Some Mine Planning Problems

Universidad Federico Santa MaríaLaboratorio de Modelamiento

Septiembre 2012

AMTC

University of Chile (1842)

AdvancedMining

TechnologyCenter (2009)

AMTC:

-Started Late 2009-64 researchers,-86 MsC & PhD students-USD 3.4MM/year budget.

- Georesources & Exploration,- Geosmetallurgical modeling, - Mine Planning, - Mine Design, - Robotics & Automation,- Image Processing and Recognition, - Energy, - Water & Environment- Extractive Metallurgy

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Delphos

University of Chile (1842)

MiningEngineeringDpt. (1853)

AdvancedMining

TechnologyCenter (2009)

Delphos (2008)

Delphos Today:-3 researchers, -2 PhD students,-5 grad students (master).

Mission:Become a bridge between academia and industry.-Original research & development.-Collaboration with other research groups (validation)-Education.

Ore located “near” the surface: material is extracted by digging from top to bottom, requires to

move WASTE.

Ore located “deep” under the surface, material is reached by

constructing tunnels and shafts. There is no waste (methods are

more selective).

Relatively cheaper (no ventilation, light, larger equipment).

OPEN PIT UNDERGROUND

Main physical constraint: SLOPE ANGLES..

Very large production.

Cost vary a lot depending on the method, but are higher (specially

investments) than open pit.

Physical constraints depend on the method too: material flow, material

management, constructability.

Relatively smaller production.

Some “vocabulary”

• Production Plan: – Is a graph in which the

production (tonnage) and concentration (grade) are represented over time (X-axis).

• Block:– Mine is discretized into an

array of “blocks”. – Each block has a position

and a set of geometallurgical attributes (constant within the block).

– The set of all blocks is called the Block Model.

• Scheduling:– A mapping of the blocks

into a production plan.0

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Periodo (años)Mineral Waste Cu Au

VAN: 840.5 MUS$

Block Model

• Block Model is constructed from samples, and using geostatistics to estimate the attributes of the rock at different locations.

• Most of the time, the block model is constructed using Kriging, which produces unbiased estimations with minimum variance.

• The optimization models presented here work on a given block model, but,this is certainly a relevant source of uncertainty.

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.

In the 60’s …• Lerchs y Grossman (1965)

present the final pit problem and an algorithm to produce nested pits (as a way to sell computers!).

• T. B. Johnson (1969) introduces a mathematical model which solution is a sequence of pits that maximizes NPV.

Example: Final Pit

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11 1 = $

j

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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.

Lerchs and Grossman

• The Final Pit Problem:– What is the set of blocks containing the maximum value?

• Develop an algorithm to solve it.• Show that, by changing prices, they can produce a

sequence of nested pits.• Notes:– They consider precedence constraints.– Only ONE value per block.– No considerations about transportation capacity or

production constraints.

Scheduling by Final Pit

• Time is introduced artificially.

• Needs to define the destination/process of the block in advance (fixed cut of grade)

• Pushbacks are selected manually.

+ price

Scheduling by final Pit (2)• Without considering the

oportunity cost, the left pit is always prefered.

• Planning with a fixed cut off grade induces certain geometries: it defines the geometric distribution of waste and mineral.

• The plan is constructed with aggregated information. In the short-term, the geometric distribution does matter.

Revenue Factor

Good things about the final pit

• If value v(i) is replaced with v’(i) (where v’(i) is smaller 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 used in commercial software for mine planning.

Bad things about the final pit

• It does not consider production and mining capacities, hence, it does not take time into 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.

The T. B. Johnson’s Model• States the problem of calculating the nested pits that maximize NPV

under the constraints of:– Slope angle (precedences)– Capacity (and demand) of transportation and processing.– Minimum and maximum cut-off grades, as a result, within ranges.

• Model decides:– What to mine and what not to mine, and when.– What to do with mined blocks.

