McGill Consulting Asif Kan Alexandre Marinho de Almeida Michael Spleit Ahmed Ragab

1. McGill ConsultingAsif KanAlexandre Marinho de AlmeidaMichael SpleitAhmed Ragab Mine Production Scheduling of a Porphyry copper deposit February 15, 2012

2. Outline • Problem overview • Objective function • Constraints to be handled • Required input data and specifications • Deliverables • Simplified prototype • Possible extensions

3. Problem overview Copper deposit Peru South America Porphyry Copper (0.4 – 1% Cu)

4. Problem overview Copper deposit A volume representation of the deposit Block model The deposit is represented discretely as 3D blocks 20m 20m Cu% 10m

5. Problem overview Aerial view of the deposit Zone 1 Zone 2 (Example) Zone 3 Zone 4

6. Problem overview Decision to mine Decision of destination Constrained Capacity Grade

7. Problem overview The average cycle time from each zone to each of 3 possible destinations is known. (12 combinations) Cycle time from Zone 1 To Mill Cycle time from Zone 1 To Dump Cycle time from Zone 1 To Leach Mill Leach Dump

8. Preprocessing Without knowing if a given block will be mined, we can preprocess its destination. Waste If Cu% < Ore cut-off grade Dump Ore If Cu% in Mill grade range Mill If Cu% < Mill minimum but in Leach grade range Leach Cu%

9. Preprocessing Pair • There are twelve cycle times (each pair makes a different cycle). • The total operating cost will be different for each of these cycles, and should also be provided by the managers.

10. Preprocessing Known for each block: Source (Zone) Destination Distance/cycle time Operating cost Block volume Density tonnes/m3 (to be provided by managers) Recovery by destination Selling price (to be provided by managers) Therefore revenue can be preprocessed Therefore the net value for each block can be calculated: Net Value ($) = Revenue ($) – operating cost ($)

11. Objective Select which blocks should be mined when. Make selection for 4 periods (4 quarters = 1 year). Goal is to make this selection in order to maximize the total money earned (NPV), while also being subject to several constraints. Start (un-mined) Q1 Q2 Q4 Q3

12. Objective For each block Given (input data) Index i, j, k for location of the block Copper grade Zone Decision Do we mine? In what period [1 to 4]? 20m 20m 10m

13. Objective Function Maximize the Net Present Value (NPV) ( ) • Where • is the Net Present Value of block i if mined in period t (discounting) • is a binary variable, it is 1 if block i is mined in period t and 0 • otherwise • Penalties are the unit cost for violating the upper and lower limits of • processing and grade requirement in each period.

14. Constraints • Types of constraint • Reserve • Forbidden blocks • Slope • Haulage capacity • Processing capacity • Grade blending

15. Constraints • Reserve constraints • A block cannot be mined more than once! <= Hard Constraint : must be achieved

16. Constraints Forbidden Blocks Some blocks may be marked forbidden. Each such block will have a constraint that it can not be mined. Hard Constraint : must be achieved

17. Constraints Slope constraints 45o Mining slope constraint due to rock stability. Each block can only be mined if the block above it and the other four blocks adjacent to that upper block are mined. Hard Constraint : must be achieved

18. Constraints • Haulage capacity constraints • The total amount of material (waste and ore) to be mined cannot be more than the total available equipment capacity for each period Hard Constraint : must be achieved

19. Constraints Haulage capacity constraints • Based on the grade of each block its destination is known • Based on the block tonnage, zone and destination, and truck capacity, we know the total time required to mine each block where t = 1,2,3,4 • Where S is the cycle time required for any given block based on the zone pair cycle time and the truck haulage capacity S = (Block Tonnage) / (Truck capacity) * (Zone Pair Cycle time) • bit is a binary variable representing whether the block is mined in period t

20. Constraints • Processing capacity constraints • Upper bound constraints • For each of Mill and Leach: • Total tonnage of ore processed cannot be more than the maximum processing capacity for that period • Lower bound constraints • For each of Mill and Leach: • Total tonnage of ore processed cannot be less than the minimum processing capacity for that period Soft Constraint : best attempt made

21. Constraints • Grade blending constraints • Upper bound constraints • For each of Mill and Leach: • Average grade of material sent to the mill has to be less than or equal to a maximum grade • Lower bound constraints • For each of Mill and Leach: • Average grade of material sent to the mill has to be greater than or equal to a minimum grade MAX: 1.2% 1.4% 1.3% 0.5% 1.07% 0.7% 0.9% 0.7% 0.77% MIN: 0.8% Soft Constraint : best attempt made

22. Data required A block model containing, for each block: integer indexes i j k; Cu grade; Zone; A binary value, equal to 1 if the block is available for mining (inside pit limits) and 0 otherwise; 12 average cycle times (hours) Number of trucks available for each quarter Number of productive hours each truck has per day Number of days per quarter Loading capacity for trucks of the fleet (tonnes) Mill and leach pad metal recoveries (g Cu / tonne ore) Mill and leach pad upper and lower production capacity limits (tonnes) Penalty cost for each tonnage above the upper or below the lower limits, for mill and leach capacities ($ / tonne) Average grade upper and lower limits, with respective penalties ($ / delta%) Quarterly discounting rate (%) Selling price of Cu ($/g) Total operating cost for each of 12 cycles ($/tonne) Specific gravity of copper ore (tonnes / m3)

23. Deliverables • The ultimate deliverables will consist of a four-period mine plan where : • Each block is assigned to a destination (Mill, Leach, Dump). • Each block is assigned to a period of 1 to 4 (or 0 if un-mined). • The total revenue and NPV of the solution will be provided. • The results will be summarized in tables and/or graphs.

24. Deliverables

25. Deliverables The total metal recovered from each zone and each quarter. Truck fleet usage for each quarter.

26. Deliverables Penalties

27. Prototype problem • Our prototype problem will consist of maximizing the NPV of 4 quarters • while only considering a reduced # of blocks and the following constraints: • Blocks can only be mined once • Slope constraints must be observed • Haulage capacity

28. Reasonable-sized problem • If and when we are able to solve the Prototype, we can add the other constraints: • Processing capacity • Forbidden blocks cannot be mined • Grade blending • If this is working, we consider a greater number of blocks (ideally the entire block model)

29. Extensions • Determine optimum fleet size. • Consider stochastic optimization taking into account grade • and market uncertainty. • Consider stockpile management. • More complex costing information (fixed and variable). • Calculate cycle times at the block level instead of zone level.

30. Asif KanAlexandre Marinho de AlmeidaMichael SpleitAhmed Ragab Thank you. Questions?