A stochastic optimization formulation for the transition from open pit to underground mining
 2.5k Downloads
 2 Citations
Abstract
As open pit mining of a mineral deposit deepens, the cost of extraction may increase up to a threshold where transitioning to mining through underground methods is more profitable. This paper provides an approach to determine an optimal depth at which a mine should transition from open pit to underground mining, based on managing technical risk. The value of a set of candidate transition depths is calculated by optimizing the production schedules for each depth’s unique open pit and underground operations which provide yearly discounted cash flow projections. By considering the sum of the open pit and underground mining portion’s value, the most profitable candidate transition depth is identified. The optimization model presented is based on a stochastic integer program that integrates geological uncertainty and manages technical risk. The proposed approach is tested on a gold deposit. Results show the benefits of managing geological uncertainty in longterm strategic decisionmaking frameworks. Additionally, the stochastic result produces a 9% net present value increase over a similar deterministic formulation. The riskmanaging stochastic framework also produces operational schedules that reduce a mining project`s susceptibility to geological risk. This work aims to approve on previous attempts to solve this problem by jointly considering geological uncertainty and describing the optimal transition depth effectively in 3dimensions.
Keywords
Mine production scheduling Stochastic optimization Stochastic mine planning1 Introduction
The transition from open pit (OP) to underground (UG) methods requires a large capital cost for development and potential delays in production but can provide access to a large supply of reserves and subsequently extend a mine’s life. Additionally, an operating mine may benefit from such a transition because of the opportunity to utilize existing infrastructure and equipment, particularly when in a remote location. Optimization approaches towards the open pit to underground transition decision (or OPUG) may commence with discretizing the space above and below ground into selective units. For surface mining, material is typically discretized into mining blocks, while underground material is frequently grouped into stopes of varying size depending on the mining method chosen. From there and through production scheduling optimization, the interaction between the OP and UG components can be modeled to realistically value the asset under study.
Historically, operations research efforts in mine planning have been focused on open pits as opposed to underground operations. Most commonly, the open pit planning process begins by determining the ultimate pit limits, and the industry standard is the nested implementation of the Lerchs–Grossman’s algorithm (Lerchs and Grossmann 1965; Whittle 1988, 1999). This algorithm utilizes a maximum closure concept to determine optimal pit limits, and a nested implementation facilitates economic discounting. For underground mine planning, optimization techniques are less advanced than those employed for open pit mines and heavily depend on the mining method used. In practice, longterm underground planning is divided into two phases: stope design and production sequencing. For stope design methods, the floating stope algorithm (Alford 1995) is the oldest computerized design tool available, although not an optimization algorithm. Mine optimization research has developed methods that schedule the extraction of discretized units in underground mines (e.g. Trout 1995; Nehring and Topal 2007) based on mixed integer programming (MIP) approaches. Nehring et al. (2009), Little and Topal (2011), and Musingwini (2016) extend MIP approaches to reduce the solution times by combining decision variables and also extend application. More recent are the efforts to develop geological riskbased optimization approaches for stope design and production sequencing; these have been shown to provide substantial advantages, including more reliable forecasts, increased metal production and higher cash flows (Bootsma et al. 2014; Carpentier et al. 2016).
Some of the world’s largest mines are expected to reach their ultimate pit in the next 15 years (Kjetland 2012). Despite the importance of the topic, there is no wellestablished algorithm to simultaneously generate an optimal mine plan that outlines the transition from open pit mining to underground (Fuentes and Caceres 2004) or approaches that can address the topic of technical risk management, similarly to approaches for open pit mining (e.g. Godoy and Dimitrakopoulos 2004; Montiel and Dimitrakopoulos 2015; Goodfellow and Dimitrakopoulos 2016; Montiel et al. 2016). The first attempt to address the OP to UG transition was made by Popov (1971), while more recently, a movement towards applying optimization techniques has been made starting with Bakhtavar et al. (2008) who present a heuristic method that compares the economic value of mine blocks when extracted through OP versus their value when extracted by UG techniques. The method iterates progressively downwards through a deposit, concluding that the optimal transition is the depth reached when the value of a block mined by UG methods exceeds the corresponding OP mining value. A major drawback of this method is that it provides a transition depth only described in two dimensions, which is unrealistic from a practical standpoint. An effort is presented in Newman et al. (2013) where the transition depth is formulated as a longestpath network flow. Each path within the network has a unique extraction sequence, a transition depth and a corresponding net present value (NPV). A major limitation of this development is again that it amounts to a 2D solution of what is a 3D problem, as the orebody is discretized into horizontal strata for the above and below ground mining components. At the same time a worstcase benchwise mining schedule is adopted for open pit production and a bottomup schedule for the underground block caving component of the mine. These highly constrained mining benchwise progressions have been demonstrated to be far from optimal (Whittle 1988) and are rarely implemented in practice. More realistic selective mining units and an optimized schedule can provide a more accurate representation of a mine’s value, and this is the approach taken by Dagdelen and Traore (2014) who further extend this OP to UG transition idea to the context of a mining complex. In this work, the authors investigate the transition decision at a currently operating open pit mine that exists within the context of a mining complex that is comprised of five producing open pits, four stockpiles and one processing