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Some Mine Planning Problems

Some Mine Planning Problems. Universidad Federico Santa María Laboratorio de Modelamiento Septiembre 2012. AMTC. AMTC: - Started Late 2009 64 researchers , 86 MsC & PhD students USD 3.4MM/ year budget . Georesources & Exploration , Geosmetallurgical modeling ,

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Some Mine Planning Problems

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  1. Some Mine PlanningProblems Universidad Federico Santa María Laboratorio de Modelamiento Septiembre 2012

  2. AMTC • AMTC: • -Started Late 2009 • 64 researchers, • 86 MsC & PhDstudents • USD 3.4MM/yearbudget. • Georesources & Exploration, • Geosmetallurgicalmodeling, • Mine Planning, • Mine Design, • Robotics & Automation, • ImageProcessing and Recognition, • Energy, • Water& Environment • ExtractiveMetallurgy University of Chile (1842) +20 Advanced Mining Technology Center (2009)

  3. Delphos University of Chile (1842) • DelphosToday: • 3 researchers, • 2 PhDstudents, • 5gradstudents (master). • Mission: • Become a bridge between academia and industry. • Original research& development. • Collaborationwithotherresearchgroups (validation) • Education. Mining Engineering Dpt. (1853) Advanced Mining Technology Center (2009) Delphos (2008)

  4. UNDERGROUND OPEN PIT Ore located “near” thesurface: material isextractedbydiggingfrom top tobottom, requirestomove WASTE. Ore located “deep” underthesurface, material isreachedbyconstructingtunnels and shafts. Thereis no waste (methods are more selective). Relativelycheaper (no ventilation, light, largerequipment). Costvary a lotdependingonthemethod, but are higher (speciallyinvestments) than open pit. Mainphysicalconstraint: SLOPE ANGLES.. Physicalconstraintsdependonthemethodtoo: material flow, material management, constructability. Verylargeproduction. Relativelysmallerproduction.

  5. Some “vocabulary” • Production Plan: • Is a graph in whichtheproduction (tonnage) and concentration (grade) are representedover time (X-axis). • Block: • Mine isdiscretizedintoanarray of “blocks”. • Each block has a position and a set of geometallurgicalattributes (constantwithinthe block). • The set of all blocks iscalledthe Block Model. • Scheduling: • A mapping of the blocks into a production plan.

  6. Block Model • Block Modelisconstructedfromsamples, and usinggeostatisticstoestimatetheattributes of the rock at differentlocations. • Most of the time, the block modelisconstructedusingKriging, which produces unbiasedestimationswithminimumvariance. • Theoptimizationmodelspresentedhereworkon a given block model, but,thisiscertainly a relevantsource of uncertainty.

  7. Mine Planning • Discipline that transforms the mine information and economic parameters into a production plan (how much to produce and when) and therefore a business plan with economical value. • The production plan is supported by a scheduling, which determines what blocks are going to be extracted, whether they will be processed or not and when all this should happen.

  8. In the 60’s … • Lerchs y Grossman (1965) presentthe final pitproblem and analgorithmto produce nestedpits (as a waytosellcomputers!). • T. B. Johnson (1969) introduces a mathematicalmodelwhichsolutionis a sequence of pitsthatmaximizes NPV.

  9. Example: Final Pit j i 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 = $ 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0

  10. Example: Final Pit • Given: • A set of blocks • Real values (associated to economic value) • Precedence A relation so • Find a pit (set of blocks compatible with the precedence) of maximal contained value.

  11. Lerchs and Grossman • The Final PitProblem: • Whatisthe set of blocks containingthemaximumvalue? • Developanalgorithmtosolveit. • Show that, bychangingprices, they can produce a sequence of nestedpits. • Notes: • Theyconsiderprecedenceconstraints. • Only ONE value per block. • No considerationsabouttransportationcapacityorproductionconstraints.

