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  • Extraction restrictions at: - bench level (geotechnical) - expansion level (geotechnical) - mining level (transportation) Open pt: same as underground but upside down? NO
  • Reasons to have stocking areas: High-grade ore has priority Mix adjustment, especially concentration of pollutants Keep plants working at capacity Waste: must be removed, transportation cost is becoming very relevant
  • Running times
  • Figure depicts an extreme example

    1. 1. Centro de Investigación de Operaciones para la Industria Minera Departamento de Ingeniería de Minas y Departamento de Ingeniería Industrial Facultad de Ciencias Físicas y Matemáticas Universidad de Chile Co-Directores: Enrique Rubio Dpto. Ing. Minas Rafael Epstein Dpto. Ing. Industrial CIOMIN
    2. 2. AGENDA <ul><li>Equipo de trabajo </li></ul><ul><li>Líneas de investigación </li></ul><ul><li>ORMNet </li></ul><ul><li>Visitas al exterior </li></ul><ul><li>Organización de seminarios </li></ul><ul><li>Financiamiento tesis </li></ul>
    3. 3. 1. EQUIPO DE TRABAJO <ul><li>Investigadores Departamento de Ingeniería Industrial: </li></ul><ul><li>René Caldentey, Felipe Caro, Eduardo Contreras, José Miguel Cruz, Rafael Epstein, Nicolás Figueroa, Marcel Goic, Juan Velásquez, Andrés Weintraub. </li></ul><ul><li>Investigadores Departamento de Ingeniería de Minas: </li></ul><ul><li>Raúl Castro, Xavier Emery, Julián Ortiz, Enrique Rubio, Sebastián Troncoso. </li></ul>
    4. 4. 2. LÍNEAS DE INVESTIGACIÓN <ul><li>Logística minera. </li></ul><ul><li>Modelos de optimización. </li></ul><ul><li>Gestión minera. </li></ul><ul><li>Gestión del cambio. </li></ul><ul><li>Tecnologías de información y comunicaciones. </li></ul><ul><li>Planificación minera. </li></ul><ul><li>Tendencia en los precios del cobre. </li></ul>
    5. 5. Optimization Approaches to Long Term Mine Planning Seminar OR in Mining, 15 th - 17 th March 2010
    6. 6. <ul><li>Background </li></ul><ul><li>Problem Formulation </li></ul><ul><li>Methodology of Analysis </li></ul><ul><li>Underground Mining </li></ul><ul><li>Open Pit Mining </li></ul><ul><li>Applications in CODELCO </li></ul><ul><li>Uncertainty in Prizes </li></ul>
    7. 7. <ul><li>Traditional approach: </li></ul><ul><ul><li>First planning of mining stage (extraction, cut-off grades, tonnage). </li></ul></ul><ul><ul><li>Then plant stage planning: mills, concentration (mill recovery, throughput). </li></ul></ul><ul><ul><li>Iterations to match supply and demand. </li></ul></ul>Background
    8. 8. Background <ul><li>Challenges: </li></ul><ul><ul><li>Integration between mining and process stages. </li></ul></ul><ul><ul><li>Planning multiple mines and plants at the same time. </li></ul></ul><ul><ul><li>Additional operational constraints. </li></ul></ul><ul><ul><li>Find the overall optimal operation policy, in the long run (20-50 years). </li></ul></ul>
    9. 9. Problem Description Optimization Approaches to Long Term Mine Planning
    10. 10. Key Elements <ul><li>Long term analysis. </li></ul><ul><li>Combined mine-plant analysis. </li></ul><ul><li>Evaluation of multiple mines at the same time. </li></ul><ul><li>Process representation. </li></ul><ul><li>Spatial and temporal definition of reserves to be extracted/processed. </li></ul><ul><li>Environmental considerations. </li></ul><ul><li>Investments in infrastructure. </li></ul>
