extraction and backﬁll scheduling in a complex...

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Vol. 44, No. 2, March–April 2014, pp. 204–221ISSN 0092-2102 (print) � ISSN 1526-551X (online) http://dx.doi.org/10.1287/inte.2013.0730

© 2014 INFORMS

Extraction and Backfill Scheduling in aComplex Underground Mine

Dónal O’Sullivan, Alexandra NewmanDivision of Economics and Business, Colorado School of Mines, Golden, Colorado 80401

{[email protected], [email protected]}

We use mathematical optimization to determine a production schedule (i.e., a schedule of extraction and sub-sequent filling of the voids) for a complex underground mine in Ireland. The goal of the mining operation is tomaximize metal production over the life of the mine, subject to constraints on maximum monthly extraction andbackfilling quantities, maximum and minimum monthly metal production, and sequencing between extractionand backfilling operations. We solve our integer programming model with a heuristic to produce a schedulethat adds value to the mining operation by (1) shifting metal production forward, (2) reducing waste miningand backfilling delays, (3) avoiding expensive mill-halting drops in ore production, and (4) enabling smootherworkforce management.

Keywords : underground mine production scheduling; integer programming applications.History : This paper was refereed.

During the 1980s, surveys demonstrated that theLisheen orebody, located in south central Ireland,

might be profitable; correspondingly, a drilling pro-gram commenced. The seventh hole, drilled in thespring of 1990, revealed nearly 6.5 meters of an ore-body containing almost 15 percent sphalerite (zinc)and three percent galena (lead). By the mid-1990s,after 550 holes had been drilled, a 22.5 million-tondeposit, which was comprised of two separate hor-izontal ore bodies of similarly high-grade zinc andlead content, was defined.

In 1999, the first ore was extracted. Since then, theLisheen mine has become one of Europe’s largest zincproducers; at the time of this writing, over three mil-lion tonnes of ore have been extracted and shippedfrom the mine. As many as 6,300 tonnes of ore aretransported to the surface daily. The mine operates sixdays a week and employs nearly 400 people. Approx-imately two years of mine life remain, although con-tinued mineral exploration at the fringes of the minemay extend that time.

The two ore bodies at Lisheen are partitioned into11 zones, which are subsequently divided into 88 pan-els. Within each panel, the ore body is further dis-cretized into (1) stopes—areas outlined solely for thepurpose of extracting ore, (2) drifts—areas that are

mined to develop access to a stope and (or) for theextraction of ore, and (3) pillars—large ore blocks thatsupport the mine infrastructure. For ease of presenta-tion, we refer to any stope, drift, or pillar as a block. Atthe time of this writing, the Lisheen deposit contained1,193 mining blocks (i.e., candidates for extraction).

A critical factor in planning production is the deter-mination of a profit-maximizing cut-off grade, whichclassifies a block as either ore or waste based on thepercentage of mineral content or grade of that block.Because Lisheen is a polymetallic deposit with a zinc-to-lead ratio of 5:1, a combined mineral content orzinc-equivalent grade is calculated for each block. Oreblocks are candidates for extraction and refinement tomineral concentrate, whereas waste blocks are onlyextracted to facilitate extraction of adjacent ore blocksand are disposed of underground.

To determine the best extraction technique toemploy in a given area of the mine, geotechnical engi-neers consider the strength of the host rock (i.e., therock that encompasses the ore body), in addition toother factors such as the ore body slope and thick-ness. With varying host rock strength and blocks thatrange in thickness from 1 to 30 meters, the Lisheenmine applies three mining methods to extract ore:room-and-pillar, long-hole stoping, and drift-and-fill.

204

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O’Sullivan and Newman: Scheduling in a Complex Underground MineInterfaces 44(2), pp. 204–221, © 2014 INFORMS 205

Benching of thicker parts

Vertical benching

Pillar

Pillar

Figure 1: In room-and-pillar mining, a self-supporting method, some pil-lars of ore are left in place to support the rock that encompasses the orebody, while the remaining ore is extracted. When only pillars remain, pil-lars are removed in a sequence that starts with the furthest pillar fromthe mine exit, allowing the host rock to cave in as the extraction proceedstoward the exit (Hamrin 1997).

Where the host rock is strong and the ore body isnot steeply angled, room-and-pillar mining is pre-ferred (see Figure 1). Areas of the mine in which theore is particularly thick and the host rock is strongare suited to the large-scale and economically effi-cient long-hole stoping method (see Figure 2). Finally,

Large-holedrilling and

blasting

Stope

Blasted oreUndercut

Loading crosscut

Transport driftDraw point

Drill access

Figure 2: In the long-hole stoping process, a sublevel drift is developedbelow the ore to be excavated. The ore is then drilled and blasted andfalls to the sublevel where load-haul-dump vehicles scoop and load it intotrucks for transport to the crusher (Hamrin 1997).

where the mine has poor host rock strength, drift-and-fill mining is practiced (see Figure 3). Figures 1–3are representative of the operations at Lisheen; how-ever, in reality, the operations are tailored. Long-holestoping dominates 70 percent of the extraction, drift-and-fill mining accounts for another 20 percent, androom-and-pillar mining constitutes the remainder.

With 12 years of production already completed, themine has an extensive network of haulage routes,which are used to transport the ore from the panelsto the crusher, a machine used to break the piecesof ore into manageable sizes before conveyance tothe surface. Once above ground, the ore is trans-ported to the mill to be refined into metal concentrate.The operational plan at Lisheen recommends fillingthe mill (i.e., continually running the mill at as closeto full capacity as possible). This requirement is dif-ficult to satisfy because the quantity and the averagequality (i.e., head grade) of the ore that enters themill during a given period must be blended withina specific range; otherwise, the process stops. Manymining companies stockpile ore to more precisely con-trol the quality and quantity of the ore that feeds intothe mill. At Lisheen, a small above-ground surge pilebuffers ore production to ensure continual mill oper-ations over holiday weekends or during unplanned

Exhaust airwayDrilling

Hydraulicsandfill

Ramp

Transport drift

Figure 3: In drift-and-fill mining, a corridor of ore, known as a drift, isremoved from the ore body. The remaining void is then filled using acement-based mixture to provide structural support before cutting anadjoining drift through the ore (Hamrin 1997).

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O’Sullivan and Newman: Scheduling in a Complex Underground Mine206 Interfaces 44(2), pp. 204–221, © 2014 INFORMS

production shutdowns. However, this stockpile is notsuited to blending operations. Therefore, with limitedstockpiling capability, mine planners must carefullyselect the blocks that they schedule for extraction atany one time so that the quantity and average headgrade of the ore reaching the surface is within themill’s acceptable limits.

After refinement at the mill, the concentrate isloaded on trucks and shipped, and the waste fromthe milling process (i.e., tailings) are discarded. Mostare deposited into a tailings pond that will be drainedand covered once mining is exhausted. Remainingtailings are used to create a cement paste for fill-ing some of the voids left by extraction, a techniqueknown as backfilling.