• Notes:– There is no fixed cut-off grade.– Pits satisfy capacity constraints from the beginning (no human-made choice)– Multiple-Destinations = Multiple possible profits, depending on the decision.

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

OPBSP

Problem is very big (ex: 50,000 blocks 10 periods = half a million binary variables, and about 750,000 constraints), and this is only

the deterministic case!

1 iif block i has been extracted between 1,2,…,T

How to solve OPBSP?

• Lagrangean relaxation

Two successfull ideas:Chicoisne (2009), Bienstock (2010)

How to solve OPBSP?

• Aggregation

FIXED

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How to solve OPBSP?

• Aggregation

Aggregated problem is “solved”

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How to solve OPBSP?

• Other possibilities (to study):– Metaheuristics– Combinations of the above

• Other associated problems– Approximation algorithms: Do it well (with a

guarrantee).

Do we want to solve OPBSP?

• OPBSP does not consider:– Design constraints– Ability to “choose your destiny”– Accesibility limitations (blocks are too small)– Blending constraints.– Uncertainty

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.

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.)

AccessMRU (Mineral Reserve Units): Clustering of blocks in terms of their attributes (Mintype, profit, anillos, etc).

BOS2

Graph structure: Slope and accessibility constraints.

Stocks: Existing stocks are considered in the scheduling process.

Multiple destinations: Modeling of the material handling system.

Acceso

WASTE OXIDE MIX SULFIDE

EN

E

Stock

Bioleach

Leaching

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CatodesWastedump

Los Colorados

Laguna Seca

Copper Concentrate

Coloso port

Multiple Destinations

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.

Underground Development & Extraction Sequencer and Scheduler

UDESS

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.

• 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 to schedule.

• Solution in a couple of hours.

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

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Problems associated to UDESS

• Extend the model, and still be able to solve it:– “Or” precedences.– Equipment assignment.

• Make it faster, bigger:– Currently, we solve it using time-aggregation.

WHAT WE ARE DOING THESE DAYSDelphos:

Planning under geological uncertainty

• Interpolation of samples introduces uncertainty.•Classical mine planning

done for mean case.•How to develop

strategies dealing with spatial variance.

“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:i) La aplicación práctica es poder generar sampling del valor del cono, en

vez de bloques al interior del cono y luego sumar.ii) Se requiere saber/aprender un poco de geoestadística.iii) 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).

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:i) La aplicación es parte de herramientas de diseño de un rajo.ii) Las rampas son los caminos por donde transitan los camiones con carga

en los rajos.iii) No confundir con los bancos, que son los encargados de modelar el talud

global del rajo.iv) La pendiente de la rampa está acotada por seguridad vial.

Feedback systems design for short & mid term mine planning

Mine Integration & Information Systems for Planning

Mining Process

Mine Planning

Feedback Information

Better Decisions

Sensing & data gathering systems

Integrated mining models development

Real options applied to mining

Real options introduce flexibility to address market

uncertainty

Option Value= NPV(with option)-NPV(without option)model 1 model 2 model 3 model 4

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Option to defer 1 year investment

Valo

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US$]

US$ 2 Billion losses in future sales due to market uncertainty could rise to US$ 4.6 Billion

Algorithms for Mine Planning

• What to extract and when?• Maximize NPV over millions of

decision variables. • How to solve this problem?

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Real options in open-pit mine planning

• Real options provides flexibility to protetct projects against uncertainty.

• Uncertaity sources:Geological Market Operational

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1 Effect of geological uncer-tainty

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Underground development sequencer and scheduling

• Traditional method: First to schedule production, then to define construction schedule.

Resources

ModelEconomic Envelop

Mining

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Sequence and

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Economic evaluatio

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Not enough infrastructure prepared

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evaluation

• Our model produces development and production schedules that are consistent with each other.

Wrap up

• Mine Planning is a big source of interesting problems of:– (Combinatorial) Optimization– Algorithmics– (Stochastic) Control– Applied Statistics and Stochastic Processes.