    11. 11. Overview of the Problem <ul><li>Input: </li></ul><ul><ul><li>Mineral resources. </li></ul></ul><ul><ul><li>Economic & external factors (geo/technological). </li></ul></ul><ul><ul><li>Regulations/policies. (company’s objectives, risk constraints). </li></ul></ul><ul><li>Output: </li></ul><ul><ul><li>Extraction plan. </li></ul></ul><ul><ul><li>Production plan. </li></ul></ul><ul><ul><li>Investment plan (infrastructure). </li></ul></ul>
    12. 12. Methodology of Analysis Optimization Approaches to Long Term Mine Planning
    13. 13. Methodology <ul><li>Math programming model (MIP) </li></ul><ul><ul><li>Optimizes long term plans and large investments in open pit and underground copper mines. </li></ul></ul><ul><li>Model has two main components: </li></ul><ul><ul><li>Extraction: ore is extracted from the mineral resource. </li></ul></ul><ul><ul><li>Transportation and processes: ore is processed and refined. </li></ul></ul><ul><li>Multi-commodity capacitated network flow problem: </li></ul><ul><ul><li>Capacitated arcs and nodes. </li></ul></ul><ul><ul><li>Products: </li></ul></ul><ul><ul><ul><li>Copper </li></ul></ul></ul><ul><ul><ul><li>Molybdenum </li></ul></ul></ul><ul><ul><ul><li>Arsenic </li></ul></ul></ul>
    14. 14. Underground Mine PRODUCTION Block caving LHD Block caving ROCK SIZE REDUCTION MAIN TRANSPORTATION (Train) PLANTS (MILLS) COMMERCIAL PRODUCTS Gravity (Tunnel) Drain stage SAG COPPER MOLY
    15. 15. Open Pit Mine Plant Mine Mine Mine Plant Plant Mine Plant Stock Stock
    17. 17. Underground Mining Optimization Approaches to Long Term Mine Planning
    18. 18. <ul><li>t : Planning periods </li></ul><ul><li>a : Exploitation sector </li></ul><ul><li>j,i : Mineral Columns </li></ul><ul><li>n : Mineral Block in a column </li></ul><ul><li>v : Process Node </li></ul>Underground mining Indices (sets) <ul><li>Extraction at: </li></ul><ul><li>Blocks  Columns  Sector </li></ul><ul><li>Flow routing </li></ul>Decision variables Blocks Sector
    19. 19. <ul><ul><ul><li>: Height extracted in column j at period t </li></ul></ul></ul><ul><ul><ul><li>: Tonnage extracted (per day) in sector a at period t </li></ul></ul></ul><ul><ul><ul><li>: Flow of product k between node v and node p at period t (generic) </li></ul></ul></ul>Underground mining Main Variables 1 If block n of the column j is extracted at period t 0 ~
    20. 20. <ul><ul><li>Each block can be exploited just once through the horizon. </li></ul></ul>Underground mining Main Constraints <ul><ul><li>Sequence in which columns can be extracted. </li></ul></ul><ul><ul><li>Blocks must be extracted bottom-up. </li></ul></ul>
    21. 21. <ul><ul><li>Relation between blocks, columns and sectors. </li></ul></ul>Underground mining <ul><ul><li>Sector extraction capacity. </li></ul></ul><ul><ul><li>Time availability constraint (at max extraction rate). </li></ul></ul>Main Constraints
    22. 22. <ul><ul><li>Maximum allowed horizontal extraction per sector (geotechnical constraint). </li></ul></ul><ul><ul><li>Flow conservation and node/arc capacities. </li></ul></ul><ul><ul><li>Maximum level of allowed contaminants. </li></ul></ul>: products at node v Main Constraints Underground mining
    23. 23. <ul><li>Regularity in heights </li></ul><ul><li>Interaction with neighborhoods </li></ul>Columns Neighborhoods Underground mining
    24. 24. Neighborhood Underground mining
    25. 25. <ul><li>Goal: To Maximize Net Present Value </li></ul><ul><ul><li>Income: sale of products and sub-products. </li></ul></ul><ul><ul><li>Cost: Extraction, transportation and process at plants. </li></ul></ul><ul><ul><li>Investment: New projects . </li></ul></ul>Underground mining
    26. 26. Open Pit Mining Optimization Approaches to Long Term Mine Planning