Aside from providing a location for the disposalof tailings, backfilling is required in some areas tomaintain structural integrity and to allow mining tocontinue. Prior to backfilling a void, it must firstbe prepared by sealing it with wooden panels andinstalling hoses that run from the underground voidto the surface. Backfill paste, mixed in a plant aboveground, is then pumped via the hoses down into thearea being filled. A void, depending on its size, mayrequire anywhere from a day to a month to fill. Oncefilling is complete, an additional 24 days are allowedfor the paste cement to set, after which time extrac-tion may begin on the adjacent blocks that require thesupport from the backfilled areas.

The quality of the schedule for the productionprocess described previously is a significant driverof the mining operation’s profitability. To produce atimely and coordinated production plan, the plan-ner at Lisheen must consider the rate of extrac-tion associated with each block’s mining method, thedependencies between activities including backfillingrequirements, and the size and grade of each block,so that the blend of ore reaching the mill is satisfac-tory. This difficult assignment is further complicatedbecause mining follows a number of narrow veinsof high-grade ore located along fault lines betweenplates of different rock strength. The interaction ofthese plates has produced an ore body with an incon-sistent distribution of metal. Engineers highlight pock-ets of high-grade material by their choice of cut-offgrade and these pockets form the basis for the cre-ation of block shapes and the selection of extraction

9%

7%

7%

12%

A B

C D

12%

9%

Cut-off grade: 7% 9% 12%

Figure 4: Cut-off grade selection is a significant factor in determining oreblock shape and size. Box A shows a section of ore body with a nonuniformdistribution of metal. Selecting a cut-off grade of seven percent resultsin a large area classified as ore (Box B), which can be divided into rea-sonably homogeneous blocks and extracted with a single mining method(e.g., the drift-and-fill process). At a cut-off grade of nine percent, thematerial classified as ore diminishes (Box C). Because drift-and-fill min-ing would now excavate too much waste rock, a different approach isused. Box D shows material classified as ore at a cut-off grade of 12 per-cent. With most of the area now defined as waste, a targeted miningmethod is suitable.

methods (see Figure 4). Hence, the grade of a block canvary greatly between adjacent blocks, and we some-times find a high-grade ore block located beside awaste block. In addition, unlike open-pit mining, inwhich the ore body is subdivided into blocks of equaldimensions, the size and shape of the blocks can varygreatly. As a result, some blocks can be extracted in aday, while excavating others requires months. Conse-quently, each panel at Lisheen has such specific miningrequirements that developing general mining rules isimpossible, even at the zone level, which would makesequencing activities straightforward.

Schedule creation is particularly difficult becauseof the intertemporal effects on future mining activityassociated with each extraction decision. These effectsbecome especially important as mining reservesapproach depletion and the variety in grade of the orethat is available to satisfy milling requirements alsodiminishes. With mine closure on the horizon, man-agement must make critical decisions about whichblocks to take and which blocks to leave behind.

In this paper, we show how we use integer pro-gramming (IP) to determine a near-optimal schedule

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for the Lisheen underground mine for its remainingyears of operation. Arguably, the mine would havebenefited greatly from an automated scheduling pro-cedure more than a decade ago. However, we becameinvolved with the project in only the past few years.As we describe in our paper, legacy limits the extentto which we can impact mining operations with ourscheduling procedures. However, we are able to dic-tate a sequence in which blocks should be extracted tosatisfy the profit-maximization goals for the remain-ing life of the mine, especially as those goals pertainto the extraction of haulage pillars along a criticalretreat path out of the mine, and the ability of themine to sustain an operationally acceptable level ofproduction for as long as possible.

Literature ReviewWilliams et al. (1973) first demonstrated the poten-tial role for optimization in underground mine pro-duction planning. Although their linear programmingapproach suits certain strategic mining problemsinvolving blending or cost reduction, the methodcannot incorporate the binary decisions requiredto enforce the logic to schedule production at anoperational level. Chanda (1990) recognizes IP as aviable method for modeling discrete decision mak-ing. He combines mixed-integer programming (MIP)and simulation to produce a schedule for six con-secutive work shifts for a copper mine in Zambia.A similar approach by Winkler (1996) employs integervariables, not only to model operational relation-ships, but also to model the costs as piecewise linearfunctions. She highlights the advantages of apply-ing MIP to underground mine scheduling and illus-trates the exponential complexity associated with thisapproach when generating a multiperiod schedule.Solving one single-period MIP model at a time, sherelies on simulation to produce a multiperiod sched-ule for a German coal mine. Trout (1995) presents amore generalized MIP approach to underground mineproduction scheduling. Introducing variable restric-tions to reduce the size of the model, he producesa schedule that maximizes the net present value ofa copper mine for a 17-period horizon. Carlyle andEaves (2001) apply a MIP model to schedule pro-duction at an underground platinum and palladium

mine. They maximize discounted ore revenue by solv-ing their model for a number of mine expansionscenarios.

Sarin and West-Hansen (2005) developed a Benders’decomposition technique to produce an exact solu-tion for their MIP model. Applying their model to acoal mining case study results in a profit-maximizing100-week schedule. Newman and Kuchta (2007) opti-mize long-term production at an underground ironore mine in Sweden. With an objective of minimiz-ing deviations from contracted production quantities,they use an aggregation heuristic to solve a MIPmodel for a 60-month horizon. This model forms thebasis for Martinez and Newman (2011) to scheduleboth long- and short-term production for the sameiron ore mine. That model provides greater resolu-tion in the near term to satisfy detailed operationalrequirements. They produce solutions for a 48-monthhorizon using a decomposition heuristic.

Similar to the work we cite previously, we formu-late, solve, and implement an IP model for produc-tion scheduling in an underground mining operation.However, although almost all underground mine pro-duction scheduling models have similarities (e.g.,blocks, periods, sequencing constraints, resourcerestrictions), each mine operates in a specific manner.For example, objectives tend to differ between mines;an appropriate objective for a mine containing pre-cious metal may be to maximize net present value,whereas the objective for a mine containing basemetal may be to minimize deviations from long-termcontracts. Some mines may stockpile to exploit blend-ing and its associated quality of output; others mayregard such a policy as a nuisance, requiring rehan-dling and its costs. The six standard undergroundmining methods have many variants. As such, eachmethod necessitates the implementation of differentoperational policies. Therefore, writing one generalmodel for underground mine production schedulingis difficult. Specific aspects relevant to our applicationare (1) a discounted objective involving metal (butno explicit economic parameters), (2) a complex andextremely ill-defined and nonhomogeneous (i.e., spa-tially, temporally, and regarding their metal content)set of areas, and (3) a mine that uses three under-ground mining methods and the associated rulesgoverning extraction and backfill, where applicable.

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We now describe the mine with respect to optimizingits production schedule in more detail.

Production Scheduling at Lisheen MineAt the Lisheen mine, ore production will continueas long as the operation is profitable. However, asat other mines, a time will come when the metalthat the mine produces will no longer be able to jus-tify the costs of its extraction and, although ore blocksremain below ground, the mine will close. This timeis called economic exhaustion. The challenge then isto extract the combination of blocks that realizes themost value from the mine before economic exhaustionis reached.

Production schedulers at Lisheen use iGantt(MineMax 2012) mine planning software to man-ually generate production schedules. This difficultand time-consuming task, performed semi-annually,requires the manual arrangement of approximately2,000 extraction and backfilling activities on a Ganttchart (see Figure 5), such that planned ore productionis sufficient to keep the mill running as close tocapacity as possible.

