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CUT-OFF GRADE OPTIMIZATION OF OPEN PIT MINES WITH MULTIPLE PROCESSING STREAMS
In this study, dynamic cut-off grades and multiple processing streams are used to
maximize the value of a mining project based on a finite resource. Optimal cut-off
policies are generated using Lane’s method for determining cut-off grade. By
maximizing the present value of future profits as a function of cut-off grade, mine project
value is increased over the traditional break-even approach. A method for determining
multiple cut-off grades at a single deposit was applied to analyze the impact that
changes in processing capacity have on NPV. It was found that additional capacity
related to a separate mill facility resulted in an economic reclassification of ore and
waste. Grade tonnage data used in the case study was simulated to represent the
geologic uncertainty associated to low-grade mineral deposits. Results from the
hypothetical case study examined in this thesis reveal that a low-grade open pit gold
mine will benefit from the use of multiple processing streams when a dynamic cut-off
policy is applied. Particularly, when incorporating a “high grade” modular processing
stream to maximize the potential revenue of the mineralized material. This means that
for a given set of design, production and geological parameters, the classification of ore
and waste is what ultimately determines the NPV of a mining project.
iii
LAY SUMMARY
The purpose of this research was to identify the effects that multiple mineral processing
streams have on the overall value of a mining project. By incorporating more than one
processing facility into a mine plan, the classification of ore and waste at a gold deposit
was improved. This allowed for each processing facility to process ore best suited for
that particular stream, based on the concentration of gold in the ore. Ultimately, by
optimizing the grade of ore sent to each facility, mine value was increased over a
project with a standalone stream.
iv
PREFACE
This thesis is original, unpublished, independent work by the author. The Algorithms
used are based on dynamic cut-off theory proposed by Lane (1) and work done by Asad
and Dimitrakopoulos (2).
v
TABLE OF CONTENTS
ABSTRACT ...................................................................................................................... iiLAY SUMMARY ............................................................................................................... iiiPREFACE ....................................................................................................................... ivTABLE OF CONTENTS ................................................................................................... vLIST OF TABLES ............................................................................................................ viiLIST OF FIGURES ......................................................................................................... viiiLIST OF SYMBOLS ........................................................................................................ ixDEDICATION .................................................................................................................. xii1 INTRODUCTION ........................................................................................................ 1
1.1 THESIS ORGANIZATION ............................................................................................... 31.2 IMPORTANCE TO INDUSTRY ....................................................................................... 41.3 RESEARCH OBJECTIVES ............................................................................................. 5
5 THE MODEL ............................................................................................................. 445.1 ECONOMIC AND OPERATIONAL INPUTS ................................................................. 45
5.2 INITIAL ESTIMATES OF NPV ....................................................................................... 475.3 LIMITING CUT-OFF GRADE ........................................................................................ 485.4 QUANTITY OF ORE, WASTE AND AVERAGE GRADE .............................................. 485.5 QUANTITY MINED, PROCESSED AND REFINED ...................................................... 495.6 ANNUAL CASH FLOW AND NPV ................................................................................. 51
6 HYPOTHETICAL CASE STUDY .............................................................................. 536.1 MODEL ASSUMPTIONS AND LIMITATIONS .............................................................. 536.2 METHOD ....................................................................................................................... 56
6.2.1 BASE CASE ........................................................................................................... 576.2.2 MODULAR CASE ................................................................................................... 59
8 CONCLUSIONS ....................................................................................................... 689 RECOMMENDATIONS ............................................................................................ 70REFERENCES ............................................................................................................... 72APPENDIX A. Derivation of Lane’s Equations .............................................................. 74APPENDIX B. Simulated Grade Tonnage Data ............................................................ 77APPENDIX C. Cut-off Policy Results ............................................................................ 83APPENDIX D. Sensitivity Analysis ................................................................................ 89
vii
LIST OF TABLES
Table 1 Estimated unit operating costs for the Gekko Python modular processing plant,
from (22). ................................................................................................................ 36Table 2 Mine design parameters for hypothetical gold mine. ......................................... 55Table 3 The complete cut-off policy for base case using grade tonnage curve 1 (GT1).
