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A variable neighbourhood descent algorithm for the open-pit mine production scheduling problem with metal uncertainty

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A variable neighbourhood descent algorithm for the open-pit mine production scheduling problem with metal uncertainty
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     A   U   T   H  O   R   C  O   P   Y A variable neighbourhood descent algorithm for theopen-pit mine production scheduling problem withmetal uncertainty Amina Lamghari 1 *, Roussos Dimitrakopoulos 1 and Jacques A Ferland 2 1  McGill University, Quebec, Canada; and  2 University of Montreal, Quebec, Canada Uncertainty is an inherent aspect of the open-pit mine production scheduling problem (MPSP); however, little isreported in the literature about solution methods for the stochastic versions of the problem. In this paper, twovariants of a variable neighbourhood descent algorithm are proposed for solving the MPSP with metal uncertainty.The proposed methods are tested and compared on actual large-scale instances, and very good solutions, with anaverage deviation of less than 3% from optimality, are obtained within a few minutes up to a few hours.  Journal of Operational Research Society advance online publication, 3 July 2013; doi:10.1057/jors.2013.81 Keywords: stochastic production scheduling; open-pit mining; uncertainty; heuristics The online version of this article is available Open Access 1. Introduction Production scheduling of open-pit mining operations is a challenging and critical issue for mining companies, a keyfactor in determining returns on investments in the order of hundreds of millions of dollars. In scheduling mine production,the mineral deposit is represented as a three-dimensional arrayof blocks. Each block has a weight and a metal content estimated using information obtained from drilling. To recover the metal, the block is fi rst mined from the ground and thenprocessed in a mill. These operations are termed mining andprocessing, respectively.Blocks are classi fi ed as ore or  waste according to their metalcontent. Ore blocks are those that have a selling revenue greater than their processing costs, while waste blocks have a totalmetal content whose selling revenue is less than the processingcosts. Both ore and waste blocks must be mined in order to gainaccess to all the ore blocks. Any block that must be mined inorder to reach another block is called the predecessor of thesecond block.Decisions on block scheduling are subject to various typesof constraints. The production schedule not only must respect the limits on extraction capacity ( mining constraints ) and thecapacity of the processor (  processing constraints ) at eachperiod of the life of the mine, but also must take intoconsideration the order in which blocks can be removed from the orebody to ensure that a block is not mined before anyof its predecessors ( slope constraints ). In addition, anyblock can be mined only once ( reserve constraints ). Theproblem is to determine which blocks to extract and whento extract them (mining sequence) in order to maximize thenet present value (NPV) of the mine while respecting thevarious constraints.A major complexity in the open-pit mine production sche-duling problem (MPSP) is that the number of blocks is large, ingeneral in the order of tens to hundreds of thousands, yielding a large-scale optimization problem. Another factor adding to thecomplexity of the mine scheduling problem is metal uncer-tainty,for the metalcontent ofthe blocks is notknown preciselyat the time decisions are made but is inferred from limiteddrilling information. Up to now, most methods developed tosolve the MPSP either ignore the metal uncertainty issue or arenot able to solve large-scale instances in which metal uncer-tainty is accounted for.Indeed, the MPSP has been frequently studied since the1960s (Newman et al  , 2010). Different methods have beenapplied to solve the deterministic version of the problem, whichassumes that all the problem parameters are well known. Thesemethods can be classi fi ed into three categories: exact methods(Dagdelen and Johnson, 1986; Caccetta and Hill, 2003;Ramazan, 2007; Boland et al  , 2009; Bley et al  , 2010), heuristicand metaheuristic methods (Gershon, 1987; Denby andScho fi eld, 1994; Ferland et al  , 2007; Chatterjee et al  , 2010),and hybrid methods (Tolwinski and Underwood, 1996; Sevim and Lei, 1998; Moreno et al  , 2010). However, the uncertainnature of the problem is ignored in the deterministic version of  *Correspondence: Amina Lamghari, COSMO  —  Stochastic Mine Planning Laboratory, Department of Mining and Materials Engineering, McGill University, FDA Building, 3450 University Street, Room 113, Montreal,Quebec, Canada H3A 2A7.  