CIOMIN

Primera Reunión de Directorio, Centro de Investigación de Operaciones para la Industria Minera Santiago, 2 de Junio de 2009Centro de Investigación de Operaciones para la Industria

MineraSantiago, 8 septiembre de 2010 11

Centro de Investigación de Operaciones para la Industria Minera

Departamento de Ingeniería de Minas y Departamento de Ingeniería IndustrialFacultad de Ciencias Físicas y Matemáticas

Universidad de Chile

Co-Directores:

Enrique Rubio Dpto. Ing. MinasRafael Epstein Dpto. Ing. Industrial

CIOMIN



AGENDAAGENDA

1. Equipo de trabajo

2. Líneas de investigación

3. ORMNet

4. Visitas al exterior

5. Organización de seminarios

6. Financiamiento tesis



1. EQUIPO DE TRABAJO1. EQUIPO DE TRABAJO

- Investigadores Departamento de Ingeniería Industrial: René Caldentey, Felipe Caro, Eduardo Contreras, José Miguel Cruz, Rafael Epstein, Nicolás Figueroa, Marcel Goic, Juan Velásquez, Andrés Weintraub.

- Investigadores Departamento de Ingeniería de Minas: Raúl Castro, Xavier Emery, Julián Ortiz, Enrique Rubio, Sebastián Troncoso.



2. LÍNEAS DE INVESTIGACIÓN2. LÍNEAS DE INVESTIGACIÓN

- Logística minera.

- Modelos de optimización.

- Gestión minera.

- Gestión del cambio.

- Tecnologías de información y comunicaciones.

- Planificación minera.

- Tendencia en los precios del cobre.

Optimization Approaches to Long Term Mine Planning

Seminar OR in Mining, 15th - 17th March 2010



- Background

- Problem Formulation

- Methodology of Analysis

- Underground Mining

- Open Pit Mining

- Applications in CODELCO

- Uncertainty in Prizes


MineraSantiago, 8 septiembre de 2010

• Traditional approach:

– First planning of mining stage(extraction, cut-off grades, tonnage).

– Then plant stage planning: mills, concentration(mill recovery, throughput).

– Iterations to match supply and demand.

7

Background



Background

• Challenges:

– Integration between mining and process stages.

– Planning multiple mines and plants at the same time.

– Additional operational constraints.

– Find the overall optimal operation policy, in the long run (20-50 years).

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Problem Description


9



Key Elements• Long term analysis.

• Combined mine-plant analysis.

• Evaluation of multiple mines at the same time.

• Process representation.

• Spatial and temporal definition of reserves to be extracted/processed.

• Environmental considerations.

• Investments in infrastructure.10



Overview of the Problem• Input:

– Mineral resources.

– Economic & external factors (geo/technological).

– Regulations/policies.

(company’s objectives, risk constraints).

•Output:– Extraction plan.

– Production plan.

– Investment plan (infrastructure).

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Methodology of Analysis




Methodology• Math programming model (MIP)

– Optimizes long term plans and large investments in open pit and underground copper mines.

• Model has two main components:– Extraction: ore is extracted from the mineral resource.– Transportation and processes: ore is processed and refined.

• Multi-commodity capacitated network flow problem:– Capacitated arcs and nodes.– Products:

Copper Molybdenum Arsenic

13



Underground MineUnderground Mine

PRODUCTION

Block caving

LHD

Block caving

ROCK SIZE REDUCTIO

N

MAIN TRANSPORTATION

(Train)

SAG

PLANTS(MILLS)

COMMERCIAL PRODUCTS

Gravity (Tunnel)

Drain stage

COPPERMOLY



PlantMine

Mine

Mine

Plant

Plant

Mine

Plant

Stock

Stock

Open Pit MineOpen Pit Mine



OPEN PIT MINES

UNDERGROUND MINES

EXTRACTION INITIALTRASNPORT

REDUCTION INSIDE MINE

MAINTRANSPORT

DRAIN STAGE

REDUCTION OUTSIDE MINE

MILLS CONCENTRATION

DEMAND

DEPOSIT OF VERY-LOW-GRADE ORE

AND STOCK

Network FlowNetwork FlowRepresentationRepresentation



Underground Mining




• t t : Planning periods : Planning periods

• aa : Exploitation sector: Exploitation sector

• j,ij,i : Mineral Columns : Mineral Columns

• nn : Mineral Block in a : Mineral Block in a column column

• v : Process Nodev : Process Node

Underground miningIndices (sets)

• Extraction at:

Blocks Columns Sector

• Flow routing

Decision variables

BlocksBlocks

SectorSector



: Height extracted in column j at period t

: Tonnage extracted (per day) in sector a at period t

: Flow of product k between node v and node p at period t (generic)

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Underground miningMain Variables

0tjY

0taExt

0tvpkf

1 If block n of the column j is extracted at period t t

jnZ0 ~



• Each block can be exploited just once through the horizon.