................................................................................................................................ 58Table 4 Calculated NPVs for both the base and modular scenarios across the set of 15
equally probable simulated grade tonnage curves. ................................................ 58Table 5 Complete cut-off policy for modular case using GT1. ....................................... 60Table 6 Base case cut-off policy when HL has capacity of 573,000 t/yr, for GT1. ......... 60Table 7 Complete break-even cut-off policy for modular case using GT1. .................... 61Table 8 Comparison of annual gold production across set of simulated grade tonnage
curves. .................................................................................................................... 64Table 9 Cut-off policy for GT13 showing how an increase in HL unit costs by 5% results
in an increase in HL COG and a decrease in CIL COG. ......................................... 66Table 10. Cut-off policy for base case, GT13. ................................................................ 66Table 11. Cut-off policy for GT13 when HL capacity is increased by +15% of base case
Figure 3.1 Graphical representation of the break-even relationship between costs and
revenue similar to (5). ............................................................................................. 16Figure 3.2 Graphical representation of balancing cut-off when the mine and the mill are
limiting. .................................................................................................................... 24Figure 3.3 Increment in present value versus cut-off grade with the mine and mill
components in balance. .......................................................................................... 26Figure 3.4 Increment in present value versus cut-off highlighting the maximum value in
the feasible region of ve when the mine and the mill are in balance. ..................... 27Figure 3.5 Simulated tonnage histogram of gold deposit. .............................................. 31Figure 3.6 Sample grade tonnage curve with cut-off grade of 4.0 g/t. ........................... 32Figure 4.1 Process flow diagram for the Gecko Python Plant, from (22). ...................... 34Figure 4.2 Cut-off and cutover grade defined by revenue earned per ton of material
processed. .............................................................................................................. 41Figure 6.1 Flow sheet diagram for hypothetical gold mine with capacity constraints
(modular case). ....................................................................................................... 54Figure 6.2 Grade tonnage distribution GT1 for hypothetical gold mine. ......................... 56
ix
LIST OF SYMBOLS
𝑃 [$] Annual profit.
𝑡 [yr] Time.
𝑇 [yrs] Time (life of project).
𝑄𝑚 [t/yr] Quantity mines.
𝑄𝑐 [t/yr] Quantity processed.
𝑄𝑟 [oz/yr] Quantity refined.
𝑀 [t/yr] Maximum annual mining capacity.
𝐶 [t/yr] Maximum annual processing capacity.
𝑅 [oz/yr] Maximum annual marketing/refining capacity.
ℎ [$/t] Rehabilitation unit cost.
𝑐 [$/t] Processing unit cost.
𝑦 [%] Average annual recovery from processing.
𝑔 [g/t] Average mineral grade.
𝑄 [tons] Total resource remaining.
𝑞 [t/yr] Rate of extraction.
𝑉 [$] Present value.
𝑝 [$/t] Cash flow from one unit of resource.
𝛿 [%] Discount rate.
𝐹 [$/-] Opportunity cost.
𝑆 [$/oz] Selling price of gold.
𝑟 [$/oz] Market/refining unit cost.
𝑥 [%] Ore to mineralized material ratio.
𝑚 [$/t] Mining unit cost.
𝑓 [$/yr] Fixed annual cost.
𝑔 [g/t] Mineral grade.
𝑔! [g/t] Mine limiting economic cut-off grade.
𝑔! [g/t] Process limiting economic cut-off grade.
𝑔! [g/t] Market/refining limiting cut-off grade.
x
𝑔!" [g/t] Mine and process balancing cut-off grade.
𝑔!" [g/t] Processing and refining balancing cut-off grade.
𝑔!" [g/t] Mine and refining balancing cut-off grade.
𝑣 [$/-] Increment in PV per unit of resource utilized.
𝑣! [$/-] Max of the min increment in PV per unit of resource utilized.
𝑊 [$] Present value one year in the future at t=t+1.
𝑇! [tons] Quantity of tons in each grade category ‘n’.
𝑇𝑂 [tons] Quantity of ore tons above cut-off grade.
