the mineral supply chain which include value of materials on each possible route and is defined as cumulative product of the economic value ($ per tonne) multiplied by the quantity (tonnes) of materials.

The reserve constraints (Eq.(5)) indicate that the quantity of material flowing from a mining block in the pit to the processing options,mining stockpile bins,waste dumps and GE techniques may not exceed the available quantity of material in that block.

Constraints (Eqs.(6) and (7)) guarantee that the quantity of materials fed from the pit to the various destinations (processing options,mining stockpile bins,GE techniques,GE stockpile bins,and waste dumps) remain within the allowable mining capacity.Constraints (Eqs.(8) and (9)) ensure that materials moving from the pit,mining stockpile bins,GE stockpile bins to processing options remain within the allowable processing capacity.Similarly,constraints (Eq.(10)) maintain the flow of materials through GE techniques within their production capacity.

Constraints (Eqs.(11) and (12)) maintain the quality of ore flowing to the processing streams or options.In this respect,each processing option requires ore between a minimum and maximum average(or head)grade.Accordingly,these constraints control the average grade of ore fed to the processing options.

Eqs.(13)–(16)apply the GE concepts outlined in Section 1(Eqs.(1)–(3);Table 1).Therefore,these constraints control the flow of high-grade ore to the expensive processing options (i.e.flotation in Fig.2) and low-grade ore to the relatively low-cost processing option (i.e.heap leaching in Fig.2).

Constraints(Eqs.(17)and(18))satisfy the precedence(or slope)constraints within the open pit.

Eq.(19) guarantees that the on-hand materials in the mining stockpile bins at the end of current period(except first period;t≥2)is equal to the quantity of the materials in the mining stockpile bins at the end of previous period plus the total quantity of materials moving from the pit to the mining stockpile bins at the current period minus the total quantity of materials fed from the mining stockpile bins to the GE type and the processing options,waste dump or GE stockpile bins at the current period.Eq.(20) is analogue to Eq.(19) for the first period.Constraint (Eq.(21))ensures that materials in the mining stockpile bins at the end of each period remains within the allowable mining stockpile bins capacity.

Eqs.(22)guarantees that the on-hand materials in the GE stockpile bins at the end of current period (except first period;t≥2) is equal to the quantity of the materials in the GE stockpile bins at the end of previous period plus the total quantity of materials moving from the pit to the GE type and then to the GE stockpile bins at the current period minus the total quantity of materials fed from the GE stockpile bins to the processing options at the current period.Eq.(23)is analogue to Eq.(22)for the first time period.Constraint(Eq.(24))ensures that materials in the GE stockpile bins at the end of each time period remain within the allowable GE stockpile bins capacity.

Inequalities (Eqs.(25) and (26)) guarantee that the power and steel consumption in various stages (e.g.SAG mill,ball mill,and flotation) of the processing plant remain within the available power and steel for these stages.

Constraints (Eqs.(27)–(32)) or administrative costs for mining,processing plant and GE components of the operation.As opposed to the operating costs,fixed costs apply on a yearly(i.e.$per year)basis,and accordingly,these constraints incur relevant fixed costs if mine GE or processing plants produce or process even a single tonne of material.

3.Pre-processing

The aim of data pre-processing is to enhance the contextually relevant information(i.e.exclude excess or irrelevant information)that leads to a reduction in the number of variables [41].In the OPMPS literature,normally model pre-processing is based on slope and capacity limitations which impose early start and late start calculations for each mining block[42–44].However,in this study the authors introduce a new pre-processing aspect that is based on suitability and stockpiling.An extensive analysis of the orebody model and the structure of the stockpiles [39] in the context of GE framework allows the following suitability rules in the developed model:(1)a suitable stockpile binSbexists for material from the blockb∈B;(2) a suitable GE techniqueGbexists for material from the blockb∈B;(3) a suitable processing optionPbexists for material from the blockb∈B;(4)a suitable GE techniqueGsexists for material from the stockpile bins∈S;(5) a suitable processing optionPsexists for material from the stockpile bins∈S;(6)a suitable GE stockpile binS’gexists for material from the GE techniqueg∈G;(7) a suitable processing optionPgexists for material from the GE techniqueg∈G;and (8) a suitable processing optionPs’exists for material from the GE stockpile bins’∈S’.

