Open-Pit Block-Sequencing Formulations: A Tutorial

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Vol. 44, No. 2, March–April 2014, pp. 127–142ISSN 0092-2102 (print) � ISSN 1526-551X (online) http://dx.doi.org/10.1287/inte.2013.0731

© 2014 INFORMS

Open-Pit Block-Sequencing Formulations:A Tutorial

W. Brian LambertDivision of Economics and Business, Colorado School of Mines, Golden, Colorado 80401, [email protected]

Andrea BrickeyDepartment of Mining Engineering, Colorado School of Mines, Golden, Colorado 80401, [email protected]

Alexandra M. NewmanDivision of Economics and Business, Colorado School of Mines, Golden, Colorado 80401, [email protected]

Kelly EurekStrategic Energy Analysis Center, National Renewable Energy Lab, Golden, Colorado 80401, [email protected]

A classical problem in the mining industry for open-pit mines involves scheduling the production of notionalthree-dimensional production blocks, each containing a predetermined amount of ore and waste. That is, givenoperational resource constraints on extraction and processing, we seek a net present value-maximizing scheduleof when, if ever, to extract each block in a deposit. We present a version of the problem, which some literaturerefers to as (CPIT). This constrained ultimate pit limit problem (i.e., open-pit production-scheduling problemvariant) produces a sequence of blocks to extract given minimum and maximum bounds on production andprocessing capacity, and geospatial precedences. Our tutorial demonstrates methods to expedite solutions forinstances of this model through variable definition, preprocessing, algorithmic choice, and the provision of aninitial feasible solution. As such, our paper is relevant for any mining practitioner interested in productionscheduling, and any operations researcher interested in a basic introduction before extending the boundaries ofalgorithmic development in this area.

Keywords : mine scheduling; mine planning; open-pit mining; surface mining; optimization; integerprogramming applications.

History : This paper was refereed.

Mine production scheduling is the process ofdetermining an extraction sequence for notional

three-dimensional blocks, each containing prespec-ified amounts of ore and waste. Large excavatorsand haul trucks extract and subsequently transportthese blocks of material to a mill (i.e., processingplant) or to a waste site, depending on the expectedprofitability of the material. The rate at which thematerial is excavated and processed depends on ini-tial capital expenditure decisions, permitting, contractagreements, projected sales, processing capacities, thepresence of equipment (e.g., haul trucks and load-ers), and the existence of infrastructure (e.g., roadsand rail lines). Processed ore can be sold according tolong-term contracts or on the spot market. Waste isleft in piles, which must ultimately be remediated orreclaimed when the mine is closed.

Researchers have studied mine production schedul-ing since the late 1960s (Johnson 1968). Commonvariables in this integer-programming formulationinclude when, if ever, to extract a block to maxi-mize the operation’s net present value (NPV). Con-straints include spatial sequencing between blocksand restrictions on available operational resources.However, hardware and software capabilities haveonly recently allowed solutions to realistic variants ofthe production-scheduling problem (Newman et al.2010). Commercial software exists to address produc-tion scheduling in open-pit mines; under some cir-cumstances it works well. Nonetheless, researchershave also shown that such software does not neces-sarily provide an optimal, or even a feasible, solu-tion for realistic variants (De Kock 2007, Gaupp 2008,Somrit 2011). These shortcomings may stem from the

127

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Lambert et al.: OPBS Tutorial128 Interfaces 44(2), pp. 127–142, © 2014 INFORMS

fact that every mine possesses different attributes andpolicies. For example, some mines may stockpile (in avariety of ways) and some mines may invoke a pol-icy whose decisions also include whether to send ablock to a waste dump or to a processing plant. Ourobjective in this paper is to provide a tutorial for min-ing practitioners and applied operations researcherswho are interested in building customized models foropen-pit mine production scheduling and (or) whoare interested in increasing the tractability of existingmodels. For a basic model, we provide fundamentalinsights often left unstated or even untested in theliterature.

This paper is organized as follows: The Model sec-tion provides a description and mathematical formu-lation of the production scheduling model variantwith which we work. In the Solution Methodologies sec-tion, we discuss methods for increasing the tractabil-ity of this model. Some of these methods are used,but are not well documented in the literature; othersare novel. In Results, we give numerical results basedon the application of these methodologies, and weconclude with a summary and directions for futureresearch in Conclusions.

ModelThis paper focuses on solving real-world instances ofan open-pit production-scheduling model, also calledthe open-pit block-sequencing (OPBS) problem. At thestrategic level, the ultimate pit limit (UPL) problem,based on the seminal work of Lerchs and Grossmann(1965), provides a solution for the ultimate pit limitcontours of a deposit with the objective of maximiz-ing total profit. This formulation includes geometryconstraints; however, it does not factor in the effects oftime, production, or blending constraints. At the tac-tical level, the OPBS problem includes the dimensionof time, answering the question of when, if at all, toextract each block. Next, we show our formulations ofthe OPBS problem and methodologies to reduce theirsolution time. See Espinoza et al. (2013) for a discus-sion of the relationship between these two problemsand their important attributes.

Three-dimensional block models used to determinethe optimal block sequence can vary in size and blockcharacteristics according to the deposit, or even thearea within a deposit, being modeled. The geology,

shape, and scale of a deposit can influence the blocksize, typically given by the dimensions of the blockin x-, y-, and z-directions. For a narrow-vein golddeposit, the blocks may be oriented along strike withblock size dependent on the width of the vein. Forother deposits in which the mineral formation ishighly dispersed, the height of the blocks may cor-respond to the scale of the equipment being utilized.In this tutorial, we depict the deposit of interest usinga three-dimensional block model, where each block’scharacteristics are known with certainty.

Block models can include extensive data represent-ing many physical properties of each block, includ-ing, but not limited to, their volume, tonnage, cost,metal recovery, and grade. In a deterministic model,such as the one we present in this paper, these dataare assumed to be certain, enabling the a priori cal-culation of each block’s value, and the calculation ofoperational resources (i.e., production and process-ing) required for its extraction and destination (e.g., aprocessing plant). The information stored in a blockmodel are estimations based on the available data,such as drillhole samples, existing operations, andmetallurgical testing. Metal recoveries represent thepercentage of metal that can be recovered from pro-cessed ore. Ore grades reported in a block model canbe in-situ grades, with a recovery percentage appliedduring revenue calculations, or as recovered gradeswith the recovery factor already applied. For this tuto-rial, we compute the revenue associated with a blockas the product of the amount of metal recovered fromthe processing plant and an estimated sale price; wecompute the operational cost as the product of theamount of ore and the mining and processing costsless the product of the amount of waste and the asso-ciated mining cost. If the difference between the rev-enue and the operational cost is negative, we classifythe block as waste and compute its extraction costas simply the product of the amount of waste andthe associated mining cost. Otherwise, the block qual-ifies as an ore block. Based on this classification, wecategorize each ore block as destined for the process-ing plant and each waste block as destined for thedump. Although this a priori categorization is gen-erally regarded as a strong assumption in short-termscheduling models, it can be an appropriate simplifi-cation for strategic block-sequencing models.

