An SIP Formulation for Production Scheduling

and Application at a Gold Mine:

An Industry Example of the Value of Stochastic Solutions

SIP – 2, Fall 2013

Contents

• Introduction

• Stochastic integer programming (SIP) model

• Risk management using the SIP model

• Case studies

• Value of the stochastic solution

• Conclusions

Introduction

• The traditional “optimum scheduling methods” are based on

mathematical models with inputs of 100% certainty.

• Uncertainty may exist from technical, environmental and

market sources. Grade variability is examined in this

presentation.

• A recently developed Stochastic Integer Programming (SIP)

model uses multiple simulated orebody models in optimising

long-term production schedules in open pit mines.

• BUT, let’s provide some background

Surface

Z

X

Ore

Body

Surface

Drill

Holes

Ore

Body

Ore body

High grade

Low Grade and waste

Drill holes

Production Scheduling - Open Pit Mine

Ultimate

Pit Pit3 Pit 2 Pit 1 Surface

bench 8

bench 6

Z bench 3

bench 1

X

Final Slope

Ore

Body

Ultimate

Pit Pit3 Pit 2 Pit 1 Surface

bench 6

Z bench 3

bench 1

X

Final Slope

Ore

Body

Production Scheduling - Open Pit Mine

Production Sequence

Ultimate Pit

>10.0

>5.00

>1.50

>0.60

>0.35

>0.01

Grade legend

5 % ?

0.01 % 10 %

5 %

?

Representation of uncertainty

Period DCF

(m$)

Ore

(Mt)

Waste

(Mt)

Gold

(Mgr)

1 14.2 1.0 0.36 1.78

2 19.0 2.0 1.4 2.9

3 22.2 3.0 2.6 4.1

An Open Pit Gold Deposit

Lode 1502

Simulation #1

Quantification of Uncertainty about a Gold Deposit

Monte Carlo Simulations

A Formula for a Block Value (Blocks Representing the Deposit) when Optimizing

BLOCK ECONOMIC VALUE =

(METAL*RECOVERY*PRICE - ORE*COSTP)

- ROCK*COSTM

• The objective function: The main objective of the long term production scheduling is to maximize net present value (NPV) of the mine.

Maximize

where

p is the maximum number of scheduling periods

n is the total number of blocks to be scheduled

Cit is the NPV to be generated by mining block i in period t

Xit is a binary variable, equal to 1 if the block i is to be mined in period t, 0 otherwise.

p

1t

n

1i

t

i

t

i X*C c4

c1 c2 c3

Orebody model

Subject to the following constraints

• Grade blending constraints

Upper bound constraints: The average grade of the

material sent to the mill has to be less than or equal to

a certain grade value, Gmax, for each period, t

where

gi is the average grade of block i

Oi is the ore tonnage in block i

0X*O*)Gmax-g(n

1i

t

iii

• Reserve constraints

• Processing capacity constraints

Upper bound: The total tonnage of ore processed cannot

be more than the processing capacity (PCmax) in any

period, t

Lower bound: The total tonnage of ore processed cannot

be less than a certain amount (PCmin) in any period, t

1Xp

1t

t

i

n

1i

maxt

ii PC)X*(O

n

tmini i

i 1

(O *X ) PC

Note:

ALL DETAILS ARE IN THE PAPERS

provided for this lecture

Particularly:

Ramazan and Dimitrakopoulos,

Optimization in Engineering, 2013

Integer Programming

An objective function

Maximize (c1x11+c2x2

1+…. ) …

Subject to

c1x11+c2x2

1+…. = b1

c1x1p+c2x2

p+…. = bp

c4

c1 c2 c3

Period 1

Period p

Orebody model

c = constant

X11 = binary variable

Models for Optimisation

The objective function now is …..