Underground miningMain Constraints

tjn

tnj ZZ )1(

ti

tj ZZ 00 ),(),( ijSECij

1tjn

t

Z nj,

tnj ,,

• Sequence in which columns can be extracted.

• Blocks must be extracted bottom-up.

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• Relation between blocks, columns and sectors.

( ) ( )

tjn kjn

k j J a n N jta t

a

Z A

ExtDP

Underground mining

• Sector extraction capacity.

1

( )

t t tj j jn jn

n N j

Y Y Z h

ta

ta

ta MAX_EXTExtMIN_EXT ta,

• Time availability constraint (at max extraction rate).

( )

t tjn jn a

n N j

Z DP

, ( )t j J a

Main Constraints



– Maximum allowed horizontal extraction per sector (geotechnical constraint).

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• Flow conservation and node/arc capacities.

• Maximum level of allowed contaminants.

0( )

t t ta j j a

j J a

MIN_AREA Z area MAX_AREA

ta,

tvp

Kk

tvpk CAPf

V

VK : products at node v

Main Constraints

' ' ' , ,okk omkt ko k

CT f MaxC k CONT t m

Underground mining



• Regularity in heights

• Interaction with neighborhoods

23

Columns

Neighborhoods

aji

ti

tj YY JIij ),( t

Underground mining



Neighborhood

24

Underground mining



•Goal: To Maximize Net Present Value

– Income: sale of products and sub-products.

– Cost: Extraction, transportation and process at plants.

– Investment: New projects .

INVVCFCIFO ..max

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Underground mining



Open Pit Mining




•Mining resource:– Expansion: exploitable “slice” of the pit.– Bench: transversal cut, characterized

by elevation and height.

expansion

bench

Open Pit Mining



• Operational constraints Min/Max extraction rates

• Only resource on the surface can be exploited.

• Safety constraints.

∆h MIN

Rock spillage

∆h MAX

Instability of walls

Open Pit Mining



etapa mina

etapa almacenamientode material.

etapa chancadoprimario

etapa preparaciónplanta

etapa planta.concentradora sx/ew

molienda lixiviación sbl

chancador

botadero stock

concentradora sx/ew

molienda lixiviación sbl

chancador

botadero stock

Mine

Storage Stage

Crushing Stage

Preparation plant

Plant

Deposits of very low-grade ore

Crushing

Grinding Leaching Dumpleachin

g

Flotation SX/EW

Stock

Network Flow & Design



Artificial Ending Bench

Artificial Starting Bench

Art

ificia

l S

tart

ing

Peri

od

Art

ificia

l E

nd

ing

Peri

od

Leads to less fractional solution than classical

formulation.

Time

Benches

Network flow formulation to extract a sequence of benches (includes safety considerations).

Network Flow Formulation



Main decision variables

wijt = 1 if benches from (i+1) to j are extracted on period t

zit ∈ {0,1} equals 1 if bench i is extracted in period t

SetsOr(a): artificial starting bench.

De(a): artificial ending bench.

Pini: artificial starting time period.

Pfin: artificial ending time period.

SU(i,t): successors of node (i,t).

AN(i,t): predecessors of node (i,t).

(1) Flow conservation:(1) Flow conservation:

(2) Each expansion can start only one extraction sequence.

(3) Each expansion can end only one extraction sequence.

( 1)( , ) ( , )

,hit ij th AN i t j SU i t

w w i t

( ) 1( ( ),0)

1Or a jj SU Or a

w a

( )

( )

1jDe a Pfinj a Or a

w a

(4) Relation between variables.

( , )

,it hjtj SU i t h i

z w i t

Network Flow Formulation



Open Pit Mining• Downstream processes: Downstream processes:

– Same as underground but with several Same as underground but with several alternatives routes.alternatives routes.