Figure 5: This example of manual scheduling using a Gantt chart shows extraction and backfilling activities ashorizontal bars. The thin lines between the bars represent the precedence relationships between the activities.To meet production targets, planners arrange these activities on the timeline.

To narrow the scope of the task, the planneradopts the following assumptions about the miningoperation:

1. Financial aspects of the operation are ignored.Planners do not explicitly consider operational costs,mineral prices, and costs associated with mine closurewhen generating schedules.

2. The goal is to maximize metal production overthe life of the mine. Although the mine plannersdo not explicitly consider financial information, theyseek to produce as much metal up front in the remain-ing life of the mine as possible. The rationale behindthis objective is that, because Lisheen sells metal onthe spot market, too much risk is associated withscheduling large quantities of metal late in the mine’slife when a drop in the spot price might render thatmetal uneconomical to mine.

3. The mining methods (i.e., room-and-pillar, long-hole stoping, drift-and-fill) for each mining area arefixed. A change in the method of extraction for anarea may require the planner to redefine the ore blockshape, establish a new order of mining, or changethe rate of mining for that area. As a result, foreach mining activity, we also assume the same fixed

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rates that Lisheen has used to generate its manualschedules.

4. The mine design and infrastructure are fixed.The planner does not consider developing new accessdrifts (i.e., passageways) to reach the ore blocks orhaulage routes to connect panels with the crusher.

5. The cut-off grade that defines a block as ore or aswaste based on the percentage of mineral content isfixed. During the engineering design of the mine, theplanner selects a cut-off grade to maximize the minevalue. Although economic conditions might indicatethat a different cut-off grade could increase the mine’svalue, the change would also require the developmentof a new engineering plan (see Figure 4).

6. The rates for extraction and backfilling activitiesare fixed. Based on the block tonnage, excavation dis-tance, or void volume, the planner can quickly cal-culate the time required to complete an activity. Thisis a standard assumption in most scheduling mod-els, and changing the rates would be tantamount tochanging a strategic decision (e.g., fleet size or workshift length).

7. Sufficient resources are in place to implement theschedule. Production at Lisheen is not limited by theavailability of equipment or labor.

With these assumptions in mind, the mine plan-ner must determine a start date for each extrac-tion and backfilling activity, given the followingconstraints:

• Monthly tonnage targets for mine productioncannot be exceeded. The mine has a production capac-ity constraint for each month based on the number ofworking areas that can be active at one time and thenumber of working days in that month. Mine produc-tion includes extraction of ore that feeds the mill, andnecessary waste.

• Metal output from the mill must be main-tained between monthly maximum and minimumlevels. These blending constraints on metal produc-tion ensure that the quantity and head grade of theore feeding the mill are maintained within operationallimits.

• A monthly limit on cement paste for backfillingmust not be surpassed. Paste availability depends onthe tailings produced as waste by the mill and thecapacity of the backfill plant that produces the paste

cement. The production of tailings is a function of thequantity and grade of ore reaching the mill, whichmeans that backfilling may be constrained by the pro-duction of ore in a previous period. However, becauseonly a fraction of the voids created by extraction needto be filled at Lisheen, the availability of tailings doesnot limit paste production.

• Sequencing constraints must be observed.The extraction of a block or backfilling of a voidmust satisfy any sequencing rules that exist betweenthat activity and any other activity. Often calledprecedences, these relationships may be definedimplicitly as a consequence of the combination ofmining methods chosen for a panel. Otherwise,precedences are defined explicitly to ensure that theschedule remains feasible.

• Once started, an activity must proceed con-tinuously until completion. The engineering designdefines blocks under the assumption that the entireblock will be extracted. Partial extraction of blocks orpartial backfilling of voids would result in structuralinstability.

Manual scheduling of ore production under theseconstraints is challenging. To satisfy the Lisheen busi-ness objectives, the planner adopts a trial-and-errorapproach to bring metal forward in the produc-tion plan, while simultaneously adhering to the oreproduction limits and grade blending requirements.For large problems, the combinational nature of thissequencing and blending problem would require theplanner to enumerate a staggering number of alter-native scenarios. Consequently, planners often havedifficulty producing schedules that satisfy these con-straints. Finally, the planner must be mindful thatextraction decisions made today can have conse-quences for future mining activities. For example, tac-tical scheduling decisions to satisfy the next week’sproduction quota may unintentionally prevent accessto an area of ore and destroy its value.

As Lisheen approaches closure, a particularly dif-ficult decision facing the planner is when, if ever, toextract ore blocks that form part of the mine’s criti-cal infrastructure. Throughout the mine, trucks carryore from the panels to the crusher along haulageroutes, which are supported by large ore blocks calledhaulage pillars. Moreover, these pillars are also can-didates for extraction, and often contain valuable

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high-grade ore. However, once a haulage pillar isremoved, the ore blocks that require the pillar toremain in place for their extraction can no longer bereached; these are termed sterilized reserves. Hence,the planner must consider the opportunity cost ofsterilizing the ore associated with the extraction of ahaulage pillar (see Figure 6).

Given the challenging nature of the task, manualscheduling can result in solutions that are far fromoptimal. Also, the time-intensive process of manualscheduling, which can require several weeks to com-plete at Lisheen, precludes any possibility of scenarioanalysis to evaluate the operation with respect toother parameter values (e.g., realization that poorer-than-expected ground quality in a panel would slowdown mining in that area and necessitate an updateof the mining rates for that zone). Finally, the plan-ner cannot examine all of the schedule’s permuta-tions; therefore, measuring the quality of the solutionis impossible.

By contrast, a mathematical optimization approachallows the planner to enumerate multiple schedules,select the one that results in the highest objectivevalue, and then show a measure of the solution qual-ity. Moreover, the planner can quickly generate an IPsolution, and thus address any changes that may arisebecause of unexpected events.

PANEL

P1

P3

P2P4

P5

N

Figure 6: This example shows a mining panel adjacent to a haulage route.The pillars that support the haulage route are shown as P1–P5. The arrowsindicate the direction of the mine exit. The haulage path collapses westof any pillar that is removed. Hence, the extraction precedence for thesepillars would require that P1 be taken first and P5 last. Removal of pillarP3, P4, or P5 would result in the destruction of the haulage path in thatarea, eliminating access to the panel.

Integer-Programming ApproachAdopting an IP approach, we determine a near-optimal schedule for the Lisheen mine. We cast theproblem mathematically, with the objective of recov-ering the maximum amount of discounted metal pos-sible from the mine over its life. We also incorporatethe same constraints and assumptions mentioned pre-viously in relation to manual scheduling.

We apply a small discount factor to the objectivesolely to encourage a solution in which metal produc-tion is brought forward in the schedule. Specifically,without this discount factor, a solution with one tonneof ore extracted either in week 1 or week 2 wouldbe equivalent. The addition of this factor produces asolution with the tonne of ore extracted in the firstweek. Such discounting can also improve the run-time performance of mathematical programs (Klotzand Newman 2013).