𝑇𝑊 [tons] Quantity of waste tons below cut-off grade.
∆ [-] Difference between TO/Tn for each process.
𝐺𝑇! [-] Grade tonnage data.
xi
LIST OF ABBREVIATIONS
CIC Carbon in column
CIL Carbon in leach
COG Cut-off grade
HL Heap leach
HLF Heap leach facility
LOM Life of mine
NLP Non-linear programming
NPV Net present value
PV Present value
ROM Run of mine
xii
DEDICATION
This thesis is dedicated to Christine A. Pettingell. Thank you for your continuous
support.
1
1 INTRODUCTION
Current trends in the gold mining industry show that weak commodity prices and an
overall decline in metal grades have resulted in less gold being mined (3). There has
also been less investment dedicated to exploration in recent years resulting in a smaller
inventory of new deposits (4). Although the majority of exploration dollars is spent in
remote areas of Latin America and other underdeveloped countries, the lack of
infrastructure has made these ore bodies increasingly challenging to mine. The question
then becomes how to mine these remote, low-grade deposits economically? Barring
any new technological breakthroughs this must be done strategically with capital cost
reduction and operational excellence.
One solution is to incorporate a modular processing stream and a dynamic cut-off grade
strategy to capture the full value of the resource being mined. By utilizing multiple
processing streams the mineralized body can be further classified into zones based on
the geology and/or mineral content that is best suited to a particular processing stream.
Modular processing provides flexibility to the mine operator to route the mined material
to the most economic recovery method no matter how small or remote the zone or
mine. A dynamic cut-off strategy refers to what material should be mined based on the
current mine design, the local geology and prevailing market conditions. This is
fundamental to maximizing the present value of the mining asset.
The simplest definition of cut-off grade (COG) is the amount of metal concentration in
2
the mineralized material that determines what is considered ore versus what is
considered waste (1). Mine operators use cut-off grades to optimize mine designs,
estimate resources and guide production. As cut-off grades decrease more material is
classified as ore and production rates increase. Consequently, the average grade of the
ore being processed decreases. The opposite occurs when cut-off grades increase. An
optimum dynamic cut-off strategy will change the classification of ore and waste
throughout the life of the project to maximize the present value of all future profits.
Factors such as, grade distribution, available capacities and variable costs for mining,
milling and refining as well as the selling price for the commodity(s) all play a crucial role
in determining optimum cut-off grades.
Modular processing technology is ideal for processing material in remote areas, which
lack the infrastructure necessary to construct and operate a conventional mill. Modular
processing plants are designed to be small and flexible, dividing the components of the
mill into separate subsystems that can be added or removed based on the needs of the
operator for the material being mined. The smaller size of the unit over its traditional
counterpart allows some modular systems to fit underground in a drive or drift, the idea
being that underground processing would lower haulage costs and increase hoisting
capacity while reducing energy consumption and in turn the environmental footprint.
There are several manufacturers that offer modular processing solutions ranging from
small skid mounted units that can process 10-20 t/hr, to larger units capable of
processing over 50tph. There is also a wide range of useful applications for these
3
plants, from gravity concentration to flotation of several different commodities in virtually
any location.
To aid in the exploitation of low-grade precious metal deposits multiple mineral recovery
process streams can be used. By implementing multiple processing streams the miner
has the ability to choose the best processing method for a particular ore type at any
given time. This strategy has been used at mining operations where the ore is classified
into different categories based on lithology or mineralization. This is common when the
deposit is composed of both sulfide and oxide ore.
The benefits of utilizing multiple processing streams combined with modular processing
technology include, but are not limited to, reduced blending requirements, additional
processing capacity, mitigation of geologic uncertainty and the ability to push revenue
forward from higher-grade material. However, the greatest benefits arise from the
options available at the time a particular area is mined. With the capability to send
material to a low or a high recovery stream over just an ore or waste pile, the miner can
have greater influence on the economics of the project through a more refined cut-off
strategy.