    27. 27. <ul><li>Mining resource: </li></ul><ul><ul><li>Expansion: exploitable “slice” of the pit. </li></ul></ul><ul><ul><li>Bench: transversal cut, characterized by elevation and height. </li></ul></ul>expansion bench Open Pit Mining
    28. 28. <ul><li>Operational constraints  Min/Max extraction rates </li></ul><ul><li>Only resource on the surface can be exploited. </li></ul><ul><li>Safety constraints. </li></ul>Open Pit Mining ∆ h MIN Rock spillage ∆ h MAX Instability of walls
    29. 29. Deposits of very low-grade ore Crushing Grinding Leaching Dump leaching Flotation SX/EW Stock Network Flow & Design Mine Storage Stage Crushing Stage Preparation plant Plant
    30. 30. Artificial Ending Bench Artificial Starting Bench Artificial Starting Period Artificial Ending Period 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
    31. 31. <ul><li>Main decision variables </li></ul><ul><li>w ijt = 1 if benches from (i+1) to j are extracted on period t </li></ul><ul><li>z it ∈ {0,1} equals 1 if bench i is extracted in period t </li></ul><ul><li>Sets </li></ul><ul><ul><li>Or(a): artificial starting bench. </li></ul></ul><ul><ul><li>De(a): artificial ending bench. </li></ul></ul><ul><ul><li>Pini: artificial starting time period. </li></ul></ul><ul><ul><li>Pfin: artificial ending time period. </li></ul></ul><ul><ul><li>SU(i,t): successors of node (i,t). </li></ul></ul><ul><ul><li>AN(i,t): predecessors of node (i,t). </li></ul></ul><ul><ul><li>(1) Flow conservation: </li></ul></ul><ul><ul><li>(2) Each expansion can start only one extraction sequence. </li></ul></ul><ul><ul><li>(3) Each expansion can end only one extraction sequence. </li></ul></ul><ul><ul><li>(4) Relation between variables. </li></ul></ul>Network Flow Formulation
    32. 32. Open Pit Mining <ul><li>Downstream processes: </li></ul><ul><ul><li>Same as underground but with several alternatives routes. </li></ul></ul><ul><ul><li>Different types of plants: flotation, leaching, bioleaching, low-grade sulfides. </li></ul></ul><ul><ul><li>Stocking areas: large amount of material is stored for future use. </li></ul></ul><ul><ul><li>Waste dumps: material without economic value. </li></ul></ul>
    33. 33. Solving the Model <ul><li>Real instances at CODELCO lead to hard problems. </li></ul>Caro et al. - Long Term Optimization of Investment & Production Plans in Open Pit & Underground Copper Mines <ul><ul><li>Given the large amount of binary variables, especially those representing the extraction stage, solving the model using a mixed integer programming routine is nonviable. </li></ul></ul>Model Nº Constraints Nº Variables 0-1 Variables Underground mine 446,521 535,639 196,386 Open pit mine 245,391 898,742 160,386
    34. 34. <ul><li>The integrality condition is relaxed and a continuous version is solved: </li></ul>Rounding Heuristic <ul><li>Heuristic consists of fixing binary variables based on the solution of continuous version. </li></ul>Fraction of block n of column j extracted at period t Underground mines
    35. 35. <ul><li>Logic based on: </li></ul><ul><ul><li>Tonnage per column to remain constant. </li></ul></ul><ul><ul><li>Fix variables in logical order. </li></ul></ul><ul><li>This process is iterative and stops when the solution is integer. </li></ul><ul><li>Since copper grade is better for lower altitude blocks, LP is reasonably good. </li></ul><ul><li>A similar heuristic is used for open pit mine problems. </li></ul>Underground mines Rounding Heuristic
    36. 36. Applications in CODELCO Optimization Approaches to Long Term Mine Planning