Maximise (s11x11+s21x2

1+….

s12x11+s22x2

1+….) …

Subject to

s11x11+s21x2

1+…. = b1

s11x1p+s21x2

p+…. = b1

s12x1p+s22x2

p+…. = b1

s1rx1p+s2rx2

p+…. = b1

Stochastic Integer Programming (SIP)

Simulated model 1

Simulated model 2

Simulated model r

Period 1

Period p

s41

s11

s21

s31

s41

s11

s21

s31

s41

s11

s21

s31

s41

s1n

s2n

s3n

Stochastic Integer Programming (SIP)

Account for uncertain inputs

Consider simulated grade realizations in the optimization

process

Minimize the risk of not meeting production targets

caused by geological variability

Objective function

SIP - Production Scheduling Model

Part 1

Part 2

Part 3

Part 4

Ut t t

*i iii 1

- E{(NPV) }MC s

+

Mtt

s ss 1

+ (SV) (P) q

P Nt t

iit 1 i 1

Max [ E{(NPV) } b

Mty ty tyty

su l slus 1

- )](c d c d

+

Mill & dump

Stockpile input

Stockpile output

Risk management

Note:

ALL DETAILS ARE IN THE PAPERS

provided for this lecture

Particularly:

Ramazan and Dimitrakopoulos,

Optimization in Engineering, 2013

Objective Function - Deviations

Rtyty ty ty

Minimise........ ruu l rlr 1

)].....(c dd c +

Part 2

Deviation from production targets ctyu and cty

l penalized by dtyru and dty

rl

for each simulation r

Stochastic Integer Programming - SIP

……

Ore

Grade 1

Metal

…

Orebody

Model 1

A production

schedule

Orebody

Model 2

Orebody

Model R

Ore

Grade 2

Metal

…

Ore

Grade R

Metal

…

- TARGET [ ]

- TARGET [ ]

- TARGET [ ]

Deviation 1

Deviation 2

Deviation R

1 2 3

4

SIP – Geological ‘Discount Rate’

Risk Management

penalized by ctyu and cty

l respectively for each simulation s

ty

uty

ud

cc

+

1

0

ty

lty

ld

cc

+

1

0

Deviations from production targets by dtysu and dty

sl are

Shortage in ore

production

Excess in ore

production

Rtyty ty ty

Minimise........ ruu l rlr 1

)].....(c dd c +

SIP – Penalties

Risk Management

d = geological ‘discount rate’

P M

ty ty ty ty

u su l sl

t 1 s 1

c d c d

+

ty

uty

ud

cc

+

1

0

P N

t t

iit 1 i 1

NPV b

Total NPV

r = economic discount rate

t

it

ir)1(

EVNPV

0

+

SIP – A Stochastic Definition of Ore

wasteis iblock ; if ,

ore is iblock ; if , VE i

iiii

iiiii

PCNRPCMC

PCNRPCMCNR

)( cost SellingPrice recGTNR iii

A probability cut-off (p) is also utilized to classify a block as ore

wasteis iblock else,

ore is iblock ,Prob if pgG offcuti

Cl3t > (Cl1

t = Cu1t) > (Cl2

t = Cu2t) > Cu3

To

nn

es

(m

illio

n)

Schedules

0.7

1.6

1.3

1.0

1 2 3

Managing Risk within a Given Period

Ore Production

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4

0

1 2 3

1

2

3

Meta

l q

uan

tity

(10

00 K

g)

Periods

Ct=Ct-1 * RDFt-1 RDFt=1/(1+r)t

r – orebody risk discount rate

Managing Risk Between Periods

Deviations from metal production target

RDF – risk discounting factor

Total blocks 22,296

Block dimensions (m) 20 x 20 x 20

Processing input capacity (PC) 18 Mtpa

Metal production capacity (MC) 28,000 Kg pa

Total mining capacity (TC) 85 Mtpa

Stockpile capacity (SC) 5 Mt

Stockpile re-handling cost 0.6 $/t

Discount rate 10 %

Mine Life 6 yrs

General information

Case Study on a Large Gold Mine

Orebody risk discounting rate 20 %

Cost of shortage in ore production 10,000 /t

Cost of excess ore production 1,000 /t

Cost of shortage in metal production 20 /gr

Cost of excess metal production 20 /gr

Number of simulated orebody models 15

The SIP specific information

Case Study on a Large Gold Mine

Periods 1 - 4 4 - 6

Total blocks 11,301 10,995

Constraints 33,273 21,363

Total variables

Binary:

53,301

18,540

37,286

9,580

T Time (hr:min:sec) <04:49:55 <37:15:33

The SIP Model Information

Supercomputing system used with parallel processors 8 in 2002

Cross-Sectional Views of the Schedules

SIP Whittle Four-X

1

2

3

4

5

6

Periods

0

- 10

- 8

- 6

- 4

- 2 To

nn

es (

mill

ion

)