1.1 THESIS ORGANIZATION
The purpose of this research was to identify the effects that multiple processing streams
have on cut-off grade policy and how cut-off strategy influences overall project value,
particularly when a modular processing plant is introduced to an open pit gold mine with
4
an existing heap leach facility.
The cut-off analysis conducted in this thesis is based on Lane’s (1) method for
optimizing cut-off grades by maximizing the present value of future cash flows
generated in a specified time period. Section 2 is comprised of a literature and
theoretical review outlining past studies on cut-off grade strategy. In section 3, the
primary methods for determining cut-off grades are discussed in detail. The concept of
modular processing and the use of multiple processing streams are examined in section
4. Section 5 contains the description and steps to the algorithm used to maximize the
value of a mine through dynamic cut-off grade optimization. In section 6, the model is
applied to a hypothetical small-scale, open-pit gold mine. Section 7 presents a
sensitivity analysis on the results obtained in section 5. In the remaining sections, 8 and
9, conclusions and recommendations of this research are presented.
1.2 IMPORTANCE TO INDUSTRY
The optimization of cut-off grades in mine planning and design is of practical and
theoretical interest. Although it is well known that a dynamic cut-off strategy can
improve the net present value (NPV) of a mining project over a break-even cut-off
model, many mining companies refrain from incorporating a robust cut-off analysis in
their valuations and long-term plans. Hall (5) suggests that junior engineers or
geologists are often determining cut-off grades based on past practices. More
importantly, the way cut-offs are determined has become indistinguishable from what a
cut-off grade is and break-even has become synonymous by default.
5
The model presented here applies Lane’s cut-off theory focusing on modular processing
technology as a secondary processing stream, which has the flexibility to be easily
expanded or contracted depending on current geologic or market conditions. A strategy
of this kind can be applied to low-grade, open pit precious metal deposits where
environmental and/or land area constraints prohibit the construction or expansion of
conventional process facilities.
1.3 RESEARCH OBJECTIVES
The objective of this research was to maximize the value of small-scale, low-grade,
open pit homogenous gold deposits using multiple processing streams and optimum
dynamic cut-off grade strategy. To accomplish this, two processing streams, a heap
leach facility (HLF) and a modular carbon in leach plant (CIL) were incorporated
simultaneously to fully exploit a set of simulated grade tonnage curves. An optimum cut-
off policy was determined and mineralized material was routed to a waste pile, a high-
grade stream or low-grade stream depending on which combination maximized the
present value of the resource.
Specific research objectives include:
1. Apply Lane’s methods for determining cut-off grade to maximize the NPV of a
gold mine when one processing stream is utilized.
2. Apply Lane’s methods to a gold mine when two processing streams are utilized.
3. Determine if Lane’s methods improve the NPV of a mine over the traditional
break-even approach.
6
2 LIERATURE REVIEW
Cut-off grade research has been an important topic for mine planners in industry and
academia ever since K. Lane published his seminal work in 1964 and subsequent text
book in 1988. The following section provides a general review of cut-off theory and
highlights the research aimed at improving the optimization of cut-off grades.
2.1 LANE’S METHOD
Kenneth Lane was the first to consider cut-off grade as a dynamic value that must be
optimized to maximize the value generated from a mine. Instead of maximizing profits,
he sought to maximize the present value of the resource as a whole by considering the
opportunity cost associated to mining at a particular cut-off. In his book “The Economic
Definition of Ore” published in 1988, he explains that due to the time value of money,
processing lower grade ore today reduces the potential future value of higher-grade ore
if processed at a later date. Therefore, by mining at higher cut-off grades early in the life
of the project the opportunity cost is minimized and NPV is increased.
Lane also considers the mining system to be comprised of three main limiting
components, the mine, the processing facility and the market. By his theory, at any
given time the system will be constrained by one or more of the limiting factors. In order
to derive optimal cut-off grades that maximize the NPV of the project, the capacities to
the limiting components must be considered and the overall system balanced.