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Lambert et al.: OPBS TutorialInterfaces 44(2), pp. 127–142, © 2014 INFORMS 129

These calculated block values, in turn, enable thegeneration and economic valuation of block extractionschedules, accounting for costs associated with dif-ferent mining and processing methods, transportationmodes, and revenues based on the expected marketvalue of the mineral. However, depending on the sizeof the deposit and the block size, block models cancontain millions of blocks, which precludes fast calcu-lations of solutions to scheduling models. To improvethe speed of these calculations, algorithms (typically,that of Lerchs-Grossman applied with an estimatedmetal price) can be used to determine the ultimate pitlimits and to reduce model sizes by eliminating thoseblocks that fall outside of the limits. When conduct-ing any model size reduction, we must ensure thatall blocks that might be included in an optimal pro-duction schedule are retained. It is possible, for exam-ple, in the presence of lower bounds on operationalresource constraints in the block extraction schedulingmodel, that certain techniques to reduce model sizemay exclude blocks that would otherwise be includedin an optimal production schedule. Therefore, if thepractitioner wishes to reduce the problem size bysolving a series of (ultimate pit limit) problems, theauthors recommend choosing higher economic (price)parameters, which would result in removing fewerblocks from the original model, and thus, a lower like-lihood of removing a block that should be containedin the optimal solution to a block extraction schedule.

Aggregating adjacent blocks is a technique to fur-ther reduce model size. The decision to aggregateblocks from a model could be based on the locationof blocks that share common characteristics (e.g., oreor waste content), deposit characteristics, the num-ber of blocks in the model, the time fidelity in theformulation, and the required accuracy necessary inthe resulting solution. Aggregation can be used whensufficient operational resources exist to extract manyblocks in a given period, and must be used if theinitial block size is not consistent with the miningmethod (e.g., the equipment is not capable of selec-tively mining at that block size). For our paper, weassume that the initial block size is suitable for theanalysis. However, our techniques are valid for anylarge instance, regardless of whether that instance wasarrived at by aggregating blocks.

Although mining only valuable ore blocks wouldbe preferable, the order in which blocks can be mined

1

2

Figure 1: Block 1 must be mined prior to mining block 2.

is dictated by a maximum allowable slope angle ofthe deposit and by the blocks’ relative positions toeach other. The slope angle is determined by geo-logical conditions, such as rock strength, and designparameters, such as roads. Typical geospatial prece-dence constraints between blocks in an open-pit minerequire that the block immediately above the blockin question be mined before the block in questioncan be mined (see Figure 1). The literature com-monly contains more elaborate precedence constraintsto preserve the necessary pit slope angles or adequateworking areas for equipment. To retain the focus ofthis tutorial on optimization modeling, we addressslope angles with the common, but simplified, “+”sign convention. Here, we assign precedences suchthat prior to mining a block b, the block b′ directlyabove b, and those blocks adjacent to and on the samelevel as b′ must first be mined (see Figure 2). Althoughthese precedence constraints are an oversimplification

4

3

65

21

Figure 2: Blocks 1–5 must be mined prior to mining block 6.

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of reality, they are commonly used in the academic lit-erature; as of this writing, they are thought to be rea-sonable approximations of reality. For cases in whichthey are not reasonable approximations, the mod-eler may specify other precedence rules without anyloss of generality. See Espinoza et al. (2013) for adiscussion. The complete set of blocks on all levelsabove b, which must be mined prior to mining b,constitute b’s complete predecessor set, or b’s prede-cessor pit. A transitivity relationship exists betweenblock precedences. By transitivity, we can say thata precedence relationship is immediate if no otherpair of precedence relationships implies it. The sub-set of blocks within the complete predecessor set thatpossesses an immediate precedence relationship withb constitutes b’s direct predecessor set.

It is important to ensure that the problem beingsolved is well defined. Different specifications of theopen-pit mining problem exist, from the strategic ulti-mate pit limit problem, which identifies the most eco-nomical subset of blocks to extract, to the tacticalprecedence-constrained production-scheduling prob-lem that identifies which blocks to extract and whenand where to send them. In this paper, we focuson the deterministic, constrained pit limit problem(CPIT), as named in Chicoisne et al. (2012), which isa simplified version of the formulation presented inJohnson (1968). We use this variant, but generalizethe problem in Chicoisne et al. (2012) by includingnonzero minimum operational resource constraints,which require some extraction and (or) processingactivities in each period. The solution to (CPIT) pro-vides a NPV-maximizing block extraction sequencethat satisfies minimum resource requirements andmaximum resource capacities, in addition to thegeospatial precedences. We also refer to (CPIT) asthe monolith, in that it encompasses the entire for-mulation, as opposed to a formulation produced viaa mathematical decomposition procedure, such asLagrangian relaxation or Benders decomposition.

The mathematical formulation of the (CPIT) prob-lem is a mixed-integer linear program. The objec-tive is to determine the period in which each blockhas to be mined to maximize the NPV over agiven time horizon. We represent the block miningprecedence requirements and the resource availabilityrequirements as constraints in the model. A detailed

mathematical formulation of the problem is availablein Appendix A.

Solution MethodologiesReducing solution time for instances of the OPBSproblem is imperative, because these can containbetween 10,000 and millions of blocks, and between10 and 100 periods, as Chicoisne et al. (2012) illus-trate. Most commercial solvers, such as CPLEX (IBM2011), utilize the branch-and-bound (B&B) algorithmto solve integer programs. Our solution method-ologies are intended to enhance the solver’s func-tionality by exploiting the special structure of theOPBS problem. The following solution methodolo-gies attempt to reduce the solution time for instancesof (CPIT) by (1) tightening the formulation; (2) solv-ing the linear programming (LP) relaxation of theinteger program at the root node with the properalgorithm and corresponding algorithmic settings;(3) reducing the problem size through variable elim-ination; and (4) providing the optimizer with an ini-tial integer feasible solution. A reduction in solutiontime corresponds to a reduction in costs associatedwith in-house or consulting engineers who developand analyze scheduling scenarios, and the reductionshould contribute to an improved ability to solvelarger-sized block models.

Improved FormulationsThe formulation described in Appendix A can beimproved by making the constraint set tighter andby replacing the set of decision variables that deter-mine when a specific block must be mined, by aset of variables that indicate the period before whicha block must be mined. We denote the formulationwith the first and second improvements, (CPITatt) and(CPITby), respectively. Appendix B provides a detaileddescription of the improved formulations.

Space precludes us from presenting all methodsthat improve our (CPIT) formulation; however, sig-nificant academic work exists based on adding validinequalities (i.e., cuts). For a formulation with binaryvariables, such as ours, a valid inequality can assumethe form of a packing constraint, explicitly preclud-ing the simultaneous assignment of a value of 1 to acombination of variables, where such an assignmenthas been determined a priori to be infeasible. For

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example, if it is infeasible to extract both blocks b

and b′ in period t because of, for example, theresources required versus those available to extractthe blocks, then the corresponding valid inequalityis wbt +wb′t ≤ 1. Although numerous types of validinequalities (e.g., covers, cliques) are available, a chal-lenge for their generation lies in identifying valid(and helpful) infeasible variable combinations. Parkand Park (1997) and Boland et al. (2012), respec-tively, identify techniques to generate valid cover andclique inequalities for precedence-constrained knap-sack problems in general, whereas Gaupp (2008)presents empirical evidence of their benefit to theOPBS problem in particular.