plant. Dagdelen and Traore (2014) take an iterative approach by evaluating a set of selected transition depths through optimizing the lifeofmine production schedules of both the open pit and underground mines using mixed linear integer programming techniques. The authors begin by using Geovia’s Whittle software (Geovia 2012) to generate a series of pits which provide an ultimate pit contour. The crown pillar, a large portion of undisturbed host material that serves as protection between the lowest OP working and the highest UG levels, is located below the ultimate pit. The location of the ultimate pit and crown pillar provide a basis for the underground mine design. Optimized lifeofmine production schedules are then created to determine yearly cash flow and resulting NPV. This procedure is repeated for progressively deeper transition depths until the NPV observed in the current iteration is less than what was seen for a previously considered transition depth, at which point the authors conclude that the previously considered depth, with a higher NPV, is optimal.
All the above mentioned attempts to optimize the OPUG transition depth fail to consider geological uncertainty, a major cause of failure in mining projects (Vallee 2000). Stochastic optimizers integrate and manage space dependent geological uncertainty (grades, material types, metal, and pertinent rock properties) in the scheduling process, based on its quantification with geostatistical or stochastic simulation methods (e.g. Goovaerts 1997; Soares et al. 2017; Zagayevskiy and Deutsch 2016). Such scheduling optimizers have been long shown to increase the net present value of an operation, while providing a schedule that defers risk and has a high probability of meeting metal production and cash flow targets (Godoy 2003; Ramazan and Dimitrakopoulos 2005; Jewbali 2006; Kumral 2010; Albor and Dimitrakopoulos 2010; Goodfellow 2014; Montiel 2014; Gilani and Sattarvand 2016; and others). Implementing such frameworks is extremely valuable when making longterm strategic decisions because of their ability to accurately value assets.
In this paper, the financial viability of a set of candidate transition depths is evaluated in order to identify the most profitable transition depth. To generate an accurate projection of the yearly cash flows that each candidate transition depth is capable of generating, a yearly lifeofmine extraction schedule is produced for both the OP and UG components of the mine. A twostage stochastic integer programming (SIP) formulation for production scheduling is presented, which is similar to the work developed by Ramazan and Dimitrakopoulos (2005, 2013). The proposed method improves upon previous developments related to the OPUG transition problem by simultaneously incorporating geological uncertainty into the longterm decisionmaking while providing a transition depth described in three dimensions that can be implemented and understood by those who operate the mine.
In the following sections, the method of evaluating a set of preselected candidate transition depths to determine which is optimal is discussed. Then a stochastic integer programming formulation used to produce a longterm production schedule for each of the preselected candidate transition depths is presented. Finally, a field test of the proposed method is analyzed as the method is applied to a gold mine.
2 Method
2.1 The general set up: candidate transition depths
The method proposed herein to determine the transition depth from OP to UG mining is based on the discretization of the orebody space into different selective units and then accurately assessing the value of the OP and UG portions of the mine based on optimized yearly extraction sequences of these discretized units. More specifically, this leads to a set of several candidate transition depths being assessed in terms of value. The candidate depth that corresponds to the highest total discounted profit is then deemed optimal for the mine being considered. Stochastic integer programming (SIP) provides the required optimization framework to make an informed decision, as this optimizer considers stochastic representations of geological uncertainty while generating the OP and UG longterm production schedules that accurately predict discounted cash flows.
2.2 Stochastic integer programming: mine scheduling optimization
The proposed stochastic integer program (SIP) aims to maximize discounted cash flow and minimize deviations from key production targets while producing an extraction schedule that abides by the relevant constraints. The OP optimization produces a longterm schedule that outlines a yearly extraction sequence of mining blocks, while UG optimization adopts the same twostage stochastic programming approach for scheduling stope extraction. The formulation for both OP and UG scheduling are extremely similar, so only the OP formulation is shown. The only difference for the UG formulation is that stopes are being scheduled instead of blocks, and yearly metal is being constrained instead of yearly waste as seen in the OP formulation.
2.3 Developing riskmanagement based lifeofmine plans: open pit optimization formulation
The objective function for the OP SIP model shown in Eq. (1) maximizes discounted cash flows and minimizes deviations from targets, and is similar to that presented by Ramazan and Dimitrakopoulos (2013). Part 1 of the objective function contains firststage decision variables, \(b_{i}^{t}\) which govern what year a given block i is extracted within. These are scenarioindependent decision variables and the metal content of each block is uncertain at the time this decision is made. The terms in Part 1 of Eq. (1) represent the profits generated as a result of extracting certain blocks in a year and these profits are appropriately discounted based on which period they are realized in.
Part 2 of Eq. (1) contains secondstage decision variables that are used to manage the uncertainty in the ore supply during the optimization. These recourse variables (d) are decision variables determined once the geological uncertainty associated with each scenario has been unveiled. At this time, the gap above or below the mine’s annual ore and waste targets is known on a scenariodependent basis and these deviations are discouraged throughout the lifeofmine. This component of the objective function is important because it is reasonable to suggest that if a schedule markedly deviates from the yearly ore and waste targets, then it is unlikely that the projected NPV of the schedule will be realized throughout a mine’s life. Therefore, including these variables in the objective function and reducing deviations allows the SIP to produce a practical and feasible schedule along with cash flow projections that have a high probability of being achieved once production commences.