To measure value, we assume a constant metalprice, which we normalize to 1 (see the formulationin the appendix). Multiplying the objective by a con-stant factor would not change the solution we pro-duce, and would obfuscate our intent, which is to pullthe blocks with the highest metal content forward inthe schedule to realize the greatest metal sales (subjectto operational constraints of the mine) in the short-est possible time. Indeed, lack of an explicit monetaryfactor happens occasionally in a base metal extractionoperation (Newman and Kuchta 2007, Martinez andNewman 2011).

Although relatively straightforward to describe inmathematical terms, the complexity of the problemmakes it difficult to solve in practice. The sequencingrules that govern the relationships between activitiesare idiosyncratic, complicated, and defined unclearly.In describing a challenge of solving integer programsfor mine scheduling, Smith (1998) emphasizes thedifficulty associated with complex precedence con-straints. These precedences—encoded in iGantt dur-ing manual scheduling—are an output from thatsoftware. However, the iGantt precedence set pro-vides few degrees of freedom, and solving our inte-ger program with these encoded precedences wouldsimply return the existing iGantt schedule. Inspec-tion of the iGantt relationships using a computer-generated block model in Maptek’s Vulcan (Maptek2012) reveals that, although maintaining a feasible

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schedule requires many rules, some rules merelyreflect a subjective choice in the order of mining cer-tain ore blocks. Specifically, the precedence rules relat-ing the extraction of infrastructural haulage pillarsto other activities in dependent panels are unneces-sarily restrictive. In most cases, the existing miningrule simply reflects the order in which the pillar waspreviously scheduled, often requiring extraction towait until all dependent panel activities have finished.As previously mentioned, the timing of the extrac-tion of these high-grade haulage pillars is a criticalaspect of the mine schedule. Hence, we want to allowour model the freedom to choose when, if ever, toextract them. To this end, we reverse the existinglogic, that a pillar must wait until the mining of thedependent panel has finished, and instead create arule that prevents any further mining in the depen-dent panel once the haulage pillar has been extracted.For each panel, we identify the critical haulage pillar

Figure 7: In this example of critical haulage routes, the dashed lines show haulage routes through a miningzone. These haulage routes are bordered by major panels containing blocks that are candidates for extraction.The pillars that support these haulage routes often consist of valuable ore blocks. However, once removed, welose access to the area east of the extracted pillar.

whose extraction would prevent any further activ-ity in that panel. We then define a constraint thatenforces this relationship, not just for that one criti-cal haulage pillar, but for all blocks along the haulageroute that, if extracted, would sterilize remaining orein that panel. In this way, we delineate critical haulageroutes throughout the mine that characterize the rela-tionships between haulage pillars and mining panels(see Figure 7).

Another difficulty in constructing the sequenc-ing constraints arises from activities for which nosequencing rules are defined. Including these or-phaned activities in our model without defining rulesfor their execution would likely produce an IP solu-tion that we could not implement. Without explicitand objective mining rules, we would have to spendconsiderable effort examining each precedence rela-tionship between a pair of activities to determine if itleads to a valid constraint or whether it is merely a

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subjective planning choice. Working with Lisheen, weredefine the precedence set by identifying and remov-ing the subjective rules and by defining new prece-dents where needed.

A complication with our IP approach also arisesfrom the heterogeneity of the ore blocks. The size ofan ore block can vary greatly, which makes deter-mining a standard time interval, or fidelity, for theproblem difficult. The fidelity of the model definesthe discrete times during which an activity can begin.In scheduling, we could assign an activity to start at aspecific month, day, or hour. However, use of discretetime intervals results in schedules in which activi-ties that, in practice, would take less than the intervalto complete appear to require the full time intervalbefore a dependent activity could begin. For example,a block that would require a day to extract in practicewould be scheduled for a month if we were solvingat monthly fidelity. Thus, the smaller the time inter-val, the closer together we can schedule activities. But,this benefit comes at a computational cost, because wemust account for finer fidelity by defining variablesfor each activity and start-time combination.

A solution produced at monthly fidelity may workwell for a mine that has homogeneous, large blocksthat require more than a month to extract; however,as the histogram in Figure 8 illustrates, Lisheen hasmany small areas that can be mined in less than a day.

350

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9,00

0

10,0

00

11,0

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e

Figure 8: In this histogram of block size, smaller blocks correspond to asmall access drift or a cross cut used to facilitate extraction of a largerblock, such as a stope. The smallest blocks can be mined in a day or less.The largest require months to extract.

Therefore, applying a model with monthly fidelityresults in schedule delays, because these small areaswould be allotted a month for extraction, unnecessar-ily extending the extraction times of dependent areas.A frequently used approach to address such fidelityproblems is to aggregate the smaller mining areasinto larger clumps of similar size. However, such anapproach reduces the resolution at which blendingdecisions can be made. As Smith (1998) points out,deposits rarely contain grade that is homogeneousenough to ignore grade blending requirements. Suchis the case at Lisheen, where the high variation in thegrades within adjacent blocks, in addition to the com-plexity of the mining precedences, precludes blockaggregation.

To produce an implementable schedule, we requirea solution at weekly fidelity for a two-year horizon(i.e., a horizon that corresponds to the projected lifeof the mine). The resulting problem contains 2,214(extraction and backfilling activities) × 105 (weeks) =

232,470 binary variables. We reduce this number byimplementing start-date restrictions for each activity;these restrictions do not compromise the quality ofthe solution, but only serve to eliminate variables,which would necessarily assume a value of zero in anoptimal, or even any feasible, solution. The manualschedule in iGantt provides a basis from which wedevelop early start-date restrictions for haulage pillarretreats. However, because we relax precedence rulesbetween panels, using the same approach for panelactivities is too restrictive. Consequently, for panelactivities, we use a variant of an existing procedurefor establishing early start dates for machine place-ment activities in a sublevel caving mine (Martinezand Newman 2011). Our variant considers blocksalready scheduled to start, the prescribed sequencein which blocks must be extracted, and mining ratesto determine that certain blocks cannot be extractedbefore a specific early start date. For example, if thefollowing three conditions hold, (1) block A must bemined to obtain access to block B, (2) block A hasstarted extraction in period 1, and (3) block A requirestwo weeks to extract, then we cannot begin to extractblock B until at least the start of the third week. Cor-respondingly, we remove variables from the formula-tion that represent block B starting to be mined eitherin week 1 or in week 2. For sequences involving more

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than a single predecessor block, we simply add thedurations of all predecessor blocks to the start dateof the first block in the sequence to determine thestart date of a given block; in these cases, all blocks inthe sequence must be mined, and at a predeterminedrate, which produces an exact result (i.e., a time beforewhich a block at the end of a precedence chain cannotpossibly begin to be extracted). This approach reducesthe problem to 56,276 variables; however, with over amillion constraints, the problem remains intractable.Therefore, not only do we use the exact approach ofeliminating variables to reduce the problem size, butwe also employ a heuristic on the reduced problemto produce a schedule in a reasonable amount of time(see Figure 9).