7
Mathematically, the objective function is represented as,
max NPV =Pt
1+ d( )tt
T
∑
subject to
Qmt ≤ M
Qct ≤C
Qrt ≤ R
where P is cash flow, d is the discount rate, t is time, Qm is the quantity of tons
mined, Qc , ore tons processed, and Qr is the ounces refined. The variables M , C and
R represent the maximum periodic capacities for the mine, the mill and refinery,
respectively.
The initial work done by Lane was based on the assumptions that there was one source
of material feeding one treatment plant. He also assumed that the ultimate pit limit had
been determined and a mine schedule planned. Furthermore, he used static prices and
costs for his economic inputs. These assumptions and limiting parameters often fail to
capture the real world complexity related to valuing actual mines in practice.
Consequently, many extensions to Lane’s work have been published aimed at
improving many of the shortcomings.
8
2.2 EXTENSIONS/MODIFICATIONS TO LANE’S METHOD
2.2.1 INCORPORATING REHABILITATION COSTS
J. Gholamnejad (6) identified that the costs associated with mining and dumping waste
represent a portion of the rehabilitation costs, and therefore must be included in the cut-
off calculation. This allows the mine planner to strategically account for not only the
returns that ore provides but also the costs that are incurred from waste. For
rehabilitation costs to be factored into Lane’s original algorithm, Gholamnejad used a
rehabilitation cost variable ' h ’ subtracted from the unit processing cost ‘ c ’ in the
numerator of the limiting economic cut-off calculation (section 3.2.2).
Incorporating rehabilitation costs can provide a more accurate estimate of the profits
obtained through a cut-off grade policy. Results suggest cut off grades determined
using a rehabilitation factor will be lower than otherwise in an effort to reduce the total
amount of waste rock sent to the waste dump (6). By including these costs into the
determination of an optimum cut-off grade Gholamnejad observed an increase in NPV
over the traditional method introduced by Lane.
2.2.2 OPTIMIZATION FACTOR ON OPPORTUNITY COSTS
Another study conducted by Bascetin and Nieto (7) use an iterative approach based on
Lane’s algorithm to determine the optimal cut-off policy for an open pit mine. However,
they introduce an “optimization factor” based on the generalized reduced gradient
algorithm to maximize the NPV of a project. The optimization factor is included in the
limiting cut-off grade calculation and serves as an additional time cost associated with
9
producing one more unit of ore. This is in addition to the opportunity cost introduced by
Lane. Their findings suggest that by including a mining cost into the optimal cut-off
grade calculation, when the concentrator is the limiting capacity, the overall NPV of a
mining project is increased over Lane’s approach.
2.2.3 NON-LINEAR PROGRAMMING
Non-linear programming (NLP) can be used to solve an objective function containing
non-linear constraints. The solution to optimizing a cut-off grade that maximizes the
expected NPV of a mining project is a non-linear objective with several linear and non-
linear constraints. The reduced gradient method of solving such a problem is outlined in
a paper written by Yasrebi et al (8). Using a cut-off model based on Lane’s algorithm
created with LINGO software, they are able to optimize a single cut-off grade for the
entire life of the project. This type of simplified calculation is not optimal because it does
not apply dynamic cut-off theory whereby cut-off grades decline as the resource is
depleted.
This approach also assumes static prices and costs that will undoubtedly change
throughout the life of the project. An attempt to combine a series of NLP equations
could provide an updateable policy.
2.2.4 VARIABLE CAPACITIES
An algorithm proposed by Abdollahisharif et al (9) examines the idea of variable
capacities on the major limiting factors; mine, mill and market. Their method attempts to
10
improve on Lane’s original algorithm, which holds mining, processing and market
capacities constant, by calculating them as variable parameters. By substituting the
variable for the maximum capacity of a constraint into the equation to find the total
quantity utilized for a particular constraint, the maximum efficiency of the investment can
be obtained. For example, consider the concentrator to be the limiting capacity for an
open pit mine. To find the quantity of material refined ‘ Qr ’ for the life of the project;
Lane introduced an equation that provides the relationship between quantity produced
and quantity refined
Qr = y * g *Qc
where ‘ y ’ is the percentage of recovered material from processing, ‘ g ’ is the weighted
average grade of the mineralized material above cut-off and ‘ Qc ’ is the total amount of
material processed over the life of the mine (LOM). By substituting the maximum
capacities for both the market and the concentrator for the quantities utilized, the
equation can be rewritten as,
C = R
y * g
where ‘ C ’ is the maximum variable capacity for the concentrator and ‘ R ’ is the
maximum variable capacity for the refinery. In this case the refinery capacity is assumed
to be equal to market demand. In Lane’s algorithm ‘ C ’, ‘ R ’ and ‘ M ’ (maximum mining
11
capacity) are constant and are determined before the calculation of an optimum cut-off
grade.