Journal of the Operational Research Society (2013) , 1  –  10 © 2013 Operational Research Society Ltd. All rights reserved. 0160-5682/13www.palgrave-journals.com/jors/     A   U   T   H  O   R   C  O   P   Y the MPSP, resulting in misleading assessments (Ravenscroft,1992; Dowd, 1994; Dimitrakopoulos et al  , 2002; Godoy andDimitrakopoulos, 2004). Studies that compare stochastic todeterministic approaches (Godoy and Dimitrakopoulos, 2004;Menabde et al  , 2007; Albor and Dimitrakopoulos, 2009, 2010;Asad and Dimitrakopoulos, 2013) indicate that stochasticapproaches show major improvements in NPV, on the order of 20% to 30%, substantially reduce risk in meeting productionforecasts, and fi nd pit limits larger than the ones found bydeterministic approaches, contributing to the sustainable utiliza-tion of mineral resources.Inthe stochastic versionsofthe problem, a scenario approachis usually used to handle the metal uncertainty. The scenariosare speci fi ed from estimates in the continuous space without any underlying scenario tree structure. Consequently, the multi-stage approach commonly used in stochastic programmingcannot be used. Three different approaches are used in theliterature. The fi rst involves formulations maximizing theexpected NPV over the scenarios describing the metal uncer-tainty while satisfying the production targets in an averagesense (Menabde et al  , 2007). The second involves formulationsmaximizing the expected NPV and minimizing deviations from production targets for each individual scenario (Ramazan andDimitrakopoulos, 2007; Albor and Dimitrakopoulos, 2010;Ramazan and Dimitrakopoulos, 2013). The third approach,illustrated in Boland et al  (2008), takes into account the metaluncertainty via a multistage stochastic programming approachwhere the missing interdependency between scenarios and thedecisions (ie, the missing tree structure) is simulated usingconditional non-anticipativity constraints.While different stochastic models have been developed,solution methods have received relatively less attention.Furthermore, most of the solution methods developed havebeen able to deal only with instances of relatively small size(typically, up to 20000 blocks). The stochastic models pro-posed in the studies by Menabde et al  (2007), Ramazan andDimitrakopoulos (2007, 2013), and Boland et al  (2008) aresolved using the mixed integer programming solver  CPLEX  .The method used in Albor and Dimitrakopoulos (2010) consistsof generating a set of nested pits, grouping these pits intopushbacks, and then generating a schedule based on the push-back designs obtained, while the method used in Lamghari andDimitrakopoulos (2012) is based on Tabu search.In this paper, we propose an ef  fi cient solution method todeal with large instances of the MPSP with metal uncertainty,where the uncertainty is addressed using a two-stage stochasticprogramming approach. Speci fi cally, we introduce a metaheur-istic method based on a variable neighbourhood descent (VND)procedure (Hansen and Mladenovic, 2001). To generate theinitial solution to be improved by this procedure, we consider two different alternatives. Both are based on a decompositionapproach separating the problem into a series of sub-problems,each associated with one period. They differ in the methodused to solve the sub-problems. In the fi rst alternative, thesub-problems are solved exactly, while in the second one, thesub-problems are solved approximately with a greedy heuristic.We evaluate and compare the performance of the two variantsof the proposed solution method on large-scale instances withup to 97307 blocks. The results indicate that both variantsgenerate very