Root Node Algorithm PerformanceAttempting to solve large-scale integer programmingproblems using the incorrect LP algorithm at the rootnode can result in excessive run time before branch-ing, whereas solving the same instance with the cor-rect LP algorithm may yield a root node solution ina matter of seconds. The two problems (primal anddual), and three algorithms (primal simplex, dual sim-plex, and barrier), yield six problem-algorithm combi-nations. Testing these six combinations is worthwhilefor practitioners who repeatedly solve instances usinga single block model. Additionally, some optimizers,such as CPLEX (IBM 2011), include a network sim-plex algorithm that is able to more rapidly solve theLP when a significant portion of that LP’s constraintmatrix forms an underlying network structure. Foran in-depth treatment of LP algorithmic selection, seeKlotz and Newman (2013a).

Variable EliminationThe optimization model need not include variablesthat must assume a value of 0 in any feasible solution;therefore, the act of identifying and eliminating suchvariables a priori results in a smaller model. In theOPBS problem, the variables ybt indicate whether toextract block b at period t. Periods t′, in which itis infeasible to extract block b, imply that the vari-able ybt′ must assume the value of 0 in any feasiblesolution, allowing us to then exclude ybt′ from theformulation. For example, consider the simple two-block model, which Figure 1 depicts for two periods,with production sufficient to extract only one block

per period. Because block 1 must be extracted priorto block 2, it is infeasible in mathematical terms (i.e.,impossible in practice) to extract block 2 in period 1.Therefore, the variable value of y21 must equal 0 inany feasible solution; thus, including y21 in the modelis unnecessary.

For each block b, there exists a period t beforewhich it is not possible to extract b while satisfying allconstraints in periods 81 0 0 t9. We refer to this period t

as block b’s true early start time (TESb). CalculatingTESb is not trivial, because it requires identifying theearliest period by which all blocks b′′ in b’s prede-cessor pit Bb may be extracted while satisfying allconstraints in each period; performing this identifica-tion could require solving an integer program for eachblock b. Although identifying TESb for each block b

may not be practical, we can still determine periodsbefore which it is impossible to extract b, and therebyidentify variables to eliminate from the formulation.This is possible by comparing the aggregate materialin the predecessor pit Bb with the aggregate opera-tional resource requirements and capacities for multi-ple periods (e.g., the aggregate capacity for t′ periodsis given by the sum of the capacities in periods {100t′}).Next, for a given block b, we review early starts (ESb),before which b may not be extracted because of maxi-mum resource capacities, and develop enhanced earlystarts (EESb), before which b may not be extractedbecause of both maximum resource capacities andminimum resource requirements. The value of TESb

may be later than these early starts; however, it can-not be earlier. Equation (1) shows the relationshipbetween these three periods before which block b maybe extracted:

ESb ≤ EESb ≤ TESb0 (1)

There also exists analogous late starts (LSb) (Gaupp2008), in which variable values are fixed to 1 in(CPITby) and eliminated in (CPITat), corresponding tothose block-period combinations for which block b

must already have been extracted. Late starts exploitthe fact that, as a result of precedence constraints,if block b is included in the predecessor pit of anotherblock b, then b may not be extracted before b is ex-tracted. This, combined with the minimum resourcerequirements, forces block b to be extracted no laterthan some period LSb. We find that although helpful

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for improving tractability in some instances, late starttimes do not generally reduce (via fixing or elimina-tion) a sufficient number of variables, such that theirinclusion has a significant performance impact.

Variable Elimination: Early Starts (ES). One nec-essary, but not sufficient, condition for block b to beextracted by t is that all of the blocks b′′ in b’s prede-cessor pit Bb be extracted no later than period t. Theearliest possible time any block b may be extracted isafter all blocks b′′ in b’s predecessor pit Bb have beenextracted, while operating at maximum productioncapacity. Once extracted, the earliest possible time anore block b may be processed is after all ore blocks b′′

in b’s predecessor pit Bb have been processed, whileoperating at maximum processing capacity. Because(CPIT) does not consider stockpiling, a block may notbe extracted prior to the more conservative (later) ofthe earliest times in which the block may be extractedor processed. More generally, a block b may not beextracted prior to the latest period in which the mostconstraining operational resource may handle all ofb’s predecessors. We call this earliest extraction timebased on maximum resource capacities the block’searly start (ES). Gaupp (2008), Amaya et al. (2009),and Chicoisne et al. (2012) employ the technique ofusing early start times to reduce variables in open-pitmodels. We present notation, used in the text in thefollowing sections, and an algorithm for determininga block’s early start time in Appendix C.

Variable Elimination: Enhanced Early Starts (EES).Although the incorporation of minimum operationalresource constraints further complicates the problemof finding an optimal solution, it also affords opportu-nities for eliminating additional variables. In general,imposing additional constraints renders more variablevalues infeasible, and by identifying those variableswhose value must equal 0 in any feasible solution, wemay further reduce the problem’s size.

From Gaupp (2008), the period ESb is determinedby the maximum resource capacities, which limit therate at which the blocks b′′ in Bb may be extractedand processed. Block b may not be extracted earlierthan ESb; however, it may also be true that block b

may not be extracted earlier than some period laterthan ESb, if block b’s predecessor set is not feasibleregarding the minimum resource requirements. In this

sense, ESb, as calculated previously, could be looseand may be tightened by accounting for additionalconstraints. Therefore, we define the enhanced earlystart time (EESb) as that period before which it isinfeasible to extract block b based on both the min-imum resource requirement and maximum resourcecapacity constraints.

Consider the determination of block 7’s early starttime in the eight-block example shown in Figure 3.All blocks weigh 10 tons, blocks 6 and 8 are oreblocks, and lower-level blocks’ predecessors includethe three overlying blocks (e.g., block 6’s predecessorsare blocks 1, 2, and 3). For the case of a maximumproduction capacity of 40 tons per period, and nominimum processing requirement (i.e., e1 = 40, e2 = 0),block 7 may be extracted during period t = 1 usingan extraction sequence of 2-3-4-7. Hence, in this case,block 7 has an early start time of ES7 = 1.

For the case in which there is an additional min-imum processing requirement of 10 tons of ore ton-nage per period (e1 = 401 e2 = 10), the one-periodextraction sequence of 2-3-4-7 does not contain anore block to process in t = 1, and is therefore infea-sible. A two-period feasible extraction sequence of1-2-3-6 in t = 1 and 4-7-5-8 in t = 2 provides totaltonnage no greater than e1, and ore tonnage no lessthan e2, in both periods. Therefore, extracting 7 int = 1 satisfies the maximum production constraint,yielding ES7 = 1, but extracting block 7 prior to t = 2violates the minimum processing constraint, yieldingEES7 = 2. Because EES7 = 2 is greater than ES7 = 1,we may eliminate additional variables, excluding w71

1 2 3 4 5

6 7 8

Figure 3: In this cross-sectional view of an eight-block pit, blocks 6 and8 are ore blocks, and precedences follow the “+” sign convention in twodimensions, similar to that depicted in Figure 2. For example, a feasibleschedule must call for extracting blocks 1, 2, and 3 prior to extractingblock 6.

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and y71 from the (CPITby) and (CPITat) formulations,respectively.