i is the block identifier;

t is a scheduling time period;

\(b_{i}^{t} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{Block}}\;i\;{\text{is}}\;{\text{mined}}\;{\text{through}}\;{\text{OP}}\;{\text{in}}\;{\text{period}}\;t;} \hfill \\ 0 \hfill & {\text{Otherwise}} \hfill \\ \end{array} } \right.\)

\(g_{i}^{s}\) grade of block i in orebody model s;

\(Rec\) is the mining and processing recovery of the operation;

T _{ i } is the weight of block i;

\(NR_{i} = T_{i} \times g_{i}^{s} \times Rec \times \left( {{\text{Price}}  {\text{Selling}}\;{\text{Cost}}} \right)\) is the net revenue generated by selling all the metal contained in block i in simulated orebody s;

MC _{ i } is the cost of mining block i;

PC _{ i } is the processing cost of block i;

\(E\left\{ {V_{i} } \right\} = \left\{ {\begin{array}{*{20}l} {NR_{i}  MC_{i}  PC_{i} } \hfill & {{\text{if}}\;NR_{i} > PC_{i} } \hfill \\ {  MC_{i} } \hfill & {{\text{if}}\;NR_{i} \le PC_{i} } \hfill \\ \end{array} } \right.\) is the economic value of a block i;

r is the discount rate;