By applying this technique and comparing the results to both Lane’s original algorithm
(1) and that offered by Gholamnejad (6), which introduces rehabilitation costs into cut-
off grade determination, Abdollahisharif et al (9) find that using variable capacities to
calculate cut-off grade provides the greatest NPV. The optimal cut-off grade becomes
much lower than the other two methods. This results in more material concentrated,
~29% more than the others, while holding the refining capacity equal to market demand
for all three methods. However, the mining throughput rate was reduced by 10%
compared to the other proposed methods.
In practice, as cut-off grade changes, so does the amount of material sent to the
processing plant and potentially the mining rate, as seen with other studies (1,2,10). In
contradiction to Abdollahisharif et al (9), Breed and Heerden (11) state that “to ensure
cut-off optimization is done correctly, the capacity constraints must be independent of
the cut-off grade”. Therefore using variable capacities to determine the optimal cut-off
strategy can only be used to determine potential capacity parameters. The variable
capacity algorithm also assumes a single metal, open pit project and does not create a
LOM cut-off value policy nor does it capture the opportunity costs associated with
mining at different cut-off grades.
12
2.2.5 STOCHASTIC PRICES
Lane’s model is based on the assumption that future prices are known. In reality
determining cut-off grades for the life of a project requires some level of price and cost
forecasting to accurately estimate the value of the project. Barr suggests in his work on
real options (12), that by using a stochastic price model of the entire futures curve and
not simply a predetermined price or even stochastic spot price model, optimal cut-off
grades are lower than otherwise. This means more mineralized material is classified as
ore. Therefore, using deterministic prices and costs lead to higher than optimal cut-off
grades, which results in misclassifications of ore and waste (13).
Although applying a stochastic price model and real options valuation to a mine is closer
to a real world scenario, the steps taken to forecast price movements are beyond the
scope of this research. Readers are directed to (14,15) for a more in depth examination
of optimizing COGs under price uncertainty.
2.2.6 MULTIPLE MILLS
Asad and Dimitrakopoulos (2) applied a heuristic process to expand on Lane’s algorithm
to optimize cut-off grade at a project with multiple processing streams. They also
account for geologic uncertainty by simulating several different, but equally probable
grade tonnage curves. Using a modified algorithm that is successful at maximizing the
NPV for the set of given grade tonnage curves, they are able to determine the optimum
cut-off grades for each processing stream. Their approach was applied to a large open
pit copper mine where they observed a 13.8% difference between the minimum and
13
maximum NPV generated from the set of simulated grade tonnage curves. They
conclude that ignoring geologic uncertainty in the planning stage can have severe
economic implications on a mining project (2).
Although Asad and Dimitrakopoulos were successful in applying Lane’s theory to a
mine with multiple processing streams, the utilization of those streams was well below
capacity. Their results suggest that out of the four processing streams incorporated, for
the entire life of the project, not one stream runs at even half of its maximum capacity.
This is not practical in a real world situation. A mill design with an annual production
capacity of 43.8 million tons of ore would not be justified if its peak production were <10
million tons per year.
The algorithm used in this thesis was based on the work done by Asad and
Dimitrakopoulos (2). The difference lies in the incorporation of modular processing as
opposed to a permanent mill. This type of technology has the flexibility to increase and
decrease capacity in small increments to account for changes in geologic or market
conditions. Therefore, when the resource is low the use of an additional mill will be
excluded from the model negating the need for a balancing system, as only one
processing stream will be limiting.
