A Mathematical Model for Practical Pushback Design

A Mathematical Model for Practical Pushback DesignOptimisation

Juan L. Yarmuch 1 Marcus Brazil2 Hyam Rubinstein3 DoreenThomas1

1Department of Mechanical Engineering, The University of Melbourne

2Department of Electrical Engineering, The University of Melbourne

3Department of Mathematics and Statistics, The University of Melbourne

November 18, 2018

DEPLAMIN Workshop “Challenges in Mine Planning”

Juan L. Yarmuch (UoM) Practical Pushback Design Optimisation November 18, 2018 1 / 16

Outline

1 Open pit design

2 What is a practical pushbacks?

3 Practical pushback optimisation

4 A small, but interesting case study

5 Conclusions


The Ultimate Pit Limit Problem (UPL)

pb : profit associated to the extraction of block b ∈ B (B set ofblocks in the block model).xb : equal to one if the block is extracted, zero otherwise.

b̂ ∈ B̂b : blocks constituting vertical precedences (upwards) for block b.

max z(x) =∑

b∈B pb · xbxb − xb̂ ≤ 0, ∀b ∈ B

xb ∈ [0, 1]

Figure 1: Geological block model. Figure 2: Ultimate pit limit contour.Juan L. Yarmuch (UoM) Practical Pushback Design Optimisation November 18, 2018 3 / 16

The Next Best Ore Problem

“There are virtually unlimited number of ways of reaching theultimate pit limit”1.

L&G introduce the parametrisation analysis, which consist on findingsmaller pits through the relaxation of the volume constrained UPL.

Figure 3: Example of the parametrisation method.

1Lerchs, H. and Grossmann, I. F. (1965). Optimum Design of Open Pit MinesJuan L. Yarmuch (UoM) Practical Pushback Design Optimisation November 18, 2018 4 / 16


Sequences obtained by using the parametrisation method (nestedpits) suffer of three major problems: The gap problem, the miningwidth problem and the connectivity problem.

Currently, mining engineers design mining pushbacks using the outputof the nested pits as a guidance, requiring a great amount ofmanual intervention in most of the cases.



Sequences obtained by using the parametrisation method (nestedpits) suffer of three major problems: The gap problem, the miningwidth problem and the connectivity problem.

Currently, mining engineers design mining pushbacks using the outputof the nested pits as a guidance, requiring a great amount ofmanual intervention in most of the cases.


Practical Pushbacks

Definitions

Practical pushbacks: are connected volumes that satisfies theminimum mining width and have a haulage ramp.

Semi-practical pushbacks: are connected volumes that satisfymininum mining width constraints.


The “art” of the Pushback Design:

Figure 4: Different practical designs from the same guidance (top view).


Our Approach

Ramps and Pushbacks

We model the practical pushback design problem as a binaryprogramming model.

Our model simultaneously optimise the ramp allocation as well as thepushback design.

Our model maximise an approximate discounted cash-flow to keepthe problem tractable.

In our model, each pushback must have a ramp that connects thebottom of the pushback to the surface.

A key idea in this formulation is to use the haulage ramp as arelative coordinate system to control the shape of the pushback.


Formulation

Variables (all binary):

xb,p: extraction of block b. 1 if block b belongs to pushback p.

yb,p: accounts for the ramp entrance that acts as a relative coordinatesystem to control the shape of the pushback.

rb,p: defines if block b belongs to the ramp of pushback p.

zs,p: represents the ramp segment which determines the ramp.

ub,b̆,p: accounts for the simultaneous extraction of a ramp entrance andthe blocks on the same level.

Obj: Max∑

p∈P δp(∑

b∈B Vb · xb,p + λp∑

b∈B∑

b̆∈B̆b

ub,b̆,pDb,b̆·)


Formulation.

Some important constraints2

Material content at each pushback.

Geotechnical constraints (slope stability).

Ramps cannot be built in the air (previously mined blocks).

Every mined block must have access to their respective ramp.

x~b,p −∑~b∈~B

∑s′∈S′

~b

zs′,p ≤ 0

Ramp continuity constraints.∑s′′∈S′′

bzs′′,p −

∑s′∈S′

bzs′,p = 0

Ramp accessibility constraints.∑ρ≤p rb,ρ −

∑ρ≤p xb+,ρ ≤ 0

2The complete formulation is in the paper: A mathematical model for the practicalpushback design problem (submitted).


Matching Factor (fm)

gi : index of the pushback assigned to block i that will be used as aguidance.

di : index of the pushback that is assigned to block i after the engineeringdesign.

mi =

{1, if gi = di ,

0, otherwise.fm =

∑i∈N mi

N

Figure 5: Guidance (topview).

Figure 6: Manual design(top view).

Figure 7: Matching factor(top view).


Case Study

Small copper deposit located in Chile.

36,000 blocks of 30x30x30 metres each.

Mining rate: 20,000 kton/year .

Processing plant capacity: 12,000 kton/year.

Minimum mining width: 60 metres.

Ramp gradient: 10%.

Discount rate (NPV): 12.5% per year.


Traditional methodology

The engineer selected 4 pit shells (over 50) as a guidance.

The engineer used Datamine Studio 3 and NPV Scheduler softwares.

Matching factor fm = 48%.

NPV = US$ 222.64 millions.


Our results

We considered 12 pushbacks with max distances between 150-400metres.

We used Vulcan for CAD design and no optimisation software forscheduling.

Matching factor fm = 89%.

NPV = US$ 235.68 millions.


Mine Schedule


Conclusions

Summary

Our model shows to be capable of capturing the operationalconstraints of the open pit mining operations.

We avoid the gap problem and we can control the mining width.

Our model produce better guidelines for practical pushback designthan the traditional methodology.

In the case study, we increased the matching factor by 85% andthe NPV of the project by 5.4%.


Documents

A Mathematical Model for Practical Pushback Design