  12. Schedulingby Final Pit • Time isintroducedartificially. • Needsto define thedestination/process of the block in advance (fixedcut of grade) • Pushbacks are selectedmanually. + price

  13. Schedulingby final Pit (2) • Withoutconsideringtheoportunitycost, theleftpitisalwaysprefered. • Planningwith a fixedcut off grade induces certaingeometries: it defines thegeometricdistribution of waste and mineral. • The plan isconstructedwithaggregatedinformation. In the short-term, thegeometricdistributiondoesmatter. Revenue Factor

  14. Good things about the final pit • If value v(i) is replaced with v’(i) (where v’(i) issmaller than v(i)), then the optimal solution in the new setting is smaller (in the sense of inclusion). This allows to create nested pits. • It is “easy”: can be solved quickly (Lerchs & Grossman 65, Hochbaum 2001-2009), even for millions of blocks. • It is the algorithm usedin commercial software for mine planning.

  15. Bad things about the final pit • It does not consider production and mining capacities, hence, it does nottake timeinto account. • It requires to make the decision about the destination of the block beforehand. • It is the algorithm used in commercial software for mine planning.

  16. TheT. B. Johnson’sModel • Statestheproblem of calculatingthenestedpitsthatmaximize NPV undertheconstraints of: • Slopeangle (precedences) • Capacity (and demand) of transportation and processing. • Minimum and maximumcut-off grades, as a result, withinranges. • Model decides: • Whatto mine and whatnotto mine, and when. • Whatto do withmined blocks. • Notes: • Thereis no fixedcut-off grade. • Pitssatisfycapacityconstraintsfromthebeginning (no human-madechoice) • Multiple-Destinations = Multiplepossibleprofits, dependingonthedecision.

  17. Open Pit block scheduling problem (still simple version) • Given: • Set of blocks with values and precedences (so graph G=(B,A)), as before. • A number T of time-periods t=1,2,…,T. • Resources r=1,2,…,R: • Capacities per resource and time-period • Resource consumption per block

  18. OPBSP 1 iif block i has beenextractedbetween1,2,…,T Problemisverybig (ex: 50,000 blocks 10 periods = half a millionbinary variables, and about 750,000 constraints), and thisisonlythedeterministic case!

  19. Howtosolve OPBSP? • Lagrangeanrelaxation Twosuccessfull ideas: Chicoisne (2009), Bienstock (2010)

  20. Howtosolve OPBSP? • Aggregation FIXED FIXED FIXED FIXED FIXED FIXED

  21. Howtosolve OPBSP? • Aggregation

  22. Aggregated problem is “solved” 2 2 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

  23. “Important” decisions are made 2 2 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

  24. Blocks at the interior are fixed 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2

  25. Blocks in the borders are then refined and re-optimized 2 2 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

  26. Final solution is reported at original block level 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 1 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

  27. Howtosolve OPBSP? • Otherpossibilities (tostudy): • Metaheuristics • Combinations of theabove • Otherassociatedproblems • Approximationalgorithms: Do itwell (with a guarrantee).

  28. Do wewanttosolve OPBSP? • OPBSP doesnotconsider: • Designconstraints • Abilityto “chooseyourdestiny” • Accesibilitylimitations (blocks are toosmall) • Blendingconstraints. • Uncertainty

  29. BOS2(M) • BOS2 is a sequencer of blocks for open-pit considering the following elements: • Capacity and Blending constraints (per period) • Slope constraints • Accessibility constraints. • Multiple possible destinations per block. • BOS2M has been extensively tested on different mines in the North of Chile. • It is the base for developing a commercial product (VMM) through a spin-off company: Cube-Mine.

  30. Short Term Mine Planing • Long-term mine planning: • NPV oriented • Done over very aggregated data. • Fixes production goals. • More understood (OR people). • … “makes short-term mine planning hard”: • Actual distribution of ore/waste within the limits defined by the long-term plan is not uniform. • Hard geometric and blending constraints. • Main limits are already defined (fixed transpor. capacity/roads) • Not very studied (in O.R.)

  31. BOS2 Access • MRU (Mineral Reserve Units): Clustering of blocks in terms of theirattributes(Mintype, profit, anillos, etc). • Graphstructure: Slope and accessibilityconstraints. • Stocks: Existing stocks are considered in theschedulingprocess. Multipledestinations: Modeling of the material handlingsystem.