– Different types of plants: flotation, leaching, Different types of plants: flotation, leaching, bioleaching, low-grade sulfides.bioleaching, low-grade sulfides.

– Stocking areas: large amount of material is Stocking areas: large amount of material is stored for future use.stored for future use.

– Waste dumps: material without economic Waste dumps: material without economic value.value.



Solving the ModelReal instances at CODELCO lead to hard problems.

Caro et al. - Long Term Optimization of Investment & Production Plans in Open Pit & Underground Copper Mines

• Given the large amount of binary variables, especially those representing the extraction stage, solving the model using a mixed integer programming routine is nonviable.

Model Nº Constraints Nº Variables 0-1 VariablesUnderground mine 446,521 535,639 196,386Open pit mine 245,391 898,742 160,386



• The integrality condition is relaxed and a continuous version is solved:

Rounding Heuristic

• Heuristic consists of fixing binary variables based on the solution of continuous version.

}1,0{tjnZ 10 t

jnZ

Fraction of block n of column j extracted at period t

Underground mines



• Logic based on:– Tonnage per column to

remain constant.– Fix variables in logical

order.

• This process is iterative and stops when the solution is integer.

• Since copper grade is better for lower altitude blocks, LP is reasonably good.

• A similar heuristic is used for open pit mine problems.

Underground mines

Rounding Heuristic



Applications in CODELCO


36



Applications in Codelco

• The model was implemented using GAMS and was solved using CPLEX.

• System implementation at CODELCO’s Divisions (MUCH):

– El Teniente.– North Division.– El Salvador.– Andina Division.

• When using our methodology, the NPV of projects have increased up to 5%.

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Uncertainty in Prizes


38



Problem

•Copper price introduces risk in mine planning.

•Previous models work with deterministic prices.– Prices represent expected values.

•Copper price is volatile and variance is not small.

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Methodology

• Prize uncertainty can be addressed by:

– Extending combinatorial models.– Real Options models.– Stochastic programming.– Robust planning.

• In order to apply these techniques we need to:

– Improve solving algorithms performance.– Reduce problem size by specialized aggregations.

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MethodologyStock

Plant

Plant

Plant

Plant

Plant

Stock

Plant

Including uncertainty in prices generates multiple scenarios.

Each scenario configures one complex combinatorial problem.



Robust Planning

•Robust Planning: Generate plans that will have a reasonable behavior under all possible scenarios.

•Robust planning represents an interesting opportunity for long term mine planning.

42



Applying Robust Planning

low highlow 9,789 10,715high 9,773 10,889

-16 -174

43

• Results (MMUS$):

Planning price

Evaluation price

• High asymmetry between strategies when evaluated in the opposing price.

• When optimistic plan (high planning price) is evaluated on low price scenario loss is lower than otherwise.



Why this asymetry?• Asymmetry depends on marginal production costs.

• There are 3 possible cases:

44

C’(x)

xB xA

PA

PB

xB xA

PA

PB

xB xA

PA

PB

C’(x) C’(x)

Loss for producing XA when market price is PB

Profit for producing XA when market price is PA



3. 3. OPERATIONS RESEARCH IN MINING NETWORK OPERATIONS RESEARCH IN MINING NETWORK ((ORMNet)ORMNet)

• Operations Research for the Mining Industry Center (University of Chile)

• Curtin University of Technology (Australia)

• The University of Melbourne (Australia)

• Colorado School of Mines (U.S.A.)

• Polytechnic School of Montreal (Canada)

• Federal University of Minas Gerais (Brasil)



4. VISITAS AL EXTERIOR4. VISITAS AL EXTERIOR

- Visitas a instituciones y centros extranjeros especializados en temas de investigación afines.



5. ORGANIZACIÓN DE SEMINARIOS5. ORGANIZACIÓN DE SEMINARIOS

- Al menos un seminario por año.

- Invitación a exponer a investigadores externos.



6. FINANCIAMIENTO TESIS6. FINANCIAMIENTO TESIS

- Financiar alrededor de 10 tesis.

- Apoyo del Instituto Milenio Sistemas Complejos de Ingeniería.



Centro de Investigación de Operaciones para la Industria Minera

Departamento de Ingeniería de Minas y Departamento de Ingeniería IndustrialFacultad de Ciencias Físicas y Matemáticas

Universidad de Chile

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