These rules eliminate unnecessary routes for flow of materials within the GE framework.Moreover,by introducing these rules,the authors avoid non-linear formulation for stockpiling given in Bley et al.[38] and rely on the stockpiling strategies (i.e.define multiple stockpile grade bins based on the material attribute and grade ranges available within the orebody model) introduced in Moreno et al.[37] and Fu et al.[39] as they are widely used in the literature as well as practiced in the mining industry.Therefore,these pre-processing rules facilitate the resolution of computational complexity associated to the proposed model without a compromise on the practical aspects of mining operations.Regarding OPMPS+GE framework,there are two types of stockpiles:(1)mining stockpile in which material moves from the mine to stockpile bins without any processing in pre-concentration (GE) techniques,and (2) GE stockpile in which the materials after processing from suitable GE technique moves to GE stockpile bins.Tables 2 and 3 present an example of the linear structure in mining and GE stockpile bins.The bins are classified based on a grade range within each material attribute (Au and Cu) and rock type(A,B,and C).

Suppose that blockbhas a grade 0.47 g per tonne of Au and 0.37%of Cu and exists in rock type C.Based on Table 2,the material from blockbmay only be sent to bin 6 (i.e.the suitable stockpile bin for blockbiss=6) and,accordingly,all other stockpile bins become irrelevant to blockb(i.e.=0).Similarly,let’s consider that after processing through GE plant,the grade of blockbis upgraded to 0.85 g per tonne of Au,and 0.81% of Cu within rock type C.Based on Table 3,the material from blockbmay be sent to GE stockpile bin 9 (i.e.the suitable GE stockpile bin for blockbiss=9) and,accordingly,all other GE stockpile bins become irrelevant to blockb(i.e.=0).

Table 2 Mining stockpile grade bins.

Table 3 GE stockpile grade bins.

Among the other types of suitability rules in this research,material suitability to GE techniques is important,where material from each block or mining stockpile bin based on its attribute has a specific RR to GE techniques.In Table 4,the authors present an example of the RR for GE techniques (differential blasting,screening and bulk sorting) based on material attribute (Au and Cu)within rock types A,B,and C.According to Table 4,there is no RR for both material attributes in rock types A and C to the differential blasting.However,the differential blasting will be suitable GE technique for any blockbthat exists in rock type B for both Au and Cu,where it has an RR of 150 and 130,respectively.

4.Numerical results

Section 4 presents an implementation of the proposed OPMPS+GE model using six different instances (with increasing number of mining blocks) of the modified Marvin orebody model reported in Redwood and Scott [9].In addition,a comparison of the proposed and traditional (without GE framework in Fig.1 models reflects the value of incorporating GE techniques in a production scheduling model.Each mining block(10 m×10 m×10 m)in this orebody model constitutes copper (Cu) and gold (Au) as material attributes.The orebody exists in oxide (OX),fresh (FR) and transitional(TR)layers within two separate rock domains(1 and 2).Note that the proposed model considers all available blocks in the orebody model as an input (as opposed to subset of blocks generated through ultimate pit limit or phase/pushback analysis as practiced in the mining industry [29]).

The GE framework in this case study includes 72 possible grade bins in mining stockpiles,2 waste dump,4 bulk sorting options(high-grade mass recovery 20%,high-grade mass recovery 40%,high-grade mass recovery 60%,high-grade mass recovery 80%;i.e.as indicated in Fig.2 and explained through an example in Table 1,the bulk sorting option 1 results in 20%high-grade ore destined to flotation and 80%low-grade ore destined to heap-leach),1 differential blasting,5 screening options (30,50,75,100,and 150 mm),307 possible grade bins in GE stockpile,2 heap leaching options and power and steel consumption along with material(Au and Cu) recoveries for 4 different size (75,106,150,and 200µm)options in processing plant stages (SAG mill,ball mill,and flotation).This leads to a complex network of 917 possible routes for the material moving from one of the mining blocks within the orebody model to various destinations within the GE framework(Fig.2).In contrast,the framework without GE (Fig.1) carries 366 possible routes.However,based on the suitability rules defined in Section 3,the number of suitable routes for a mining block would be less than the possible routes.Table 5 presents a comparison of the possible and suitable routes for each instance of the orebody model.

An application of the pre-processing rules suggested in Section 3 results in the reduction in number of decision variables (continuous and discrete).This enhances the suitability of the exact methodto solve all instances of the problem through coding in the General Algebraic Modelling System (GAMS) for application of CPLEX as MIP solver.Table 6 outlines the gains received from the preprocessing rules in the context of problems size(number of continuous and discrete variables as well as constraints)for both OPMPS+GE and without GE models.These models are then implemented using a personal computer with 2.9 GHz Intel®CoreTMi7-7820HQ processor and 32 GB RAM memory.The authors utilized the optimality gap along with maximum CPU time of 48 hr as the stopping criteria for solving these models.

Table 4 GE response rankings (RR) by coarse-separation process and material attribute.