good solutions in reasonable computationaltimes. The fi rst variant, where the sub-problems are solvedexactly,slightly outperformsthesecond oneinterms of solutionquality, while the second variant, where the sub-problems aresolved with a heuristic, requires in general signi fi cantly lesscomputational time.The remainder of the paper is organized as follows: InSection 2, the approach used to deal with metal uncertainty isoutlined, and a mathematical formulation of the problem isintroduced. The following section brie fl y describes the methodsused to generate the initial solution. Section 4 summarizes thevariable neighbourhood procedure used to improve the initialsolution. Computational results on real-life data are reportedand discussedin Section5. Finally, some conclusionsare drawnin Section 6. 2. Mathematical formulation Referring to the description given in the previous section, theproblem can be formulated as a two-stage stochastic program-ming model (Birge and Louveaux, 1997). In the fi rst stage, onedetermines a set of blocks to be mined at each period in order tosatisfy the reserve constraints , the slope constraints , and the mining constraints . The metal content of each block, determin-ing whether the block is ore or  waste , is uncertain at this stage.In the second stage, when the blocks are mined, the metalcontent becomes known. In some periods, the ore blocksavailable, requiring processing, may have a total weight exceeding the mill capacity (ie, the processing constraints maybe violated). To deal with such a situation, some recourseactions are available, but they induce a cost. The problem isthen to identify a schedule that maximizes the expected NPV of the mining operation minus the expected recourse costsincurred due to the violation of the processing constraints .The following notation is used to formulate the mathematicalmodel: ● T  : the number of periods over which blocks are beingscheduled (horizon). ● t  : period index, t  = 1, … , T  . ● W  t  : maximum amount of rock ( ore and waste ) that can bemined during period t  (mining capacity in tons). ● Θ t  : maximum amount of  ore that can be processed in the millduring period t  (processing capacity in tons). ●  N  : the number of blocks considered for scheduling. ● i : block index, i = 1, … ,  N  . ● P  i : the set of predecessors of block  i ; that is, blocks that haveto be mined to have access to block  i . ● P  i : the set of direct predecessors of block  i ; that is, P  i = f  p 2 P  i : p is on the level immediately above i g : 2 Journal of the Operational Research Society     A   U   T   H  O   R   C  O   P   Y ● Γ i : the set of successors of block  i ; that is, γ  ∈ Γ i if  i ∈ P  γ  .Figure 1 gives a two-dimensional illustration of the sets P  i , P  i , and Γ i . ● w i : the weight of block  i (in tons). ● v i : the economic value of block  i . It is a random variabledepending on the metal content of the block. ● v t i = v i = ð 1 + d  1 Þ t  : the discounted economic value of block  i if mined during period t  ( d  1 being the discount rate per period).Note that it is assumed that  ore blocks are processed duringthe same period when they are mined and that the pro fi t isalso generated during that period. ● θ  i : a random variable indicating if block  i is an ore or a  waste block  θ  i = 1 if block  i isan ore block  ; 0 otherwise : ( It is a random variable because the metal content of a block determines whether the block is ore or  waste . ● c t  = c ð 1 + d  2 Þ t  : unit surplus cost incurred if the total weight of  ore blocks mined during period t  exceeds the processingcapacity, Θ t  ( c being the undiscounted surplus cost, and d  2 represents the risk discount rate).The variables used to formulate the problem are as follows: ● A binary variable is associated with each block  i for eachperiod t  :  x  t i = 1 if block  i isminedbyperiod t  ; 0 otherwise : ( This means that if block  i is mined in period τ  , then x  it  = 0 for all t  − 1, … , τ  − 1 and x  it  = 1 for all t  = 1, … , T  . If  i is not minedduring the horizon, then x  it  = 0 for all t  = 1, … , T  . ● A random variable d  t  is associated with each period t  .It measures the surplus in ore mined during period t  .The mathematical