\(E\left\{ {\left( {NPV_{i}^{t} } \right)} \right\} = \frac{{E\left\{ {V_{i}^{0} } \right\}}}{{\left( {1 + r} \right)^{t} }}\) is the expected NPV if the block i is mined in period t;

N is the number of selective mining units available for scheduling;

z is an identifier for the transition depth being considered;

P _{ z } is the number of production periods scheduled for candidate transition depth z.

s is a simulated orebody model;

S is the number of simulated orebody models;

w and o are target parameters, or type of production targets; w is for the waste target; o if for the ore production target;

u is the maximum target (upper bound);

l is the minimum target (lower bound);

\(d_{su}^{to} ,d_{su}^{tw}\) are the excessive amounts for the target parameters produced;

\(d_{sl}^{to} , d_{sl}^{tw}\) are the deficient amounts for the target parameters produced;

\(c_{u}^{to} ,c_{l}^{to} ,c_{u}^{tw} ,c_{l}^{tw}\) are unit costs for \(d_{su}^{to} ,d_{sl}^{to} ,d_{su}^{tw} ,d_{sl}^{tw}\) respectively in the optimization’s objective function.
OP Constraints

W _{ tar } is the targeted amount of waste material to be mined in a given period;

O _{ tar } is the targeted amount of ore material to be mined in a given period;

O _{ si } is the ore tonnage of block i in the orebody model s;

Q _{UG,tar} is the yearly metal production target during underground mining;

MCap _{ min } is the minimum amount of material required to be mined in a given period;

MCap _{ max } is the maximum amount of material that can possibly be mined in a given period;

l _{i} is the set of predecessor for block i.
ScenarioDependent
ScenarioIndependent
Constraints (2) and (3) are scenariodependent constraints that quantify the magnitude of deviation within each scenario from the waste and ore targets based on firststage decision variables (\(b_{i}^{t}\)). Constraints (4)–(6) contain only firststage decision variables (\(b_{i}^{t}\)) and thus are scenarioindependent. The precedence constraint (4) ensures that the optimizer mines the blocks overlying a specific block i before it can be considered for extraction. The reserve constraint (6) prevents the optimizer from mining a single block i more than once.
The size of OP mine scheduling applications cause computational issues when using commercial solvers since it can take long periods of time to arrive at or near an optimal solution, if able to solve (Lamghari et al. 2014). In order to overcome these issues, metaheuristics can be used. These are algorithms which efficiently search the solution space and have the proven ability to find high quality solutions in relatively small amounts of time (Ferland et al. 2007; Lamghari and Dimitrakopoulos 2012; Lamghari et al. 2014). To be effective these algorithms must be specifically tailored to match the nature of the problem being solved. In the context of mine production scheduling, the tabu search algorithm is well suited, and a parallel implementation is utilized here to schedule the open pit portion of the deposit for each transition depth that is considered (Lamghari and Dimitrakopoulos 2012; Senecal 2015). For more details on tabu search, the reader is referred to the Appendix.
2.4 Developing riskmanaging lifeofmine plans: underground optimization formulation
The UG scheduling formulation is very similar to the OP formulation. Both have objective functions which aim to maximize discounted profits, while minimizing deviations from key production targets. The UG objective function is similar to that proposed for the OP scheduling function in Eq. (1), except the binary decision variables can be represented using \(a_{j}^{t}\) which designates the period in which extractionrelated activities occur for each stope j. As well, recourse variables in the second portion of the objective function aim to limit deviations from the ore and metal targets, as opposed to the ore and waste targets in the OP objective function. Since UG mining methods have a higher level of selectivity than OP mining, waste is often not mined, but rather left in situ and only valuable material is produced. Therefore, it is more useful to constrain the amount of yearly metal produced in a UG optimization. Underground cost structure is viewed from a standpoint of cost per ton of material extracted. This standard figure contains expenses related to development, ventilation, drilling, blasting, extracting, backfilling and overhead. In terms of size and complexity, the UG scheduling model presented here is simpler than the OP model. The reduced size is due to only considering longterm extraction constraints and a small number of mining units that require scheduling. This allows for the schedule to be conveniently solved using IBM ILOG CPLEX 12.6 (IBM 2011), a commercially available software which relies on mathematical programming techniques to provide an exact solution.
UG Constraints
ScenarioDependent:
ScenarioIndependent
3 Application at a gold deposit
Size of orebodies and life of mine length at each transition depth
Transition Depth 1  Transition Depth 2  Transition Depth 3  Transition Depth 4  