Deviations from Production Targets

Ore Production

1 2 3 4 5 6

SIP model

WFX

Periods

Deviations from Production Targets

1 2 3 4 5 6

Periods

0

- 4

Me

tal q

ua

ntity

(1000 K

g)

- 8

- 12

- 16

- 20

Metal Production

SIP model

WFX

0

To

nn

es

(m

illio

n)

1 2 3 4 5 6

2

4

6 Stockpile’s Profile

Available ore at

the end of each

period

1 2 3 4 5 6 Periods

0

To

nn

es

(m

illio

n)

1

2

3 Ore taken out

from the

stockpile

4

5

SIP model

WFX

1 2 3 4 5 6 Periods

0

200 $ (

mill

ion)

400

600

800

1000

SIP model WFX

Cumulative NPV values

SIP model WFX

Average NPV values

Uncertainty is Good: Traditional vs Risk-Based

Stochastic Integer Programming

$723 M

Risk Based

$609 M

Traditional

Difference = 17%

Geological Risk

Discounting= 20%

Value of SIP Solution (VSS)

ESPI = Expected Solution of Perfect Information

15 Scenarios, 15 schedules = average NPV (a ‘theoretical NPV’ value which one to use?)

EVS = Expected Value Solution

15 Scenarios, Expected value scenario, 1 Schedule tested

with 10 Scenarios = NPV

ESS = Expected Stochastic Solution

14 Scenarios, 1 Schedule tested with 14 Scenarios = NPV

Value of SIP Solution (VSS)

EVPI = Expected Value of Perfect Information

EVPI = ESPI - EVS

VSS = Value of Stochastic Programming or

Solution (VSS)

VSS = ESS – EVS ≥ 0

COST of IGNORING Uncertainty

Value of SIP Solution (VSS)

ESS = Expected Stochastic Solution = 723 million $

VSS = Value of Stochastic Programming or Solution

VSP = ESS – EVS = 64 million $ 10%

COST of IGNORING Uncertainty

EVS = Expected Value Solution = 659 million $

Note:

ALL DETAILS ARE IN THE PAPERS

provided for this lecture

Particularly:

Dimitrakopoulos and Ramazan,

Mining Technology, 2008

Some Key Comments

The new SIP production scheduling model:

• Uses individual realisations, thus explicitly accounts for

geological risk

• Allows the risk management at three levels:

1. manage the magnitude of risk within a period

2. manage the variability of risk

3. control the risk distribution between time periods

• Maximises NPV for a desired risk profile

• The SIP is efficient: Contains less binary variables than

traditional MIP models

A Second Case Study

Disseminated low-grade copper deposit

Orebody dips mainly N180/60S

185 DH in a pseudo-regular grid of 50x50m2

Mineralized envelop defined using the drill core logs

Direct block simulation

20 simulations, directly generated on a 20x20x10m3 mining

block size

Generates equally probable scenarios of the deposit

Stochastic Simulations

1

2

…

n

Parameters for the SIP

Total blocks 15,391

Block dimensions (m) 20 x 20 x 10

Processing input capacity (PC) 7.5 Mtpa

Total mining capacity (TC) 28 Mtpa

Economic discount rate 10 %

Cost of shortage in ore production 10,000 /t

Cost of excess ore production 1,000 /t

Cut-off 0.3% Cu

Number of simulated orebody models 20

Ore Production Risk Profile

Waste Production Risk Profiles

2

3

4

5

6

7

8

9

0 2 4 6 8 10

Period

Ore

M x

to

nn

es

Maximum

Minimum

Expected

Conventional schedule

Risk Analysis - Conventional Schedule

The Value the SIP Solution

29%

The Role of Geological Discount Rate

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8

Ore

(to

nn

e)

Production period (year)

30% geological discount rate

min

average

max

Cross-sectional Views of Schedules

20% geological discount rate

30% geological discount rate

Value of SIP Solution (VSS)

ESS = Expected Stochastic Solution = 298 million $

VSS = Value of Stochastic Programming or Solution

VSS = ESS – EVS = 60 million $ or 25%

COST of IGNORING Uncertainty

EVS = Expected Value Solution = 238 million $

Note:

The End

(of this lecture only)