The preceding EES discussion focuses only on theminimum processing requirement, because satisfy-ing the minimum production requirement is alwayspossible. This follows because extracting any block,ore or waste, from anywhere in the pit, includingsurface blocks unconstrained by precedence require-ments, contributes to satisfying minimum production.Therefore, for a case in which minimum processing isbinding in the ESth

b period (i.e., the period correspond-ing to ESb) and minimum production is not satisfied,extracting any blocks external to Bb can be used toincrease total tonnage and satisfy minimum produc-tion during the ESth

b period.Unlike in the small preceding example, we can-

not generally determine block b’s EESb by inspection.In Appendix D, we present both an algorithm fordetermining a block’s enhanced early start time andassociated notation, which is also used next. A posi-tive residual processing requirement (e2b) in the ESth

b

period after extracting Bb (i.e., e2b = e2 ·ESb − S2b > 0)indicates that EESb ≥ ESb. However, eliminatingthe variable wb1ESb

(or yb1ESb) requires proving that

EESb > ESb, which is not always trivial. Specifically,determining EESb can be complicated for the situa-tion in which e2b > 0, and there also exists a positiveresidual production capacity (e1b) in the ESth

b period,(i.e., e1b = e1 · ESb − S1b > 0).

For the case in which e1b ≥ e2b, it may still be pos-sible to extract sufficient ore blocks outside of Bb tosatisfy e2 in the ESth

b period. Adequate residual pro-duction capacity exists to extract the ore blocks nec-essary to satisfy the residual processing requirement,if those ore blocks are accessible and can be extractedin a feasible sequence. Therefore, showing EESb > ESb

generally requires proving that there does not exist anextraction sequence of blocks external to Bb contain-ing sufficient ore blocks to meet or exceed e2b, whilenot exceeding e1b. In the worst case, this is an expo-nentially complex problem of enumerating each pos-sible block sequence, likely requiring the solution ofan integer program. Therefore, for the case in whiche1b ≥ e2b, without solving an integer program, we mustmake the conservative assignment of EESb = ESb, andare unable to eliminate the variable wb1ESb

(or yb1ESb).

For the case in which e1b < e2b, even if sufficient oreblocks are available, it is impossible to satisfy e2 inthe ESth

b period without violating e1, guaranteeing thatEESb > ESb. Although EESb may exceed ESb by morethan one period, establishing this requires proving thenonexistence of feasible extraction sequences in suc-cessive periods. Therefore, in this situation, EESb >ESb, but we cannot prove by how much; therefore,we make the conservative assignment EESb = ESb + 1,and eliminate the variable wb1ESb

(or yb1ESb).

Again, it is possible to relax the assumption of time-invariant maximum resource capacities, although wedo not include the corresponding more general algo-rithm here. However, a formula analogous to Equa-tion (C1) would require interaction between differentresources (e.g., r and r ′). This complicates a genericformula, because it would then be highly dependenton the resources to which r and r ′ refer; in our case,r and r ′ have a very specific relationship, not a gen-eral relationship. Our formulation contains resource r(i.e., production) with a nonzero value for all blocksand resource r ′ (i.e., processing) with a nonzero valuefor some of the blocks. And, by comparing r and r ′

in a particular way (i.e., the lower bound on r ′ withthe upper bound on r) we can, under certain cir-cumstances, increase the start time of a block by oneperiod.

Initial Integer Feasible Solution (IIFS)Providing the optimizer with an IIFS should expeditethe B&B search for an optimal solution both by allow-ing an initial lower bound to preclude evaluation ofdominated solutions (i.e., solutions not guaranteed tobe feasible, but if feasible, are guaranteed to havean objective function value lower than that achievedwith the IIFS), and by assisting the optimizer to moreeffectively tailor its search strategies. The challengeis identifying such a solution. Some heuristics to thisend are as follows:

• Greedy algorithms can quickly identify an initialsolution to a problem specification in which resourcesare only constrained by maximum capacities (i.e., nominimum requirements exist). For example, we mayorder blocks, descending by profit, and then chooseto extract each block, along with its predecessor set,if sufficient resource capacity exists. We repeat thisprocedure for each period, creating a block extractionschedule for the entire time horizon.

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• Chicoisne et al. (2012) and Moreno et al. (2010)extend this simple approach by exploiting the prece-dence constraints’ underlying network structure toinduce a topological sorting of nodes (i.e., blocks).This sorting may be based on any attribute, includ-ing LP relaxation values, as demonstrated by theirExpected-Time TopoSort heuristic. Using these tech-niques, Moreno et al. (2010) show that initial feasiblesolutions may be obtained quickly for large data sets.

• Although greedy heuristics work well whenthe problem specification includes only maximumresource capacities, finding a feasible solution issignificantly complicated by including minimumresource requirements. A sliding time-window heuris-tic (STWH) (Pochet and Wolsey 2006) may be morelikely to determine an integer feasible solution than asimple greedy approach. With the STWH, a subprob-lem is formed by partitioning the set of periods in thehorizon T into windows (i.e., T ≡ T Fix ∪ T Cont ∪ T Int),within which variable values are fixed (T Fix), or theirintegrality is either relaxed (T Cont) or enforced (T Int).Solving this subproblem produces a partial solution,which is integer feasible for variables in T Int. Then,a subsequent subproblem is formed by changing thepartition of T (i.e., the window slides), so integral-ity is enforced in a separate set of periods, and theprocedure is repeated. Finally, linking the partial inte-ger feasible solutions from each subproblem creates acomplete solution. Although this approach does notguarantee an integer feasible solution for a monolithwith minimum resource requirements, even an infea-sible solution may assist the optimizer in its searchfor an optimal solution. See Cullenbine et al. (2011)for further discussion.

• An alternative approach leverages the seminalwork of Lerchs and Grossmann (1965) by finding,within the entire pit, a maximum-value subpit con-taining sufficient ore and material to fulfill the totalresource requirements over the problem’s time hori-zon. Then, an integer program may identify an integerfeasible sequence within that subpit, which satisfiesthe resource constraints in each period. This resultingsolution, combined with setting variable values to 0for all blocks outside the subpit, forms a complete ini-tial feasible solution, as Lambert and Newman (2013)fully explain.

ResultsWe empirically test the strategies outlined in the pre-vious section using 12 open-pit mine block modelsderived from two master data sets. The first masterdata set is an open-pit mine block model consistingof 10,819 blocks, approximately 5,600 tons each, afterthe elimination of irrelevant blocks (e.g., waste blocksat the bottom level of the pit, and air blocks of 0 ton-nage used for graphical display). From this masterdata set, referred to as 10k, we perturb each block’smineral content by ±5 percent to generate seven moreblock models, identified as 10kA110kB1 0 0 0 110kG; wecollectively refer to them as the 10k data sets.

The second master data set is the well-knownMarvin test data set available from Whittle soft-ware, which Norm Hanson created while at the RoyalMelbourne Institute of Technology (Hanson 2012).This fictional, yet realistic, data set models typi-cal deposits found in New South Wales, Australia,and contains 53,668 blocks, approximately 62,400 tonseach, after eliminating air blocks from the top of thepit. We use four smaller subsets from this full Marvinfor test cases, two each of size 18,300 (18kA and 18kB),and 25,620 (25kA and 25kB) blocks. We refer to thesecollectively as the Mrv data sets. Table 1 displayssummary characteristics of all data sets.

Number of blocks Tonnage (Mtons)

Data set Total Ore Waste∑

b∈B a1b∑

b∈B a2b

10k 10,819 1,423 91396 60057 709210kA 10,819 1,423 91396 60057 709210kB 10,819 1,414 91405 60057 708710kC 10,819 1,414 91405 60057 708710kD 10,819 1,410 91409 60057 708510kE 10,819 1,418 91401 60057 708910kF 10,819 1,423 91396 60057 709210kG 10,819 1,420 91399 60057 709010k avg 10,819 1,418 91401 60057 708918kA 18,300 2,032 161268 11145047 14606218kB 18,300 1,436 161864 11137042 10303925kA 25,620 2,248 231372 11595062 16200325kB 25,620 2,805 221815 11601013 199095Mrv avg 21,960 2,130 191830 11369091 153000

Table 1: We test our solution strategies on these 12 data sets with charac-teristics including the number of blocks by type (e.g., total, ore, waste),and the tonnage required to extract all blocks and process all ore blocks.