Number of OP blocks  64,255  72,585  80,915  89,245 
Number of UG stopes  418  356  340  311 
Production years through OP  7  8  9  10 
Production years through UG  7  6  5  4 
Economic and technical parameters
Metal price  $900/oz 
Crown pillar height  60 ft 
Economic discount rate  10% 
Processing cost/ton  $31.5 
OP mining cost/ton  $1.5 
UG mining cost/ton  $135 
OP mining rate  18,500,000 t/year 
UG mining rate  350,000 t/year 
OP mining recovery  0.95 
UG mining recovery  0.92 
3.1 Stochastic optimization results and risk analysis
In order to evaluate the risk associated with stochastic decision making, a risk analysis is performed on the lifeofmine plans corresponding to the optimal transition depth stated above. Similar analysis has been done extensively on open pit case studies (Dimitrakopoulos et al. 2002; Godoy 2003; Jewbali 2006; Leite and Dimitrakopoulos 2014; Ramazan and Dimitrakopoulos 2005, 2013; Goodfellow 2014). To do so, a set of 20 simulated scenarios of the grades of the deposit are used and passed through the longterm production schedule determined for the optimal transition depth, which in this case is Transition Depth 2. This process provides the yearly figures for mill production tonnages, metal production and cash flow projections for each simulation if the schedule was implemented and the grades within a given simulation were realized.
3.2 Comparison to deterministic optimization result
The increased NPVs seen for the stochastic approach are due to the method’s ability to consider multiple stochastically generated scenarios of the mineral deposit, so as to manage geological (metal grade) uncertainty and local variability while making scheduling decisions. Overall, the stochastic scheduler is more informed and motivated to mine lower risk, high grade areas early in the mine life and defer extraction of lower grade and risky materials to later periods.
4 Conclusions and future work
A new method for determining the optimal OPUG transition depth is presented. The proposed method improves upon previously developed techniques by jointly taking a truly threedimensional approach to determining the optimal OPUG transition depth, through the optimization of extraction sequences for both OP and UG components while considering geological uncertainty and managing the related risk. The optimal transition decision is effectively described by a transition year, a threedimensional optimal open pit contour, a crown pillar location and a clearly defined underground orebody. In the case study, it was determined that the second of four transition depths evaluated is optimal which involves transitioning to underground mining in period 9. Making the decision to transition at the second candidate transition depth evaluated results in a 13% increase in NPV over the worstcase decision, as predicted by the stochastic framework. Upon closer inspection through risk analysis procedures, the stochastic framework is shown to provide a more realistic valuation of both the OP and UG assets. In addition to this, the stochastic framework produces operationally implementable production schedules that lead to a 9% NPV increase and reduction in risk when compared to the deterministic result. It is shown that the yearly cash flow projections outlined by the deterministic optimizer for the underground mine life are unlikely to be met, resulting in misleading decision criteria. Overall, the proposed stochastic framework has proven to provide a robust approach to determining an optimal open pit to underground mining transition depth. Future studies should aim to improve on this method by considering more aspects of financial uncertainty such as inflation and mining costs.
Notes
Acknowledgements
The work in this paper was funded from the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant 41127010, and the COSMO consortium of mining companies—AngloGold Ashanti, Barrick Gold, BHP Billiton, De Beers, Newmont Mining, Kinross Gold and Vale. We thank the reviewers for their valuable comments.