  32. Multiple Destinations

  33. Multiple Destinations • Depending on the processing line, a block will contribute a different value for the project (different costs, different recoverings, etc.) • The block will also use different resources (example, transportation or capacities). • Different blending constraints apply. • Advantages (to grade or attribute base preassignment): • Capacities are used better. • We do not induce a geometry of the extraction based on predefinition of block destination. • Higher value for the project.

  34. Underground Development & Extraction Sequencer and Scheduler UDESS

  35. UDESS • The Underground Development Extraction Scheduling and Sequencing is a tool that determines, for a set of construction and extraction activities linked by constructability and operational precedences, what is the optimal time to perform each of them, so that this complies with the precedence constraints as well as resource availability. • UDESS has been tested on different data-sets at the Proof-Of-Concept Level for Open-Stope mines and Panel Caving (Palabora). • Currently we are working with El Teniente to test it on one of their mines. • Yamana Gold is also interested in using this as a software.

  36. Panel caving mine, 74 drawpoints to be scheduled on 55 monthly periods. • From Mine2-4D (software): 2,256 construction activities + 74 production activities (one per drawpoint), which are reduced to 1022+74 toschedule. • Solution in a couple of hours.

  37. There are 85 columns to be scheduled for extraction up to 14 years, and 574 development structures.

  38. Problemsassociatedto UDESS • Extendthemodel, and stillbeabletosolveit: • “Or” precedences. • Equipmentassignment. • Makeitfaster, bigger: • Currently, wesolveitusing time-aggregation.

  39. Delphos: WHAT WE ARE DOING THESE DAYS

  40. Planning under geological uncertainty • Interpolation of samples introduces uncertainty. • Classical mine planning done for mean case. • How to develop strategies dealing with spatial variance.

  41. “Distribución de una medida sobre una distribución espacial" Objetivo: Describir la distribución de probabilidad que sigue una medida en un conjunto de puntos, de una variable con distribución espacial. Ejemplo: valor de un cono de extracción de mineral en un yacimiento. Notas: • La aplicación práctica es poder generar sampling del valor del cono, en vez de bloques al interior del cono y luego sumar. • Se requiere saber/aprender un poco de geoestadística. • Para caracterizar la distribución, es posible trabajar usando desde sampling hasta intentar deducir una distribución explícita (poco probable). En cualquier caso, dar propiedades de ella (cotas, peso de colas).

  42. Tema 2: "Parametrización en cilíndricas de un switchback" Resumen: Escribir en coordenadas cilíndricas una rampa en switchback, paramétricamente en función de la pendiente de la rampa, velocidad de apertura helicoidal y ancho de la rampa. Notas: • La aplicación es parte de herramientas de diseño de un rajo. • Las rampas son los caminos por donde transitan los camiones con carga en los rajos. • No confundir con los bancos, que son los encargados de modelar el talud global del rajo. • La pendiente de la rampa está acotada por seguridad vial.

  43. Mine Integration & InformationSystemsforPlanning Feedback systems design for short & mid term mine planning Mining Process Better Decisions Feedback Information Mine Planning Sensing & data gathering systems Integrated mining models development

  44. Real options applied to mining Realoptions introduce flexibility to address market uncertainty US$ 2 Billionlosses in future sales duetomarketuncertaintycouldriseto US$ 4.6 Billion OptionValue= NPV(withoption)-NPV(withoutoption)

  45. Algorithms for Mine Planning 3 1 2 What to extract and when? Maximize NPV over millions of decision variables. How to solve this problem?

  46. Real options in open-pit mine planning • Real options provides flexibility to protetct projects against uncertainty. • Uncertaity sources: Real Options

  47. Underground development sequencer and scheduling Not enough infrastructure prepared • Our model produces development and production schedules that are consistent with each other. • Traditional method: First to schedule production, then to define construction schedule.

  48. Wrap up • Mine Planningis a bigsource of interestingproblems of: • (Combinatorial) Optimization • Algorithmics • (Stochastic) Control • AppliedStatistics and StochasticProcesses.

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