model can be summarized as follows:max  E  X  N i = 1 v 1 i x  1 i " # +  E  X T t  = 2 X  N i = 1 v 1 i ð  x  t i -  x  t  - 1 i Þ " # -  E  X T t  = 1 c t  d  t  " # (1)(  M  ) Subject to  x  t  - 1 i ⩽  x  t i i = 1 ; ¼ ;  N  ; t  = 2 ; ¼ ; T  (2)  x  t i ⩽  x  t  p i = 1 ; ¼ ;  N  ;  p 2 P  i ; t  = 1 ; ¼ ; T  (3) X  N i = 1 w i  x  1 i ⩽ W  1 (4) X  N i = 1 w i ð  x  t i -  x  t  - 1 i Þ ⩽ W  t  t  = 2 ; ¼ ; T  (5) X  N i = 1 θ  i w i  x  1 i - d  1 ⩽ Θ 1 (6) X  N i = 1 θ  i w i ð  x  t i -  x  t  - 1 i Þ - d  t  ⩽ Θ t  t  = 2 ; ¼ ; T  (7)  x  t i = 0 or 1 i = 1 ; ¼ ;  N  ; t  = 1 ; ¼ ; T  (8) d  t  ⩾ 0 t  = 1 ; ¼ ; T  : (9)The objective function (1) includes two terms to maximizethe expected NPV of the mining operation and to minimize theexpected recourse costs incurred whenever the processingconstraints are violated due to metal uncertainty. Constraints(2) guarantee that each block  i is mined at most once duringthe horizon ( reserveconstraints ). The miningprecedence( slopeconstraints ) is enforced by constraints (3). Constraints (4) and(5) impose an upper bound W  t  on the amount of rock ( ore and waste ) mined during each period t  ( mining constraints ).Constraints (6) and (7) are related to the requirements on theprocessing levels (  processing constraints ). The target is to havethe total amount of  ore mined during any period t  be smaller than Θ t  ; otherwise, the surplus penalty cost is equal to c t  d  t  .The model is a two-stage stochastic programming model. Thevariables x  it  specifying the mining sequence are the fi rst-stagedecision variables, and the random variables d  t  measuring thesurplus in ore production are the second-stage decision vari-ables. d  t  depend on both the realization of the metal content andthe fi rst-stage decisions.To transform this stochastic model into an equivalent deter-ministic one, assume that  S  possible scenarios are availablewhere each scenario s speci fi es the metal content of each block.Furthermore, assume that the probability of occurrence of scenario s is π  s , with P S s = 1 π  s = 1. Let  v ist  , θ  is , and d  st  denoterespectively a realization of the random variables v it  , θ  i , and d  t  . Figure 1 Illustration of the sets P  i , Γ i , and P  i . Amina Lamghari et al  — Variable neighbourhood descent algorithm 3     A   U   T   H  O   R   C  O   P   Y Then, the srcinal model (1) – (9) can be reformulated as follows: max X  N i = 1 X S s = 1 π  s v 1 is  x  1 i + X T t  = 2 X  N i = 1 X S s = 1 π  s v t is ð  x  t i -  x  t  - 1 i Þ - X T t  = 1 X S s = 1 π  s c t  d  t s (10) (DE) Subject toConstraints (2) – (5), (8), and X  N i = 1 θ  is w i  x  1 i - d  1 s ⩽ Θ 1 s = 1 ; ¼ ; S  (11) X  N i = 1 θ  is w i ð  x  t i -  x  t  - 1 i Þ - d  t s ⩽ Θ t  t  = 2 ; ¼ ; T  ; s = 1 ; ¼ ; S  (12) d  t s ⩾ 0 t  = 1 ; ¼ ; T  ; s = 1 ; ¼ ; S  : (13)The model (DE) utilizes a limited number of scenarios,each specifying the metal content of each block, which thenare used to calculate the corresponding economic value.A natural question to address is how to choose the number of scenarios to consider. This is a well-studied topic in stochasticoptimization (Dupacova  et al  , 2003) and other  fi elds (Scheidt and Caers, 2009). The case studies presented in this paper use20 scenarios because past work, such as the work in Albor andDimitrakopoulos (2009), indicates that after about 15 simulatedrepresentations of an orebody, stochastic schedules converge toa stable fi nal physical schedule as well as stable forecasts of production performance. This behaviour is not surprising; whilesimulated scenarios represent a mineral deposit at the support-scale of mining blocks, each with a volume of a few cubicmeters, a mine ’ s production schedule represents a grouping of a few thousand of these mining blocks in just one time (mining)period under various constraints. Thus, as the support-scale of a mine ’ s schedule is orders of magnitude larger than that of thesimulated representations of the mineral deposit being sched-uled, the stochastic schedule becomes insensitive to additionalscenarios after a relatively small number of scenarios. In thispaper, the set of suf  fi cient scenarios is provided, and theobjective is to design an ef  fi