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We formulate our models in the AMPL program-ming language, version 11.11 (AMPL 2009) and thenpass this formulation to the CPLEX solver, version12.2.0, (IBM 2011), which runs on a Sun X4150, with2 × 2083 GHz processors, and 16 GB RAM.

Root Node Algorithm PerformanceThe algorithm selection for solving the LP relaxationof the problem instance at the root node can dra-matically impact overall solution time. Consider theresults displayed in Table 2 from solving the LP relax-ations with a horizon of � = 10. For the 10k data sets,solving with CPLEX default parameter settings (i.e.,dual simplex performed on the dual problem as inIBM 2011) finds no feasible solution within 1,800 sec-onds for any of the instances tested, whereas solvingthe primal problem using the primal simplex methodfinds an optimal solution within 1,800 seconds for sixof the eight instances. The barrier algorithm gener-ally performs best on the 10k sets, finding the optimalsolution in less than 900 seconds. The Mrv instancesexhibit the best performance using the primal simplexmethod, whereas the barrier algorithm is unable to

4P 5—Primal problem 4D5—Dual problemCPLEX

Data set default PS∗ DS∗ Barrier PS DS Barrier Netopt

10k †b 998a ‡a 807 †a †a 831 99210kA †b 11606a ‡a 732 †a †a 740 1158710kB †b 11298a ‡a 707 †a †a 737 1129110kC †b 11478a ‡a 712 †a †a 805 1147010kD †b 11093a ‡a 733 †a †a 852 1107510kE †b †a ‡a 732 †a †a 743 1197610kF †b 11464a ‡a 730 †a † 691 1145110kG †b †a ‡a 684 †a † 758 2103618kA 352b 105 11380a † †a †a † 8518kB 210b 86 11374a ‡ †a †a † 6625kA ‡b 135 11086a ‡ †a † † 10725kB ‡b 604 11008a ‡ †a † ‡ 577

Table 2: When solving problem instances for � = 10 years with no ES vari-able reductions, we obtain varying LP relaxation solution times (in sec-onds) by using different problem-algorithm combinations.

aCPLEX introduces a perturbation, suggesting degeneracy.bCPLEX reports default settings result in solution being found with dual

simplex on the dual problem; however, CPLEX uses different algorithmson multiple threads during concurrent optimization.

†Time limit of 1,800 secs in phase I.‡Time limit of 1,800 secs in phase II.∗PS: primal simplex; DS: dual simplex.

find an optimal solution within 1,800 seconds. A com-mon approach that gives good performance acrossall of our instances suggests CPLEX’s network sim-plex algorithm (Netopt), which is able to extract andexploit the problem’s underlying network structure.

The difference in performance between the primalsimplex method and barrier algorithm on the Mrvand 10k data sets can be explained as follows: usingthe primal simplex on the latter data sets requires asignificant amount of solution time in phase I, hasdifficulty because of degeneracy, and fails to utilizea favorable pricing scheme until a significant num-ber of phase II iterations have passed. By contrast,the same algorithm on the Mrv data sets spends littletime in phase I, exhibits no degeneracy, and recog-nizes a good pricing scheme (i.e., devex) upon switch-ing to phase II. Using the barrier algorithm on the 10kdata sets requires little time in crossover (i.e., find-ing a basic solution from one lying on the boundary),whereas the Mrv data sets require a long time forthis procedure. Arguably, alternate parameter settings(e.g., those to address degeneracy up front, or those todiminish the time spent in crossover) might help mit-igate the performance discrepancies between the twoalgorithms applied to these two data sets. However,experimentation at this level was beyond the scope ofour numerical testing.

“By” vs. “At” FormulationIn solving integer programs, the tighter the bound(i.e., LP relaxation objective function value), the bet-ter the B&B algorithm performance (Bertsimas andWeismantel 2005, p. 11). In Table 3, we present LPrelaxation objective function values 4zLP) to quantifythe benefit of tighter formulations. The second andthird columns contain LP relaxation values from theat formulation with no variable reductions, and withour greatest variable reductions from enhanced earlystarts, respectively. Although this preprocessing doestighten the bound, using either the att or by formu-lation (because they provide the same LP relaxationobjective function value, and as such, are equivalentlystrong), in conjunction with these reductions, tight-ens the bound to a much greater extent. As such,we achieve all further results using the by (equiv-alently, att) formulation. We present the percentage

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Linear program relaxationobjective function value ($ M)

Data set zLP 4CPITat5 zEES

LP 4CPIT at5 zEES

LP 4CPIT by51 �4at1 by5 (%)

10k 25080 18084 10097 410810kA 25093 18087 11001 410710kB 25092 18090 10098 410910kC 25080 18077 10091 410910kD 25079 18076 10096 410610kE 25071 18077 10090 420010kF 25092 18095 11001 410910kG 25077 18083 10096 4108

10k avg 25083 18084 10096 4108

18kA 146057 146019 133008 90018kB 117033 117033 114026 20625kA 143015 143015 140078 10725kB 132069 123082 85070 3008

Mrv avg 134094 132062 118046 1100

Table 3: When solving problem instances for � = 10 years, using bothvariable reductions (EES ) and the by formulation provide tighter LP relax-ation values than solving the same problem instances with no variablereductions and the at formulation.

1zEESLP 4CPIT att

5 possesses the same objective function value aszEESLP 4CPIT by

5.

Number of variables Solution time (secs)1 Solution time reduction (%)

Data set NoES 2 ES EES NoES ES EES 4NoES − ES 5/NoES 4ES − EES 5/ES

10k 108,190 421185 401373 161876 41886 31678 71% 25%10kA 108,190 421185 401362 171026 71072 31502 58% 50%10kB 108,190 421185 401366 291059 31897 31538 87% 9%10kC 108,190 421185 401379 211535 41191 31529 81% 16%10kD 108,190 421185 401353 161750 41714 41278 72% 9%10kE 108,190 421185 401371 241024 61256 31983 74% 36%10kF 108,190 421185 401380 181044 31976 51814 78% −46%10kG 108,190 421185 401359 161356 41377 41223 73% 4%10k avg 108,190 421185 401368 191960 41921 41068 74% 17%18kA 183,000 1781711 1721113 ‡ ‡ 351449 ∗ ∗

18kB 183,000 1781818 1721129 1819733 2713994 261553 −44% 3%25kA 256,200 2291808 2161586 ‡ ‡ ‡ ∗ ∗

25kB 256,200 2301013 2161634 ‡ ‡ ‡ ∗ ∗

Table 4: When solving problem instances for � = 10 years within a time limit of 10 hours, IP solution times aregenerally shorter with a greater number of variable reductions.

1Solution times do not include the time required to calculate ESs and EESs. We generate both ESs and EESsfor all blocks in a given data set in fewer than 10 seconds.