References
 Albor F, Dimitrakopoulos D (2010) Algorithmic approach to pushback design based on stochastic programming: method, application and comparisons. Min Technol 199:88–101Google Scholar
 Alford C (1995) Optimisation in underground mine design. In: Proceedings of the application of computers and operations research in the mineral industry (APCOM) XXV, Brisbane, pp 213–218Google Scholar
 Bakhtavar E, Shahriar K, Oraee K (2008) A model for determining the optimal transition depth over from open pit to underground mining. In: Proceedings of the 5th international conference and exhibition on mass mining, Luleå, pp 393–400Google Scholar
 Bootsma MT, Alford C, Benndorf J, Buxton MWN (2014) Cutoff gradebased sublevel stope mine optimisation—introduction and evaluation of an optimisation approach and method for grade risk quantification. In: Proceedings of the orebody modelling and strategic mine planning, AusIMM, Perth, pp 281–290Google Scholar
 Carpentier S, Gamache M, Dimitrakopoulos R (2016) Underground longterm mine production scheduling with integrated geological risk management. Min Technol 125(2):93–102CrossRefGoogle Scholar
 Dagdelen K, Traore I (2014) Open pit transition depth determination through global analysis of open pit and underground mine production scheduling. In: Proceedings of the orebody modelling and strategic mine planning, AusIMM, Perth, pp 195–200Google Scholar
 Datamine Software (2013) Studio 5 manual. www.dataminesoftware.com/
 Dimitrakopoulos R, Farrelly C, Godoy MC (2002) Moving forward from traditional optimisation: grade uncertainty and risk effects in open pit mine design. Trans IMM Min Technol 111:82–89CrossRefGoogle Scholar
 Ferland JA, Amaya J, Djuimo MS (2007) Application of a particle swarm algorithm to the capacitated open pit mining problem. Stud Comput Intell (SCI) 76:127–133Google Scholar
 Fuentes S, Caceres J (2004) Block/panel caving pressing final open pit limit. CIM Bull 97:33–34Google Scholar
 Geovia (2012) Geovia Whittle. www.3ds.com/productsservices/geovia/products/whittle
 Gilani SO, Sattarvand J (2016) Integrating geological uncertainty in longterm open pit mine: production planning by ant colony optimization. Comput Geosci 87:31–40CrossRefGoogle Scholar
 Godoy M (2003) The effective management of geological risk in longterm production scheduling of openpit mines. Ph.D. Thesis, University of Queensland, BrisbaneGoogle Scholar
 Godoy M, Dimitrakopoulos R (2004) Managing risk and waste mining in longterm production scheduling. SME Trans 316:43–50Google Scholar
 Goodfellow R (2014) Unified modeling and simultaneous optimization of open pit mining complexes with supply uncertainty. Ph.D. Thesis, McGill University, MontrealGoogle Scholar
 Goodfellow R, Dimitrakopoulos R (2016) Global optimization of open pit mining complexes with uncertainty. Appl Soft Comput J 40(C):292–304. doi: 10.1016/j.asoc.2015.11.038 CrossRefGoogle Scholar
 Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, OxfordGoogle Scholar
 Hansen P (1993) Model programs for computational science: a programming methodology for multicomputers. Concurr Comput 5(5):407–423Google Scholar
 IBM (2011) IBM ILOG CPLEX optimization studio, CPLEX user’s manual 12. IBM Corporation, pp 1–148Google Scholar
 Jewbali A (2006) Modelling geological uncertainty for stochastic short term production scheduling in open pit metal mines. Ph.D. Thesis, University of Queensland, Brisbane, QldGoogle Scholar
 Kjetland R (2012) Chuquicamata’s life underground will cost a fortune, but is likely to pay off for Codelco. Copper Investing News. Retrieved from: http://copperinvestingnews.com/12788chuquicamataundergroundminingcodelcochileopenpit.html
 Kumral M (2010) Robust stochastic mine production scheduling. Eng Optim 42:567–579MathSciNetCrossRefGoogle Scholar