cient method to solve the proposedmathematical model.If the mining and the processing constraints are eliminated,and if the scheduling horizon reduces to a single period, thenthe model (DE) reduces to the classical maximum closure prob-lem. This problem is reducible to the minimum-cut problem (Picard, 1976), and thus it can be solved ef  fi ciently using any of the known polynomial maximum- fl ow algorithms. However, asBienstock and Zuckerberg (2010) note on page 3, ‘ it can beshown by reduction from max clique that adding a singlecardinality constraint to a maximum closure problem is enoughto make it NP-hard ’ . Because the MPSP is more complex thana constrained maximum closure problem, and as real-worldMPSP instances are very large, having typically tens tohundreds of thousands blocks, it is most likely not appropriateto solve these large-scale realistic instances using an exact method. Instead, we propose a metaheuristic solution methodwhere an initial feasible solution is fi rst obtained and thenimproved with a  VND procedure. The methods used to generatethe initial solution as well as the VND procedure are describedin the following sections. 3. Generating the initial solution We propose to generate the initial solution using two different heuristics based on a decomposition approach where theglobal problem is divided into smaller sub-problems, eachassociated with a period t  ( t  = 1, … , T  ). The sub-problems aresolved sequentially in increasing order of  t  , and their solutionsare combined to generate the initial solution. The heuristicsdiffer in the method used to deal with the sub-problems. In the fi rst one, the sub-problems are solved exactly, while in thesecond one, the sub-problems are solved approximately witha greedy heuristic. Note that since the sub-problems aresmaller than the global problem, an exact method can be usedto solve them. 3.1. Sub-problem formulation The sub-problem associated with period t  consists of determin-ing a set of blocks B  t  to be mined in period t  . Let us denote by R t  = {block  i : x  it  = 0} the set of blocks not mined yet (if  t  = 1,then | R t  | =  N  ; otherwise, jR t  j =  N  - j S τ  < t  B  τ  j ). In order tosatisfy the reserve constraints , the blocks to be included in B  t  should be selected from  R t  . The sub-problem associated withperiod t  can then be summarized as follows:max X i 2R t  X S s = 1 π  s v t is  y i - c t  X S s = 1 π  s d  s (14)( SP t  ) Subject to  y i ⩽  y  p i 2 R t  ; p 2 P  i \R t  (15) X i 2R t  w i  y i ⩽ W  t  (16) X i 2R t  θ  is w i  y i - d  s ⩽ Θ t  s = 1 ; ¼ ; S  (17)  y i = 0or1 i 2 R t  (18) d  s ⩾ 0 s = 1 ; ¼ ; S  (19)where  y i = 1 if block  i isincludedintheset  B  t  ð i : e :; isminedinperiod t  Þ ; 0 otherwise : 8<: Recall that if block  i is mined in period t  , then x  i τ  = 0 for all τ  = 1, … , t  − 1 and x  i τ  = 1 for all τ  = 1, … , T  . 4 Journal of the Operational Research Society     A   U   T   H  O   R   C  O   P   Y Logical implications of the constraints are used to generatevalid inequalities to strengthen the formulation above and makethe sub-problems easier to solve. To be more speci fi c, consider any block  i ∈ R t  . On the one hand, the slope constraints requirethat to include i in B  t  , we must also include all blocks j  ∈  N  i = P  i ∩ R t  (the set of blocks that are predecessors of  i and not mined yet). On the other hand, i should not be included in B  t  if  w i + P  j  2N  i w  j  > W  t  because this would lead to violation of the mining constraints . Hence, we add the following constraints tothe model ( SP t  ):  y i ⩽ e i i 2 R t  (20)where e i is a parameterequal to 1 if the extraction ofblock  i willnot lead to violation of the mining constraints and 0 otherwise;that is, e i = 1 if  w i + P  j  2N  i w  j  ⩽ W  t  ; 0 otherwise : 8<: 3.2. Sub-problem solution methods Two different methods are introduced to solve the sub-pro-blems. In the fi rst method, the formulation (14) – (20) is solvedusing the branch-and-cut algorithm (BC) implemented in themixed integer programming solver  CPLEX  .The second methodis a sequentialgreedyheuristicprocedure(GH) where at each iteration we try to include in the set  B  t  aninverted cone formed by a  ‘ base ’ block