2NoES indicates no variable reductions.3Heuristic found first and final integer solution after 18,844 seconds.4Heuristic found first and final integer solution after 27,350 seconds.‡Time limit (36,000 secs) reached with no integer solution.∗Unable to calculate due to lack of solution time values.

reduction in the LP relaxation value, which quantifiesthe relative strength of (CPITby) over (CPITat) as:

�4at, by5= 100% ·zEES

LP 4CPITat5− zEESLP 4CPITby5

zEESLP 4CPITat50 (2)

Variable EliminationOur data sets each contain a significant number ofvariable reductions attributable to ES and EES, asTable 4 shows. Although instances with fewer vari-ables and constraints generally lead to faster solu-tion times than instances with no variable reductions(NoES), occasionally CPLEX’s proprietary heuristicsare able to find solutions more quickly than wouldbe suggested by theory (e.g., in the cases of 10kFand 18kB). However, Table 4 demonstrates an addi-tional savings of 17 percent in solution time, on aver-age, for the 10k data sets beyond what the early startreductions alone can achieve. We find the numberof variable values fixed by late starts, introduced byGaupp (2008), does not justify the effort required fortheir calculations. For example, in the 10k master data

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set, only 48 of the 108,190 variable values may befixed because of late starts. Hence, none of our runsinclude variables fixed to their LS values. The relevantCPLEX parameter settings for all runs include primal(i.e., set up the primal formulation), predual − 1 (i.e.,solve the primal problem), startalgorithm 3 (i.e., usethe network simplex algorithm to solve the LP relax-ation at the root node), subalgorithm 1 (i.e., use the pri-mal simplex algorithm to solve the linear programsat all nodes of the B&B tree, other than at the rootnode), mipemphasis 1 (i.e., emphasize feasibility overoptimality when searching through the B&B tree),varsel 4 (i.e., branch in the tree based on pseudo-reduced costs), probe − 1 (i.e., preclude spending timeto explore binary variable assignment implications,and then preset variables accordingly), mipgap 0.02(i.e., solve the problem to within two percent of opti-mality); the meanings of these parameter settings arespecified in IBM (2011).

Initial Integer Feasible Solution (IIFS)Providing the optimizer with an IIFS expedites theB&B algorithm by precluding the optimizer fromevaluating dominated solutions and by enhancing theoptimizer’s search path (Klotz and Newman 2013b).

Objective function value ($ M) Solution time (secs)

Data set zEESLP zEES

STWH Mono with no IIFS STWH+Mono with IIFS IIFS % reduction

10k 10097 10064 31678 21323 37%10kA 11001 10065 31502 21523 28%10kB 10098 10058 31538 21595 27%10kC 10091 10027 31529 21145 39%10kD 10096 10072 41278 9241 78%10kE 10090 10048 31983 21447 39%10kF 11001 10054 51814 31430 41%10kG 10096 10061 41223 21285 46%10k avg 10096 10056 41068 21334 42%18kA 133008 125020 351449 241256 32%18kB 114026 112073 261553 117651 93%25kA 140078 † ‡ † ∗

25kB 85070 † ‡ † ∗

Mrv avg2 123067 118096 311001 131011 62%

Table 5: We see improvements in both solution quality and time (secs) for the monolith when provided with anIIFS from the STWH for � = 10, and solved to within a mipgap tolerance of two percent.

1STWH generated initial solution within 2% mipgap for monolith.2Only includes completed runs for Mrv data sets.†Time limit (36,000 secs) reached during STWH with no integer solution.‡Time limit (36,000 secs) reached with no integer solution.∗Unable to calculate as a result of a lack of solution time values.

Additionally, the optimizer can leverage aggressivecut generation to tighten the upper bound and exploitthe use of its own heuristic more frequently to searchfor better integer solutions. Our technique requirestwo phases; in the first, we generate the IIFS withthe STWH; in the second, we solve the monolith withthat IIFS. Deducting the total time required for thesetwo phases from that required to solve the monolithwithout an IIFS yields the reduction in solution timeattributable to the use of an IIFS.

Implementing the STWH requires selecting thenumber of windows, and in each of those windows,how many periods in which to enforce integrality andto fix variable values. Enforcing integrality in fewerperiods requires more windows of relatively easierinteger programs to solve. Conversely, enforcing inte-grality in more periods requires fewer windows ofrelatively more difficult integer programs to solve. Wefind the best STWH solution times for � = 10 resultfrom implementing five windows, fixing and enforc-ing integrality in two periods (i.e., �T Fix� = �T Int� = 2)for each window.

The results in the objective function value columnsof Table 5 show that the STWH generates good qual-ity solutions. The solution time columns of Table 5

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show that generating the IIFS and then solving themonolith with that IIFS reduces solution time relativeto solving the monolith outright. These reductions insolution time are 42 and 62 percent, on average, forour 10k and Mrv data sets, respectively.

ConclusionsPractitioners face numerous challenges when imple-menting mathematical programs to solve difficultreal-world problems, such as the OPBS problem.This paper demonstrates a variety of techniques thatexpedite solutions: (1) alternative variable definitionsenabling a tighter formulation, (2) algorithmic selec-tion for solving the root node linear programmingrelaxation, (3) variable elimination, and (4) providingthe solver with an initial feasible solution. Althoughwe show that all techniques can be effective, the suc-cess of their implementation can be highly depen-dent on the specific problem instance (e.g., data setcharacteristics, time horizon). For example, recall fromTable 2 that when solving the root node LP relax-ation for the 10k data sets, the barrier algorithm per-forms best, whereas for the Mrv data sets, primalsimplex performs best. Also, although fixing variablevalues via late starts did not justify their calculationfor our data sets, they may provide justifiable ben-efits for problem instances with tight lower boundson resource constraints or for instances that representmines with different topologies.

Theory suggests that the aforementioned tech-niques should reduce solution time; however, this isnot always the case when using a modeling languageand commercial optimizer. Next, we describe some ofthe more frustrating, and sometimes surprising, diffi-culties we encountered in our implementation.

Optimizer Presolve and Heuristics. Commercialoptimization software may improve overall perfor-mance with various presolve activities (e.g., problemsize reduction, generation of the equivalent dual, val-idation of mixed-integer programming (MIP) starts),and with heuristics that modify search strategies. Pre-solve can provide significant improvements in per-formance; however, in doing so, the problem maybe modified in ways the user might not prefer. Forexample, one method to reduce time spent in solvingthe LP relaxation of a MIP’s root node is to pass the

optimizer an optimal basis, previously determinedwhen finding an IIFS. However, to do so requiresturning off the MIP presolve to ensure no basic vari-ables are eliminated via, for example, CPLEX’s prob-ing feature, which fixes deduced variable values, priorto solving the root relaxation. Hence, the user savestime otherwise spent solving the LP at the root node,but loses time savings that might have been achievedduring B&B.

Heuristics (see Achterberg 2007), can also signif-icantly improve performance. However, by defini-tion, their application does not constitute an exactapproach, and as such, provides no guarantee ofperformance. Introducing a technique that theoreti-cally should provide superior performance does notalways result in reduced solution time, because aheuristic implemented within the solver’s B&B algo-rithm may get lucky and find a solution more quickly(e.g., see the 10kF results in Table 4). Still, superiorformulations, problem reduction techniques, and pro-viding the solver with an initial solution all result in asimpler problem to solve; therefore, they should gen-erally yield faster solution times.