 Lamghari A, Dimitrakopoulos R (2012) A diversified Tabu search for the openpit mine production scheduling problem with metal uncertainty. Eur J Oper Res 222(3):642–652CrossRefzbMATHGoogle Scholar
 Lamghari A, Dimitrakopoulos R, Ferland AJ (2014) A variable neighborhood descent algorithm for the openpit mine production scheduling problem with metal uncertainty. J Oper Res Soc 65:1305–1314CrossRefGoogle Scholar
 Leite A, Dimitrakopoulos R (2014) Mine scheduling with stochastic programming in a copper deposit: application and value of the stochastic solution. Min Sci Technol 24(6):255–262Google Scholar
 Lerchs H, Grossmann IF (1965) Optimum design of open pit mines. Can Inst Min Metall Bull 58:17–24Google Scholar
 Little J, Topal E (2011) Strategies to assist in obtaining an optimal solution for an underground mine planning problem using Mixed Integer Programming. Int J Min Miner Eng 3(2):152–172CrossRefGoogle Scholar
 Montiel L (2014) Globally optimizing a mining complex under uncertainty: integrating components from deposits to transportation systems. Ph.D. Thesis, McGill University, MontrealGoogle Scholar
 Montiel L, Dimitrakopoulos R (2015) Optimizing mining complexes with multiple processing and transportation alternatives: an uncertaintybased approach. Eur J Oper Res 247:166–178CrossRefzbMATHGoogle Scholar
 Montiel L, Dimitrakopoulos R, Kawahata K (2016) Globally optimising openpit and underground mining operations under geological uncertainty. Min Technol 125(1):2–14CrossRefGoogle Scholar
 Musingwini C (2016) Optimization in underground mine planningdevelopments and opportunities. J South Afr Inst Min Metall 116(9):809–820CrossRefGoogle Scholar
 Nehring M, Topal E (2007) Production schedule optimisation in underground hard rock mining using mixed integer programming. In: Project evaluation conference, pp 169–175Google Scholar
 Nehring M, Topal E, Little J (2009) A new mathematical programming model for production schedule optimisation in underground mining operations. J S Afr Inst Min Metall 110:437–446Google Scholar
 Newman A, Yano C, Rubio E (2013) Mining above and below ground: timing the transition. IIE Trans 45(8):865–882CrossRefGoogle Scholar
 Popov G (1971) The working of mineral deposits. Mir Publishers, MoscowGoogle Scholar
 Ramazan S, Dimitrakopoulos R (2005) Stochastic optimisation of longterm production scheduling for open pit mines with a new integer programming formulation. Orebody Modell Strateg Mine Plann AusIMM Spectr Ser 14:385–391Google Scholar
 Ramazan S, Dimitrakopoulos R (2013) Production scheduling with uncertain supply: a new solution to the open pit mining problem. Optim Eng 14:361–380CrossRefzbMATHGoogle Scholar
 Senecal R (2015) Parallel Implementation of a tabu search procedure for stochastic mine scheduling. M.E. Thesis, McGill University, MontrealGoogle Scholar
 Soares A, Nunes R, Azevedo L (2017) Integration of uncertain data in geostatistical modelling. Math Geosci 49(2):253–273MathSciNetCrossRefzbMATHGoogle Scholar
 Trout P (1995) Underground mine production scheduling using mixed integer programming. In: Proceedings of the application of computers and operations research in the mineral industry (APCOM) XXV, Brisbane, pp 395–400Google Scholar
 Vallee M (2000) Mineral resource + engineering, economic and legal feasibility = ore reserve. Can Min Metall Soc Bull 93:53–61MathSciNetGoogle Scholar
 Whittle J (1988) Beyond optimisation in open pit design. In: Proceedings Canadian conference on computer applications in the mineral industries, pp 331–337Google Scholar
 Whittle J (1999) A decade of open pit mine planning and optimization—the craft of turning algorithms into packages. In: Proceeding of 28th computer applications in the mineral industries, pp 15–24Google Scholar
 Zagayevskiy Y, Deutsch CV (2016) Multivariate geostatistical gridfree simulation of natural phenomena. Math Geosci 48(8):891–920MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.