in R t  and all itspredecessors not mined yet. Let us analyse a typical iteration.Let  V  t  = W  t  - P b 2B  t  w b and O t s = max f Θ t  - P b 2B  t  θ  bs w b ; 0 g be the residual mining capacity and the residual processingcapacity under scenario s , respectively. For each block  i ∈ R t  ,we denote: ● Ω i = { i } ∪ {  j  : j  ∈  N  i } the set formed by block  i and all itsunmined predecessors (ie, the inverted cone whose base is i ). ● α  i = P k  2 Ω i w k  : the total weight of blocks in Ω i . ●  β  is = P k  2 Ω i θ  ks w k  : the total weight of  ore blocks in Ω i under scenario s . ● γ  i = P k  2 Ω i P S s = 1 π  s v t ks : the total expected discounted eco-nomic value of blocks in Ω i . ● ξ i = γ  i - c t  P S s = 1 π  s max f  β  is - O t s ; 0 g : the total contributionof blocks in Ω i to the objective function (14).Consider the set  A  = { i ∈ R t  : α  i ⩽ V  t  } of blocks i ∈ R t  whoseweight plus the weight of their predecessors not mined yet doesnot exceed the residual mining capacity. Clearly, only coneshaving blocks in A  as their base can be added to B  t  whilesatisfying the slope constraints and the mining constraints .Select the block  i * ∈ A  maximizing the value of  ξ i . Ties arebroken up randomly. Remove all blocks k  ∈ Ω i * from  R t  , addthem to B  t  , and update V  t  , O st  for each s = 1, … , S  , and theset  A  . This process is repeated until the mining constraints areapproximately satis fi ed; that is, until X i 2B  t  w i ⩽ δ  W  t  (21)where δ  is a random number in the interval [ δ  1 , δ  2 ], and δ  1 and δ  2 are parameters of the procedure in [0,1].Note that choosing blocks along with their predecessors(an inverted cone at each iteration) allows a look ahead featuregenerating better solutions than the myopic approach of choos-ing blocks one by one. 4. Improving the initial solution As mentioned before, the initial solution x  = S T t  = 1 B  t  isimproved by applying an adaptation of the VND methodproposed by Hansen and Mladenovic (2001). The basic idea of  VND is to combine different descent heuristics based ondifferent neighbourhood structures to escape from local optima.In the following, we fi rst describe the neighbourhood structuresused in our adaptation of the VND method. Next, we outline theprocedure used to improve the solution x  . 4.1. Neighbourhood structures Three neighbourhood structures are used in our adaptationof the VND method. The fi rst structure tries to swap twoblocks scheduled in consecutive periods. For example, a  waste block scheduled to be mined in period 1 and having nosuccessors scheduled in period 1 could be left behind to bemined in period 2. In its place, an ore block scheduled tobe mined in period 2 could be mined in period 1 provided that its predecessors are scheduled in period 1. The mining capacityshould not be exceeded in either period as a result of the swap.The second and third structures try to make room for newblocks ina given period byscheduling some blocks tobe minedin that period backward or forward while satisfying the slope and the mining constraints . A more formal description of thethree neighbourhood structures as well as an outline of thestrategy used to explore them is given below. ●  N  1 (  Exchange or  Swap ): Let  i and j  be two blocks mined inperiods t  and ( t  +1), respectively. An exchange consists of replacing B  t  and B  t  +1 by ( B  t  − { i })+{  j  } and ( B  t  +1 − {  j  })+{ i },respectively. The exchange of two blocks is feasible if theresulting solution is feasible; that is, only if it satis fi es the slope and the mining constraints . Figure 2 gives a two-dimensional illustration of an exchange move involving twoblocks, i and j  , with T  = 2. ●  N  2 ( Shift-after  ): Let  i be a block mined in period t  , andlet  I  = { i } ∪ {block  γ  : γ  ∈ Γ i ∩ B  t  } denote the set including i and its successors mined in the same period. A shift-after consists of replacing B  t  and B  t  +1 by B  t  −  I  and B  t  +1 +  I  ,respectively. Clearly, the slope constraints are satis fi ed inthe resulting solution since the blocks are moved along with Amina Lamghari et al  — Variable neighbourhood descent algorithm 5
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