Optimizer Parameter Settings. Commercial solversafford the user some control, via the use of algo-rithmic parameter settings, over how an algorithmis executed. Although these user-adjusted parameterscan impact solution time, each parameter’s individ-ual effectiveness is not always explicitly clear. Inter-nal heuristics, influenced directly by the parametersettings and only indirectly by the user, determinethe actual search path, which has a large impact onsolution time. Therefore, modifying any one param-eter setting, in conjunction with other parameter set-tings, may alter the heuristic’s search path, resultingin a significant change in solution time. This indirect,interactive nature of the numerous parameters, eachwith their various settings, can make it difficult tofind the single combination providing the best perfor-mance (i.e., the fastest solution time).

CPLEX includes a tuner tool (IBM 2011) to helpidentify that best combination of parameter settings.The tuner, however, can recommend very different set-tings for slight problem variations (i.e., the same dataset with different time horizons, or slightly differentdata sets, such as 10kA and 10kC, with the same time

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horizon). These different recommended parameter set-tings are not just artifacts of the tuner, but resultfrom the varying search paths previously mentioned.A reasonable approach is to obtain initial settings fromthe tuner and then test modified parameter settingsbased on the user’s knowledge of the problem. Forexample, tuning for some of the 10k data sets sug-gests precluding initial probing and using pseudo-reduced costs to select the subnode variable on whichto branch (CPLEX probe − 1 and varsel 4, respectively).Although these perform better than CPLEX defaultsettings, even faster solution times result from addinga search strategy of emphasizing feasibility over opti-mality (CPLEX mipemphasis 1). This follows from theinherent difficulty of finding feasible solutions to theOPBS problem. To select a consistent group of settingsto employ when solving all models, the user mustsometimes accept significant performance trade-offs,such as those that Table 2 discusses.

The aforementioned lack of direct user control overthe solver suggests that reductions in solution timemay not all be the result of our techniques. Althoughtheory supports our results, it is possible that ourtechniques happen to provide (this magnitude of)reductions because of the solver’s internal algorithmsand heuristics. A valid avenue for future research isto implement the techniques described in this paperwith other solvers to verify similar solution-timereductions.

Solving Linear Programs. As discussed in the RootNode Algorithm Performance section, effectively solv-ing LPs may significantly reduce solution times. Forexample, the number of LPs solved in the monolithalone versus with an IIFS provided by the STWH dif-fers significantly, on average. Solving the monolithrequires solving one root relaxation, whereas solv-ing the monolith with an IIFS (via our variant ofthe STWH) requires solving the monolith’s root relax-ation, in addition to the five root relaxations requiredfor the STWH (one for each window). In the STWH,the cumulative time required to solve the root relax-ation LPs can be greater than that required to exe-cute the B&B algorithm. For example, in one instance(10kG), the STWH solves in 1,940 seconds, but onlyexecutes B&B for 40 seconds. This means 96 per-cent of the time is spent performing presolve, execut-ing heuristics, and solving LPs. We find it helpful to

approach solving the OPBS problem by explicitly rec-ognizing the separate and very significant challengeto solving these LPs.

Despite performance irregularities and the intrac-tability of many large detailed models that use currentstate-of-the-art hardware and software, computingpower has made great strides in the past severaldecades (Bixby and Rothberg 2007). Knowing how tocarefully formulate models and appropriately invokesolvers can transform an intractable instance intoa solvable one. In this tutorial, we concentrate on(CPIT), a specific variant of the OPBS problem. Thereader should be able to glean computational insightsnot only for this problem variant, but for other integerprogramming mine production scheduling problems.

AcknowledgmentsWe wish to acknowledge Dr. Ed Klotz of IBM for hispatience, expertise, and willing assistance regarding expla-nations of the interaction between CPLEX parameters, inter-pretations of node logs, and clarification on the subtletiesinvolved with solving the root relaxation of an integer pro-gram with CPLEX.

Appendix A. Mathematical Formulation of theConstrained Pit Limit Problem (CPIT)Indices and sets:

• b ∈ B: set of all blocks b,• b′ ∈ Bb : set of blocks that must be extracted directly

before block b, (i.e., b’s direct predecessors),• b′′ ∈ Bb : set of all blocks which must be extracted

before block b, (i.e., b’s predecessor pit: Bb ⊆Bb),• t ∈ T : set of periods t,• r ∈R: set of operational resources r (1 = production of

ore and waste, 2 = processing of ore).

Data:• � : length of time horizon (i.e., �T � ≡ �),• vbt : NPV generated by extracting block b in period t ($),• arb : nonnegative amount of operational resource r used

to process block b (tons),• er/er : nonnegative, per-period minimum required

usage/maximum usage capacity for operational resource r(tons).

Decision Variables:

ybt =

{

1 if block b is extracted at time t,

0 otherwise.(A1)

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Objective Function:

4CPIT 5 max∑

b∈B

∑

t∈T

vbtybt (A2)

Constraints:

ybt ≤∑

t′≤t

yb′t′ ∀ b ∈ B1 b′∈ Bb1 t ∈ T 1 (A3)

er ≤∑

b∈B

arbybt ≤ er ∀ r ∈R1 t ∈ T 1 (A4)

∑

t∈T

ybt ≤ 1 ∀ b ∈ B1 (A5)

ybt ∈ 80119 ∀ b ∈ B1 t ∈ T 0 (A6)

The objective function (A2) seeks to maximize the NPVof all extracted blocks. Although intuitive, our maxi-mization of NPV is not a formulation requirement, andwithout loss of generality, we could consider other objec-tive functions provided they are linear. Constraints (A3)ensure block b is not extracted at period t unless everyblock b′ in b’s direct predecessor set (Bb) is extracted at,or prior to, period t. Constraints (A4) enforce minimumresource requirements and maximum resource capacities ineach period. Although our definition of r considers onlytwo resources, this is for expository purposes only; with-out loss of generality, the formulation can accommodateother side constraints in the form of knapsacks. Regardingthe data used in these knapsack constraints (i.e., arb , er , er ),without loss of generality in the formulations we consider,we express the data in tonnages. Constraints (A5) preventany block b from being extracted more than once, whereasconstraints (A6) restrict all decision variables to be binary.

Appendix B. Improved FormulationsThe (CPIT) decision variables ybt , as defined in Equa-tion (A1), assume a value of 1 if block b is extracted attime t, and 0 otherwise. We therefore designate the (CPIT)problem thus formulated and presented in objective func-tion (A2) and constraints (A3) to (A6) as (CPIT at). Althoughformulating (CPIT) with variables defined this way appearsin previous work, for example, Dagdelen and Johnson(1986), Ramazan (2007), and Osanloo et al. (2008), the for-mulation may be tightened. Specifically, constraint (A3) pre-vents block b from being extracted in the single period tprior to b’s direct predecessors b′, which implies that block bmay also not be extracted in all periods prior to t. There-fore, summing ybt′ over all t′ ≤ t, as in constraint (B1), isvalid and is stronger than constraint (A3) because of theadditional variables (with positive coefficients) included onthe left side of the constraint

∑

t′≤t

ybt′ ≤∑

t′≤t

yb′t′ ∀ b ∈ B1b′∈ Bb1 t ∈ T 0 (B1)

For an in-depth comparison of OPBS models and formu-lations, see Newman et al. (2010); regarding dominance ofvalid inequalities, see Wolsey (1998, p. 141). Replacing con-straint (A3) in (CPIT at) with constraint (B1) provides atighter formulation, which we refer to as the tightened atformulation, and denote as (CPIT att). This tightened formu-lation yields a lower (tighter) LP relaxation value for theroot node, thereby leaving a smaller integrality gap for aninteger programming solver to close.