Table 5 Number of possible and suitable routes for each block models.

Table 7 summarizes the computational effort required to solve each instance of the problem.As indicated,OPMPS+GE models derive relatively higher NPV (at a discount rate of 10%) as compared to the traditional framework.In addition,the exact method(CPLEX) generated an optimal solution for five instances only (64,130,543,5256,and 12142),as it could not offer an optimal solution for an instance of the problem with 20393 blocks within 48 hr (two days).Fig.3 is the graphical (three-and two-dimensional views)presentation of the optimal as well as practical plan derived for the largest instance of the problem (12142 blocks).

Table 8 illustrates a year-by-year summary of material flow within the OPMPS+GE framework for the largest instance of the problem (12142 blocks).As shown in Table 8,during year 1,the optimal plan schedules a flow of 3725814 tonne of ore from the mine to the processing options (Pit-PP),1665572 tonnes of ore from the mine to the GE techniques to the processing options(Pit-GE-PP),and 571600 tonnes of waste to the waste dumps(Pit-WD).In addition,the plan manages (storage and retrieval)potential ore at mining and GE stockpiles during year 1.It schedules 5254214 tonnes of material from the mine to the mining stockpile bins (Pit-Mst).Then,2825907 tonnes of this material from mining stockpile moves to the processing plant through GE techniques (Mst-GE-PP).The mining stockpile then retains 718364 tonnes(Cs)in its inventory(to be processed in the following years)as it sends the remaining 1709943(5254214-2825907-718364=1709943) tonnes to the GE stockpile after processing from the GE techniques (Mst-GE-GEst).

Table 6 Problem characteristics.

Table 7 Numerical results.

Table 8 Details of production plan OPMPS+GE for 12142 blocks.

Fig.4.Production plan OPMPS+GE for 12142 blocks.

It is evident that at the end of year 1,the optimal plan does not retain materials in GE stockpile (Css) because its full inventory(1709943 tonnes) of materials flows to the processing options(GEst-PP).Therefore,a total of 9,927,236 (3725814+1665572+2825907+1709943=9927236) tonnes of ore is scheduled for processing options during year 1 with 62.47% (1665572+2825907+1709943=6 201422;6201422÷9927236=62.47%)of this ore is routed through the GE techniques.It is also observed that during this planning horizon the optimal plan does not schedule flow of materials from:(1)the mine to GE techniques to waste dump (Pit-GE-WD);(2) mining stockpile to processing options(Mst-PP);(3) mining stockpile to GE techniques to waste dump(Mst-GE-WD);and (4) the mine to GE techniques to GE stockpiles(Pit-GE-GEst).Fig.4 reflects a graphical presentation of the optimal plan given in Table 8.

5.Conclusions

This study shares a mixed integer programming based mathematical formulation,namely OPMPS+GE,that integrates open pit mine production scheduling with GE.The proposed model considers screening,differential blasting,and bulk sorting as GE techniques within the mineral value chain.Followings are the key achievements and conclusions of this study.

(1) An application and validation of the model using modified Marvin data generates better optimal plans (up to 11.85%higher NPV) for the proposed OPMPS+GE model as compared to the traditional OPMPS framework.

(2) This confirms the value of incorporating GE based preconcentration techniques within an open pit mining operation framework.

(3) Data pre-processing strategies contribute significantly(up to 94% and 28% reduction in number of continuous variables and constraints,respectively) towards reducing the size of the problem.

However,the model is computationally complex and accordingly exact method has limited application.Therefore,future studies shall focus on the development and implementation ofmetaheuristic or hyper-heuristic methods to solve the large-scale realistic instances of the proposed problem.More specifically,the recent researches like Goodfellow and Dimitrakopoulos [17,18]and Lamghari and Dimitrakopoulos [21] are a guide and motivation towards the development of such solution methods as these studies have successfully implementedmeta-heuristic or hyperheuristic algorithms for stochastic version of the large scale OPMPS models.In addition,the proposed model may be extended in future to overcome the limitations identified in Section 2.

Acknowledgements

Cooperative Research Centre for Optimising Resource Extraction(CRC ORE)provided the funding(Project ID Code:P4-007;Curtin University Grant # 58994) for the work contributed to this paper.The collaboration between the authors would not have been possible without the financial support from CRC ORE.CRC ORE is part of the Australian Government’s CRC Program,which is made possible through the investment and ongoing support of the Australian Government.The CRC Program supports industry-led collaborations between industry,researchers and the community.The authors are thankful to the CRC ORE management (especially,Paul Ravel,Michael Scott and Luke Keeney) for their valuable collaboration and technical comments.