We can solve smaller instances of the (reduced)problem by including fewer activities in the model.We take advantage of this insight to find a solu-tion that includes all the activities by decomposingthe original problem into a series of smaller prob-lems, which we solve in stages. With this approach,we begin by separating the extraction activities intosets of progressively lower metal content. In practice,we might use two or three sets for panel activitiesand three to five sets for pillar activities; however,to illustrate the approach here, we consider only twosets: high-grade and low-grade activities. Consideringonly the subset of high-grade activities, we solve the

High-gradepanel block

High-gradehaulage pillar

Low-gradehaulage pillarLow-grade

panel block

Solve 1

Solve 2

H H H

4 8 12

4 8 12

Weeks

Weeks

H

HHH

HH HL

LLLLL

H H

Figure 9: In this example of the heuristic solution method, we solve the problem twice at weekly fidelity. Solve 1shows the solution for a 12-week horizon during which only high-grade panel and pillar activities are consideredfor scheduling. The times at which these blocks are extracted constitute the schedule at this point. The solutionresulting from Solve 2 shows that blocks that were scheduled within Solve 1 are now required to remain sched-uled at the equivalent week when solving this problem, but now low-grade panel blocks and pillars are alsoincluded.

model (P ), given in the appendix, for those activitiesat a weekly time fidelity for all weeks in the plan-ning horizon. The solution yields a production sched-ule for the high-grade activities on a weekly basis.We then include the set of low-grade activities in thenext model run, fixing those high-grade activities tothe corresponding weekly start dates from the firstsolve. The net result is that we now obtain a solu-tion that includes some subset of the low-grade activi-ties, which are scheduled around the fixed high-gradeactivities. Note that to enforce precedence betweensuccessive solves, we must insert placeholders for thetime required to extract the predecessors absent in thecurrent solve whose successors are present in the cur-rent solve.

This approach alone will not produce a near-optimal solution for Lisheen, because the problem’scomplexity is determined largely by the number ofhaulage pillars included in the model. Haulage pillarscomplicate the problem because (1) they have few, ifany, precedence requirements, and can consequentlybe scheduled for extraction at any point during thehorizon, and (2) the value of adding a haulage pil-lar to the schedule must be weighed against the totalvalue of the dependent panel blocks that would besterilized as a result. If we exclude the haulage pillars,we can solve the model for the entire horizon in amatter of minutes; however, as we introduce haulage

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pillars, the problem’s complexity increases quicklyuntil the problem becomes intractable. The key tosolving the problem in stages is to find the rightbalance between the number of pillar activities andthe number of other extraction and backfilling activ-ities included in each stage of the heuristic. If theinitial solve includes too few pillars, then that solu-tion contains too much panel mining in the schedule,leaving less opportunity to accommodate remainingpillars during the next solve stage. If it includes toomany pillars, the problem becomes intractable, or, inthe tractable case with fewer panel blocks, the solu-tion contains some pillars that are scheduled far tooearly, sterilizing dependent panel blocks and prevent-ing them from entering the schedule during the nextstage of the heuristic. For a given horizon, we usea trial-and-error approach to determine the numberof activities and the ratio of pillar-to-panel activitiesto include at each stage in the heuristic. The numberof stages depends on the length of the horizon, withlonger horizons requiring a more gradual introduc-tion of pillars and, thus, more stages.

ImplementationOur schedule, as it appears in the output file from theCPLEX solver (IBM 2011), is merely a list of sched-uled activities and their corresponding start weeks.In this format, the schedule is difficult for Lisheen tovalidate and impractical to implement. Therefore, weintegrate our solution with Lisheen’s iGantt schedul-ing software to present our schedule in a way that isfamiliar to Lisheen’s planners.

iGantt provides effective schedule visualization andreporting features that the planners use each day.The software displays the schedule graphically as aGantt chart (see Figure 5), i.e., a timeline of scheduledactivities and their interdependencies. In addition,iGantt can present a three-dimensional block modelanimation of the schedule, enabling the planner toquickly identify sequencing problems or other infeasi-bilities. Lisheen management uses iGantt’s reportingcapability extensively, for example, to produce short-term work schedules, to summarize historical backfillpaste usage, or to forecast future resource require-ments (e.g., the number of required cable boltingbits). Thus, implementing our schedule using iGantt

has the added benefit of providing continuity in themines’ daily managerial operations.

We transfer our schedule to iGantt by creating animport file of the data, with rows of activities andcolumns of activity attributes (e.g., tonnage associatedwith an extraction activity). To this file, we add astart-date column, which we populate with the weekduring which each activity begins according to oursolution. Importing this information into iGantt asa comma-separated-value file is straightforward, butthe resulting schedule that iGantt creates is incor-rect because the default functionality of the softwarechooses a start date for each activity that differs fromthe IP schedule’s start date. We override these auto-mated start dates by using a customized script thatforces iGantt to adopt the IP activity start dates; theseare given in terms of a start week, whereas iGantt’stime fidelity is given in milliseconds. We convert thetime at which an activity starts based on the begin-ning of the week in which it is scheduled in the inte-ger program to milliseconds from the beginning of thehorizon, and use that converted date in iGantt. We donot specify activity end dates to import into iGantt;instead, we let iGantt set them based on the activityrates. Admittedly, we are not scheduling at the levelof fidelity that iGantt allows or that manual sched-ulers at the mine could use with the iGantt software.However, mine planners can use discretion to shiftour dates within an allowable window, and a sched-ule at the level of milliseconds is impractical. Shiftingactivities outside of the allowable window (i.e., man-ually moving activities in iGantt forward to begin atthe end-date of the preceding activity) would likelyviolate ore production and (or) milling constraints.Although our schedule may leave gaps in time (e.g.,in our model, we record an activity that requires fivedays as requiring a week), our schedule provides aconservative estimate of the metal producible withinthe given horizon, and may be more realistic underthe unanticipated conditions that necessarily prevailin an underground mining operation.

Review and validation are possible once we formatour solution into an iGantt schedule. Using the iGanttvisualizer, the planner examines our schedule for fea-sibility. At this point, our schedule may be infeasi-ble from a practical standpoint because of missingor invalid sequencing constraints. The complexity of

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the sequencing rules makes it difficult to spot minorconflicts or omissions a priori; however, we highlightthese mistakes in the animated iGantt schedule. Withguidance from Lisheen, we correct any illogical oroverlooked sequencing constraints in our integer pro-gram and continue the process of review and correc-tion until we obtain a workable schedule.

ResultsSolving our IP model for a 104-week horizon, wecompare our solution with the manually generatedschedule on the basis of, for example, metal output(from the mill), ore production (based on extractedmaterial), and backfill paste usage (see Table 1). Withrespect to the objective of maximizing metal produc-tion over the life of the mine, the integer programsolution shows 0.17 percent less metal assigned forrecovery than the manual approach. When taken inisolation, this shortfall in total metal output is indica-tive of a poor result. However, we must consider anumber of other measures before we can properlyassess the quality of the IP solution.