Another formulation-tightening approach is to definedecision variables that assume a value of 1 if block b ismined by period t and 0 otherwise, as in Expression (B2):

wbt =

{

1 if block b is extracted by time t1

0 otherwise.(B2)

This requires a variable substitution, where the by variablevalues may be translated to their corresponding at variablevalues using Equation (B3):

ybt = wbt −wb1 t−1 ∀ t ∈ 811 0 0 0 1 T 91

where wb0 ≡ 01∀ b ∈ B0 (B3)

For example, if a solution requires block b to be extracted inperiod t = 3 during a time horizon of T = 5, the at variablevalues for block b are:

yb1 = 01 yb2 = 01 yb3 = 11 yb4 = 01 yb5 = 01

while the corresponding by variable values for block b are:

wb1 = 01 wb2 = 01 wb3 = 11 wb4 = 11 wb5 = 10

The corresponding by formulation is as follows:

Objective Function:

4CPIT by5 max∑

b∈B

∑

t∈T

vbt

(

wbt −wb1t−1

)

(B4)

Constraints:

wbt ≤wb′t ∀ b ∈ B1b′∈ Bb1 t ∈ T 1 (B5)

er ≤∑

b∈B

arb(

wbt −wb1t−1

)

≤ er ∀ r ∈R1 t ∈ T 1 (B6)

wb1t−1 ≤wbt ∀ b ∈ B1 t ∈ T 1 (B7)

wbt ∈ 80119 ∀ b ∈ B1 t ∈ T 3wb0 ≡ 0 ∀ b0 (B8)

The objective function value (B4) and constraints (B6)and (B8) of (CPIT by) are identical to expressions (A2),(A4), and (A6) of (CPIT at), respectively, where the at vari-ables (ybt) are replaced either with their by variable coun-terparts (wbt) or with their transformation (wbt − wb1t−1).The number of precedence constraints (B5) equals thatin constraint (A3); however, the by variable definition inconstraints (B5) yields a less dense representation in theconstraint matrix than that in constraint (A3). The prob-lem (CPIT by) does not require constraints equivalent toconstraint (A5), but instead contains the additional con-straints (B7), which enforce the by behavior. Although

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(CPIT by) has an additional �B� · �T − 2� constraints more than(CPIT at) (where �N � denotes the cardinality of the set N ),our results show that (CPIT by) provides a tighter LP relax-ation bound than does (CPIT at), and consequently improvesoverall performance when solving (CPIT).

The problems (CPIT by) and (CPIT att) produce the sameLP relaxation objective function values; however, (CPIT by)has a less dense constraint matrix than does (CPIT att).More importantly, constraints (B5) make explicitly clear theunderlying network structure. This insight, and use of theby variables, was first proposed in Johnson (1968).

Appendix C. Early Start Algorithm

Early Start (ES) Algorithm —Additional Notation• Srb : Total tonnage of blocks in the set Bb consuming

operational resource r (tons).• ESrb : Earliest start time of block b based on the maxi-

mum capacity of operational resource r .• ESb : Earliest start time of block b based on the most

constraining operational resource.

Early Start (ES) AlgorithmDESCRIPTION: An algorithm to calculate the earliest time ablock may be extracted and, if applicable, processed, basedon maximum resource capacities.

NOTE: We use the ceiling function (�a/b�), which roundsa/b up to the next largest integer if a/b is fractional, andreturns a/b otherwise.

INPUT: B, er ≥ 0, and arb ≥ 0, Bb1∀ b ∈ B.OUTPUT: ESb1∀ b ∈ B, the earliest time block b may be

extracted and, if applicable, processed, based on maximumresource capacities. The “←” notation indicates the assign-ment of maxr 8ESrb9 to ESb .{

for (all blocks b ∈ B) {Srb =

∑

b′′∈Bb

arb′′

ESrb = �Srb/er�

ESb ← max4ESrb5}

}

Note that we could relax the assumption of a constantoperational resource capacity for all periods and, based on ageneralization of the previous computation, define the earlystart time for block b as follows:

ESb = maxr∈R

min{

t 2∑

b′′∈Bb

arb′′ ≤

t∑

t′=1

ert′

}

0 (C1)

Appendix D. Enhanced Early Start Algorithm

Enhanced Early Start (EES) Algorithm—AdditionalNotation

• Trb : Number of periods for which the minimumresource requirement er is supportable by Srb (periods).

• e1b : Residual production capacity after extracting S1b inESb periods (tons).

• e2b : Residual processing requirement after extractingS2b in ESb periods (tons).

• EESb : Period before which it is infeasible to extractblock b given minimum resource requirement and maxi-mum resource capacity constraints (period).

Enhanced Early Start (EES) AlgorithmDESCRIPTION: An algorithm to calculate the earliest timea block may be extracted and, if applicable, processed, con-sidering the aggregate minimum and maximum resourcecapacities over ESb periods.NOTE: We use the floor function (�a/b�), which rounds a/bdown to the next smallest integer if a/b is fractional, andreturns a/b otherwise.INPUT: B1 e11 e21 and ESb1 S1b1 S2b1 ∀ b ∈ B.OUTPUT: EESb , the time period before which it is infeasi-ble to remove block b considering the aggregate minimumresource capacities, ∀ b ∈ B.{

for (all blocks b ∈ B) {

T2b = �S2b/e2�

If ESb ≤ T2b then {EESb ← ESb

}

Else {

e1b = e1 ∗ESb − S1b

e2b = e2 ∗ESb − S2b

If e1b < e2b then {

EESb ← ESb + 1

}

Else {

EESb ← ESb

}

}

}

}

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W. Brian Lambert is a senior consultant with Minemax,Inc. He holds a BA in general physics from the Universityof California, Los Angeles, an MS in operations researchfrom the Naval Postgraduate School, and a PhD in mineraland energy economics from the Colorado School of Mines.Prior to joining Minemax, Inc., he consulted with life sciencecompanies as a decision and R&D portfolio analyst withKromite, and was an operations analyst and naval aviatorin the United States Marine Corps.

Andrea Brickey is a licensed professional mining engi-neer and is a PhD candidate in mining engineering at theColorado School of Mines in Golden, Colorado. She consultsfor a global mining company and has worked extensively inAfrica and North and South America mining copper, gold,silver, nickel, phosphate, and coal.

Alexandra M. Newman is an associate professor in theDivision of Economics and Business at the Colorado Schoolof Mines. She holds a BS in applied mathematics from theUniversity of Chicago, an MS in operations research fromthe University of California, Berkeley, and a PhD in indus-trial engineering and operations research from the Univer-sity of California, Berkeley. Prior to joining the ColoradoSchool of Mines, she held the appointment of research assis-tant professor in the Operations Research Department at theNaval Postgraduate School. Her primary research interestslie in optimization modeling, particularly as it is appliedto logistics, the energy and mining industries, and mili-tary operations. She is serving as associate editor of Oper-ations Research and Interfaces, and is an active member ofINFORMS.

Kelly Eurek is an energy analyst in the Strategic EnergyAnalysis Center at the National Renewable Energy Labora-tory. He holds a BS in mechanical engineering and an MSin engineering and technology management from the Col-orado School of Mines. His primary research interests lie inoptimization modeling as it is applied to the electric powersector.

Documents

Open-Pit Block-Sequencing Formulations: A Tutorial