    37. 37. Applications in Codelco <ul><li>The model was implemented using GAMS and was solved using CPLEX. </li></ul><ul><ul><li>System implementation at CODELCO’s Divisions (MUCH): </li></ul></ul><ul><ul><li>El Teniente. </li></ul></ul><ul><ul><li>North Division. </li></ul></ul><ul><ul><li>El Salvador. </li></ul></ul><ul><ul><li>Andina Division. </li></ul></ul><ul><li>When using our methodology, the NPV of projects have increased up to 5%. </li></ul>
    38. 38. Uncertainty in Prizes Optimization Approaches to Long Term Mine Planning
    39. 39. Problem <ul><li>Copper price introduces risk in mine planning. </li></ul><ul><li>Previous models work with deterministic prices. </li></ul><ul><ul><li>Prices represent expected values. </li></ul></ul><ul><li>Copper price is volatile and variance is not small. </li></ul>
    40. 40. Methodology <ul><li>Prize uncertainty can be addressed by: </li></ul><ul><ul><li>Extending combinatorial models. </li></ul></ul><ul><ul><li>Real Options models. </li></ul></ul><ul><ul><li>Stochastic programming. </li></ul></ul><ul><ul><li>Robust planning. </li></ul></ul><ul><li>In order to apply these techniques we need to: </li></ul><ul><ul><li>Improve solving algorithms performance. </li></ul></ul><ul><ul><li>Reduce problem size by specialized aggregations. </li></ul></ul>
    41. 41. Methodology Stock Plant Plant Plant Plant Plant Stock Plant Including uncertainty in prices generates multiple scenarios. Each scenario configures one complex combinatorial problem.
    42. 42. Robust Planning <ul><li>Robust Planning : Generate plans that will have a reasonable behavior under all possible scenarios. </li></ul><ul><li>Robust planning represents an interesting opportunity for long term mine planning. </li></ul>
    43. 43. Applying Robust Planning <ul><li>Results (MMUS$): </li></ul>Planning price Evaluation price <ul><li>High asymmetry between strategies when evaluated in the opposing price. </li></ul><ul><li>When optimistic plan (high planning price) is evaluated on low price scenario loss is lower than otherwise. </li></ul>  low high low 9,789 10,715 high 9,773 10,889   -16 -174
    44. 44. Why this asymetry? <ul><li>Asymmetry depends on marginal production costs. </li></ul><ul><li>There are 3 possible cases: </li></ul>C’(x) x B x A P A P B x B x A P A P B x B x A P A P B C’(x) C’(x) Loss for producing XA when market price is PB Profit for producing X A when market price is P A
    45. 45. 3. OPERATIONS RESEARCH IN MINING NETWORK ( ORMNet) <ul><li>Operations Research for the Mining Industry Center (University of Chile) </li></ul><ul><li>Curtin University of Technology (Australia) </li></ul><ul><li>The University of Melbourne (Australia) </li></ul><ul><li>Colorado School of Mines (U.S.A.) </li></ul><ul><li>Polytechnic School of Montreal (Canada) </li></ul><ul><li>Federal University of Minas Gerais (Brasil) </li></ul>
    46. 46. 4. VISITAS AL EXTERIOR <ul><li>Visitas a instituciones y centros extranjeros especializados en temas de investigación afines. </li></ul>
    47. 47. 5. ORGANIZACIÓN DE SEMINARIOS <ul><li>Al menos un seminario por año. </li></ul><ul><li>Invitación a exponer a investigadores externos. </li></ul>
    48. 48. 6. FINANCIAMIENTO TESIS <ul><li>Financiar alrededor de 10 tesis. </li></ul><ul><li>Apoyo del Instituto Milenio Sistemas Complejos de Ingeniería. </li></ul>
    49. 49. Centro de Investigación de Operaciones para la Industria Minera Departamento de Ingeniería de Minas y Departamento de Ingeniería Industrial Facultad de Ciencias Físicas y Matemáticas Universidad de Chile