Most significantly, the integer program discountsfuture production so that we can bring forward metalin the schedule. This satisfies management’s desireto produce as much metal up front in the remain-ing life of the mine as possible. As we have men-tioned, Lisheen management is not comfortable withscheduling large quantities of metal late in the mine’slife because a low future spot price might precludeeconomic viability of the metal’s extraction. Althoughthe integer program may not increase total metal pro-duction, it does bring a significant quantity of metal

Manualschedule IP schedule IP gain (%) Comment

Metal output 3211154 3201621 −0017 No IP improvement(tonnes)

Ore production 214441446 214141298 −1023 IP improvement(tonnes)

Backfill paste 9471104 8101273 −14045 IP improvementused (cu.m)

Waste mined 1151790 1141424 −1018 IP improvement(tonnes)

Table 1: The table shows a comparison of the IP and manually generatedsolutions for a 104-week horizon.

3,000,000

2,500,000

2,000,000

1,500,000

1,000,000

500,000

–Year 1 Year 2 Total

Met

al (

tonn

es)

Manual schedule

Integer program schedule

Figure 10: In this example of annual metal production, the integer pro-gram brings metal forward in the production schedule.

forward in the schedule, increasing metal productionby 10.51 percent in the first year (see Figure 10).

Following the IP schedule would also reduce costsbecause the decreased metal yield is accompanied byan even greater reduction in mining activity, with1.23 percent less ore production and 1.18 percent lesswaste mining than the manual schedule generates.Therefore, the integer program schedules more effi-cient extraction; it also schedules significantly fewerbackfill activities, corresponding to a 14.45 percentreduction in paste consumption over the remaininglife of the mine. By releasing labor to more prof-itable extraction activities and by reducing wait timefor backfill set up, pouring, and setting, this reduc-tion in backfilling enables the planner to include valu-able extraction activities that would otherwise havebeen precluded. In addition, although a backfilledvoid generally requires one month to set, at Lisheen,oxides often interact with the backfill paste, prevent-ing or delaying hardening. Consequently, backfill-ing can sometimes fail to harden or can require twomonths or longer to complete, again forcing extrac-tion activities outside of the schedule, resulting in aloss of value. Thus, scheduling less backfill activityreduces the potential for schedule delay.

When concurrently scheduled, backfill activities canalso be responsible for large temporary drops inore production which, for a number of reasons, areproblematic. In particular, because it is highly ineffi-cient for the mill to operate below a minimum ore

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threshold, low production levels result in mill shut-downs and costly restarts. To boost production inlow-production periods, Lisheen planners use a trial-and-error approach to shift production activities inthe manual schedule. However, the planner cannotbe sure if rescheduling alone can rectify the drop inproduction. For Lisheen, production below approxi-mately 80,000 tonnes of ore results in mill shutdown,and the manual schedule falls below this level for twomonths at the start of the second year (see Figure 11).By contrast, the IP model sets a lower bound for pro-duction; as a result, it suffers no mill-halting dropsin ore production. Also, because lower productionrequires fewer workers, management would be forcedto reassign miners to other duties or worse, lay offpart of the workforce, a decision that would bringunion action against Lisheen.

Not only does the IP schedule avoid dips inproduction, but it also schedules ore productionmore consistently than the manual method, with anaverage month-to-month change in production of10,132 tonnes compared with 14,796 tonnes for themanual schedule. Thus, the IP solution enables minemanagement to create a more steady work flow thanthe manual plan does.

160,000

140,000

120,000

100,000

80,000

60,000Ton

nes

ore

40,000

20,000

00 20 40 60 80 100

Weeks

Manual schedule

IP schedule

Min. production

Figure 11: In this ore production example, the IP solution has more consistent month-to-month production thanthe manually generated schedule. Production in the IP schedule remains above 80,000 tonnes, allowing the millto operate continually. By contrast, the manual schedule suffers from low production starting in week 48, whichwould result in a temporary mill shutdown and labor problems.

Scenario AnalysisThe integer program produces schedules in fewerthan 20 hours. This is significantly shorter than themanual approach, which can require several weeksto complete. Consequently, the IP method is use-ful for scenario analysis or for expediting the gen-eration of new schedules to account for unforeseenevents or changes in engineering design. Becausethe mine is approaching the end of production, weexamine one important set of scenarios: alternativeclosure dates.

The closure date that defines the end of the mine’soperational life is a best estimate from the manualscheduling process. It is the point in the schedule atwhich economic exhaustion occurs and the mine canno longer produce enough metal to justify its opera-tional costs. The time at which Lisheen reaches eco-nomic exhaustion depends on future metal prices andcosts, including significant mine closure and reha-bilitation expenses. Although our model does notexplicitly incorporate these financial components, itcan generate feasible production alternatives, thusenabling management to understand how the sched-ule might change by shortening or lengthening themine’s life, and to make more informed mine closure

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25,000

20,000

15,000

10,000

5,000

–0 20 40 60

Weeks

80 100 120

Ton

nes

met

al

Long horizon

LOM base case

Short horizon

Figure 12: We compare IP solutions for varying time horizons against the IP solution for the current mine life.Increasing the mine life by another three months would be a viable option for Lisheen. Similarity in the productiontonnage in each week during the first year is a consequence of the low degrees of freedom in the precedencerules. Later, when haulage pillars become candidates for extraction, weekly metal production differs more acrossscenarios, reflecting the more flexible precedence constraints on these pillars in our model.

decisions. To provide insight, we run scenarios withdifferent horizons (see Figure 12).

With our previous results for a 104-week mine lifeproviding a base case, we solve our integer programfor two mine-life scenarios: (1) an early-closure optionwith a 92-week horizon, and (2) an extended horizonof 122 weeks. We compare the results of these scenar-ios to our base case on metal production (see Table 2).

The early-closure scenario is motivated by theobservation that after the first 20 weeks of the104-week IP schedule, ore production and metaloutput never approach maximum levels again. Con-sequently, Lisheen management is interested in exam-ining the possibility that, if the base-case schedulehas sufficient production or output slack, a solutionfor a shorter horizon might produce a similar orgreater quantity of metal than the base-case scenario.

Scenario 92 weeks 104 weeks 122 weeks

Short horizon 2801095 n/a n/aLife-of-mine base case 2881165 3201621 n/aLong horizon 2791532 3111511 3421614

Table 2: The table shows IP mine-life scenarios by comparing metalproduction.

Specifically, we shortened the horizon to 92 weeksbecause it corresponds to the end of a calendaryear. Simply taking the base-case solution for thefirst 92 weeks would produce an infeasible sched-ule because it would omit the backfill activitiesrequired before closure. The schedule optimized overthe 92-week horizon precludes these infeasibilities,and moves a small quantity of metal forward inthe schedule. However, with this schedule, the totalmetal produced is 8,070 tonnes lower than the amountscheduled over the same period in the base-casesolution. Shortening the horizon does not increasescheduled metal, thus illustrating Lisheen’s limitedproduction flexibility. With no significant gains usingthe early-closure option, we turn our attention to theextended horizon scenario.

The 122-week extended mine-life scenario proves tobe a more interesting case. Although scheduled metalproduction is 9,109 tonnes less than the productionamount corresponding to the base case at week 104,the extended schedule includes 21,993 tonnes moremetal than the base case by the end of its horizon.However, two additional periods of low metal pro-duction accompany this gain in metal. Although theseproduction drops are not low enough to stop the mill,and implementing the schedule would delay mineclosure, the results show that management should

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consider a tradeoff between (1) the gains from alonger mine life with higher total metal production,and (2) the early-closure risks associated with dipsin metal production and potentially low metal spotprices in the future.

