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

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

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Mine Production Scheduling

of a Porphyry copper deposit

February 15, 2012

- Problem overview
- Objective function
- Constraints to be handled
- Required input data and specifications
- Deliverables
- Simplified prototype
- Possible extensions

Problem overview

Copper deposit

Peru

South America

Porphyry Copper (0.4 – 1% Cu)

Problem overview

Copper deposit

A volume representation

of the deposit

Block model

The deposit is represented discretely

as 3D blocks

20m

20m

Cu%

10m

Problem overview

Aerial view of the deposit

Zone 1

Zone 2

(Example)

Zone 3

Zone 4

Problem overview

Decision to mine

Decision of destination

Constrained

Capacity

Grade

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

Preprocessing

Without knowing if a given block will be mined, we can preprocess its destination.

Waste

If Cu% < Ore cut-off gradeDump

Ore

If Cu% in Mill grade range Mill

If Cu% < Mill minimum but

in Leach grade rangeLeach

Cu%

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.

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 ($)

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

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

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.

Constraints

- Types of constraint
- Reserve
- Forbidden blocks
- Slope
- Haulage capacity
- Processing capacity
- Grade blending

Constraints

- Reserve constraints
- A block cannot be mined more than once!

<=

Hard Constraint : must be achieved

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

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

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

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

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

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

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)

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.

Deliverables

Deliverables

The total metal recovered from each zone and each quarter.

Truck fleet usage for each quarter.

Deliverables

Penalties

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

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)

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.

Thank you.

Questions?