Although our model can solve a new scenarioinstance in less than a day, we require additional timeto make data changes, prepare the model, and importour solution into iGantt. Using our approach, alter-ing parameter values is an easy and quick process;however, changing precedence relationships betweenactivities, a more involved task, can take a numberof days to update. For example, a change in the cut-off grade would require engineers to reexamine and,in some cases, redefine block shapes, mining meth-ods, and precedence relationships between activities.Upon completion of this study, we would requireup to a week to make the corresponding changes.For a commercial application, we could eliminate thispreparation time by writing a software interface toinstantly transfer changes made in iGantt to our IPmodel. If we add this graphical user interface to theoptimizer, nontechnical users could perform scenarioanalysis on production scheduling. However, becauseour academic scope includes limited scenario analy-ses, developing a commercial interface for our modelwould be tangential to our study.

InsightsWith the IP results in hand, the planners at Lisheencan see that the current mine design and precedencerelationships do not allow for significant metal gainsfrom scheduling changes alone. In particular, the IPschedules highlight the valuable blocks that wouldalways be excluded from an economic productionschedule, and no amount of manual rearranging ofactivities in iGantt would bring these blocks forwardinto the schedule. This insight has prompted plannersto explore alternative methods for capturing thesehigh-value blocks. These methods can involve chang-ing the engineering design (e.g., selecting a differ-ent mining method), developing new access, or evenblasting a new haulage route through a previouslybackfilled area (see Figure 13).

For the planners, the IP solutions also highlightmyopic and subjective decisions made during man-ual production scheduling. For example, the integer

Figure 13: The IP schedule identifies four high-grade haulage pillars asblocks that would never be mined given the current mine design. The pil-lars form part of an existing main haulage route (solid arrows) connect-ing the crusher with a large ore zone to the west (not shown). Under theoriginal plan, mining these pillars would only be possible once all activ-ity in the western zone has finished. However, the IP results show thatmore profitable higher-grade pillars along the same haulage route wouldalways be extracted in preference to these four pillars. Realizing this,Lisheen added a bypass (dashed arrows) to incorporate these pillars intothe schedule.

program scheduled an area of very low-grade orethat Lisheen had determined was uneconomical tomine. Planners had overlooked that production froma number of very high-grade blocks in different pan-els was occasionally scheduled simultaneously, result-ing in too high a head grade for ore feeding the mill.To maintain feasibility, low-grade ore is, at such times,required to balance the head grade. Additionally, alarge decrease in ore production occurred at the startof the second year in the manual schedule. The IPschedule brought forward a retreat of haulage pil-lars, which were to be extracted late in the manualschedule to cover this deficit. The planner had notconsidered using these pillars to prevent the produc-tion decrease because of subjective reliance on a pre-vious mine design. The mine configuration had sinceincluded new haulage routes and precedence rules,but the cognitive bias that those pillars could not betaken early remained in the planner’s mind, prevent-ing him from determining a solution for the shortfallin production.

Insights gained from the IP results have promptedLisheen to take immediate action. In particular, the IP

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schedules have highlighted a number of panels thatmust be brought into production earlier to pull depen-dent blocks into the mining schedule. These areasare currently being mined in accordance with the IPschedule. Lisheen is in the process of changing theengineering design in many areas of the mine. Whenthese designs are complete, we will incorporate thechanges into the IP model and generate a new IPschedule. In this way, our work with Lisheen will con-tinue to be an iterative process between mine designand schedule optimization.

ConclusionsOur IP approach produces near-optimal productionschedules for a complex underground mine. Over a104-week horizon, our schedule contains ore produc-tion that is more consistent with managerial goalsthan the mine’s previous manual scheduling methodwas. Although both the manually produced and theIP-produced schedules are in iGantt for presentationpurposes, our IP schedules add value to the miningoperation by (1) shifting metal production forward inthe schedule, (2) reducing waste mining and backfill-ing delays, (3) avoiding expensive mill-halting dropsin ore production, and (4) enabling smoother work-force management. In addition, the integer programprovides the mine with a scenario analysis capability,which has been instrumental in management’s deci-sion to revise much of the underground engineeringdesign to increase the mine’s operational life.

Although multiple open-pit optimization softwarepackages are on the market, the idiosyncrasies ofunderground operations have, so far, made thecreation of such a general software package forunderground mining too challenging. However, theapproach that we outline here can be adapted to otherroom-and-pillar underground operations, especiallyas they near the end of their operational lives.

Appendix. Model FormulationThe primary elements of the model are as follows.

Indicesa: Mining or backfilling activity, a= 11 0 0 0 1n.

t1 t: Period, t = 11 0 0 0 1 T .

SetsAM : Set of all extraction activities.AB : Set of all backfilling activities.T: Set of all periods.Ta: Restricted set of periods in which activity a can start.Pa: Set of activities that must precede activity a.Pa: Set of activities, each of which must precede activity a,

if it occurs.Pa: Set of activities that must not precede activity a.

Parametersvatt : Volume of ore obtained in period t, given that we

started extraction activity a at time t (tonnes).ga: Average percentage grade of the ore produced from

extraction activity a.e1 e: Maximum and minimum allowable tonnage of ore

excavated in a month, respectively (tonnes).g1 g: Maximum and minimum allowable metal produced

by the mill in a month, respectively (tonnes).tma : Number of periods required for extraction activity a

(months).tba : Number of periods required for backfilling activity a

(months).patt : Paste applied in period t, given that we started back-

filling activity a at time t (cubic meters).p: Available paste for backfilling in each month (cubic

meters).r : Discount rate used to decrease the value of the metal

produced in future periods.

Decision Variable

yat =

{

1 if activity a starts during period t,

0 otherwise0

Objective Function

(P) max{

∑

a∈AM

∑

t∈Ta

∑

t∈T

gavattyat41 + r5−t

}

1

subject to∑

t∈Ta

yat ≤ 1 ∀a ∈AM∪AB1 (1)

e ≤∑

a∈AM

∑

t∈Ta

vattyat ≤ e ∀ t ∈T1 (2)

g ≤∑

a∈AM

∑

t∈Ta

gavattyat ≤ g ∀ t ∈T1 (3)

∑

a∈AB

∑

t∈Ta

pattyat ≤ p ∀ t ∈T1 (4)

yat ≤∑

u∈Ta′ 2 u≤t−tma′

−tba′

ya′u

∀a′∈ Pa1 a ∈AM

∪AB1 t ∈ Ta1 (5)

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yat ≤∑

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1 −∑

u∈Ta′

ya′u

)


∪AB1 t ∈ Ta1 (6)

yat ≤ 1 −∑

u∈Ta′ 2 u≤t+tma +tba

ya′u


∪AB1 t ∈ Ta1 (7)

yat binary ∀a ∈AM∪AB1 t ∈T0 (8)

We present the IP formulation for the original problem atmonthly fidelity. The decision variables dictate whether webegin an activity during a specified period. In theory (i.e., iniGantt), these activities are scheduled at the beginning of theperiod. However, because most activities do not consumean integral number of periods to complete, in practice, theactivity can begin at any point during the period in whichit is scheduled to start, as long as it is finished in an integernumber of periods no larger than the ceiling of the activ-ity’s duration. Our variable definition necessarily discretizestime, and at a fidelity that is arguably coarser than that atwhich the mine would implement the schedule. However,we require binary variables to enforce the precedence con-straints, and the number of these variables is proportionalto the product of the number of activities and the numberof periods we consider. A finer time fidelity (approachinga continuum) might more closely approximate the way inwhich mine operations are conducted, but would renderthe model so intractable that it would produce no resultsquickly enough to be practical. Although not mathemati-cally portrayed here, we can reduce the number of variables(from the product of the number of activities and the num-ber of periods we consider) by employing early start datesfor each activity, as discussed in the Integer ProgrammingApproach section.

We maximize the discounted value of extracted metalover the horizon, where we discount consistent with theperiod in which we extract the metal. Note that we effec-tively assume an arbitrary constant metal price (given indollars or euros per tonne), which we can omit from theobjective without changing the optimal solution. As a result,our objective is given in tonnes, rather than in a monetaryunit. Constraints (1) ensure that we never schedule a min-ing or backfilling activity more than once. Constraints (2)require that the scheduled production in any period is nomore than the production capacity and no less than whatthe mill requires to operate for that period. Constraints (3)force the model to maintain metal output within the oper-ational limits of the mill. Constraints (4) track the use ofpaste fill in each period and ensure that its use does notexceed its availability. Constraints (5)–(7) are the sequencingconstraints that enforce precedence rules between activities.

The first set of sequencing constraints, (5), ensures thatan activity a cannot begin until all of its predecessor activ-ities a′ ∈ Pa have completed. For example, block A must bemined to access block B, and block B must be mined to

access block C. Therefore, we cannot mine block C withoutmining block A and block B first. The second set of sequenc-ing constraints, (6), ensures that the order of mining in apanel be maintained, regardless of whether all predeces-sors of a block a′ ∈ Pa have been extracted. In this example,blocks A and B do not both need to be mined if we mineblock C. Assume, without loss of generality, that we mineonly block A. Then, we only impose the constraint that thisblock must be mined before block C. Although all of theblocks’ precedences were initially characterized accordingto constraints (5), relaxing many to follow constraints (6)allows for a feasible, yet better, solution. Sequencing con-straints (7) ensure that an activity a cannot occur once ahaulage pillar a′ ∈ Pa, on which the activity depends, isextracted.

We model our integer program using the AMPL pro-gramming language, version 20090327 (Fourer et al. 2003)and solve it with the CPLEX solver, version 12.3 (IBM 2011)on a machine with two dual Intel Xeon X5570 quad coreprocessors and 48 GB of RAM.

AcknowledgmentsWe thank the Lisheen Mine for providing us with thisproject and supporting the research. In particular, we thankTom Bailey and Noeleen Fox for their time and commit-ment. We also thank Brian Hall of AMC Consultants for hiscomments and advice. Finally, we thank Dr. Tim Kaiser ofthe High Performance Computing Group at the ColoradoSchool of Mines for his resources.

References

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Chanda E (1990) An application of integer programming and simu-lation to production planning for a stratiform ore body. MiningSci. Tech. 11(2):165–172.

Fourer R, Gay DM, Kernighan BW (2003) AMPL: A ModelingLanguage for Mathematical Programming (Thompson Learning,Pacific Grove, CA).

Hamrin H (1997) Guide to Underground Mining Methods and Applica-tions (Atlas Copco, Stockholm).

IBM (2011) ILOG CPLEX. Accessed December 1, 2012, http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/index.html.

Klotz E, Newman AM (2013) Practical guidelines for solving diffi-cult mixed integer linear programs. Surveys Oper. Res. Manage-ment Sci. 18(12):18–32.

Maptek (2012) Vulcan. Accessed December 1, 2012, http://www.maptek.com/products/vulcan.

Martinez MA, Newman AM (2011) A solution approach for opti-mizing long- and short-term production scheduling at LKAB’sKiruna mine. Eur. J. Oper. Res. 211(1):184–197.

MineMax (2012) MineMax iGantt. Accessed December 1, 2012,http://www.minemax.com/downloads/Minemax-iGantt-Brochure.pdf.

Newman A, Kuchta M (2007) Using aggregation to optimize long-term production planning at an underground mine. Eur. J.Oper. Res. 176(2):1205–1218.

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Verification LetterTom Bailey, Chief Technical Services Engineer Lisheen

Mine, Killoran, Moyne, Thurles, Co. Tipperary, Ireland,writes:

“Manually scheduling ore production is a difficult andlabor intensive part of our mining operation and usu-ally takes three or more weeks to complete. Maximizingmetal extraction over the life of the mine becomes increas-ingly challenging as the number of available working areasdeclines. The production profile eventually decreases to apoint beyond which it is no longer profitable to operate,and this defines the life-of-mine (LOM). At the end of theLOM, there is still ore remaining in the mine at varyingZinc grades. The optimizer generates a mining schedulethat maximizes the metal extracted over the LOM, whilstminimizing the mineral resources left in the ground afterclosure.

“We are currently using the optimizer to assist in our pro-duction planning for the last two years of the mining opera-tion at Lisheen. The optimized schedules generally supportthe decisions encoded in the manual schedule. However,in addition to the primary objective of maximizing metalextraction, the optimization tool provides us with the fol-lowing benefits: (i) a way to quickly examine alternativeend-of-mine-life scenarios, e.g., different mine closure dates

(ii) less change in the production tonnage from one weekto the next, enabling us to generate more consistent laborshifts; and (iii) the identification of manually scheduledinfeasibilities, e.g., too much high-grade ore scheduled forsimultaneous extraction.

“The optimizer has been particularly useful in identify-ing areas that potentially will not be mined no matter howquickly we mine in the neighboring vicinity. With this infor-mation now to hand, we have initiated new plans to acceler-ate the extraction in these areas. This has resulted in c.120ktat 13 percent ZnEq of extra ore now being included in ourrevised LOM Schedule.

“We will continue to employ the optimizer both topull forward high grade areas and to identify areas thatneed additional mining to free them for mining before themine closes.”

Dónal O’Sullivan earned a BA in economics and a BA inengineering from Brown University, and he has completeda PhD in mineral and energy economics at the ColoradoSchool of Mines. Before graduate school he specializedin electric power consulting and power market modeling.While at Mines, he taught an undergraduate course in eco-nomics and technology. He is an economic consultant withGalt and Company.

Alexandra Newman is an associate professor in the Divi-sion of Economics and Business at the Colorado School ofMines. She holds a BS in applied mathematics from the Uni-versity of Chicago, an MS in operations research from theUniversity of California, Berkeley, and a PhD in industrialengineering and operations research from the University ofCalifornia, Berkeley. Prior to joining the Colorado School ofMines, she held the appointment of research assistant pro-fessor in the Operations Research Department at the NavalPostgraduate School. Professor Newman’s primary researchinterests lie in optimization modeling, particularly as it isapplied to logistics, the energy and mining industries, andmilitary operations. She is serving as associate editor ofOperations Research and Interfaces, and is an active memberof INFORMS.

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