14
3 METHODS
The analysis and optimization of cut-off grades is essential to maximizing the value
generated from a mine. The term “cut-off grade” takes on several definitions depending
on how it is applied. Taylor defines cut-off grade at an ore deposit as any mineral grade
that, for any specific reason, is used to separate two courses of action (16). This could
include whether or not to mine a unit of material or which recovery process is best
suited for that material. Another definition considers cut-off grade the level of mineral
concentration that dictates whether mineralized material is deemed ore or waste (1). In
general, cut-off grades are primarily used to classify material at a mine.
Currently, there are two main methods used to determine cut-off grade. The break-even
method, which considers only financial factors and Lane’s method which attempts to
maximize the NPV of the project subject to mine, mill and market constraints. The
following sections introduce these methods with examples.
3.1 BREAK-EVEN METHOD
Many mining companies use a break-even analysis to determine cut-off grade. This
method considers the prices and costs and average recoveries related to mining and
processing. The break-even cut-off grade is where the costs of producing a salable
product are equal to the revenue earned from that product (5,17).
Breakeven COG = Costs
Commodity Price * Recovery
15
Most commonly this is used to distinguish between ore and waste at the mining level.
However, depending on the costs included in the calculation, the break-even cut-off
analysis can be applied to many areas of the project.
• Marginal break-even cut-off considers the variable costs of mining and milling
• Mine operating break-even cut-off assumes total mining costs and milling costs
• Site operating break-even cut-off includes total mining, total milling and total site
administration costs
Although these calculations are used for different applications they are all based on a
break-even principle, by which the revenues earned are equal to the costs of producing.
Consider an example provided by Hall (5) for a simple break-even COG calculation
where,
• Selling price = $10 /g
• Recovery = 90%
• Total costs = $60 /t
$60$10 * 90%
= 6.67 g / t
Since recovery is 90% and the selling price is $10/g, the revenue earned is $9/g. By
applying a total cost break-even COG of 6.67g/t the revenue earned is equal to the
costs of producing that revenue. Figure 3.1 below is a graphical representation of the
relationship between total cost and revenue.
16
Figure 3.1 Graphical representation of the break-even relationship between costs and revenue similar to (5).
Here, total costs of $60/t are assumed to be independent of grade and are therefore
represented as a straight line. The revenue function increases with grade at a rate of
$9/g. The point where the total cost and the revenue functions meet is the break-even
cut-off grade of 6.67g/t.
This method is widely accepted by the mining industry to ensure the operation remains
profitable. However, it fails to maximize the value of the material being mined. Since the
break-even model only considers price and costs, other factors such as variability in
geology and operational capacities that have an influence on revenue are overlooked.
Ignoring such factors can lead to lower than optimum cut-off grades resulting in a lower
overall NPV.
0
20
40
60
80
100
0 1 2 3 4 5 6 7 8 9 10
$/to
n
Grade (g/t)
Breakeven Cut-off Grade
Total Cost
Revenue
17
3.2 LANE’S METHOD
The need for optimal cut-off grade calculations based on an available resource and the
capacities that limit the extraction and production of that resource is a major challenge
in the mining industry. Lane was the first to introduce an algorithm to calculate cut-off
grades that maximize the present value of cash flows from a mining project. He derived
a set of equations to calculate a cut-off grade specific to which capacity(s) in the mining
system are limiting output. Furthermore, he outlined a method for calculating the
opportunity cost associated with mining at a particular cut-off to determine a complete
cut-off policy for the life of the project.
Until Lane published his initial work on cut-off theory in 1988, cut-off grades were
calculated using the costs of mining and processing the ore and the selling price of the
commodity. Lane proposed that cut-off grade is a function of not only costs and prices
but of capacities that limit mining, processing and refining. Understanding that the
primary objective of most mine operators is to maximize the present value of all future
profits (1), Lane suggested that to maximize the value of an exhaustible resource an
exploitation track that maximizes the present value of the project at all times must be
employed.
3.2.1 EXPLOITATION STRATEGY
The present value of an operation based on a finite resource is calculated as the total of
the future cash flows discounted back to the present. At a mine, minerals are
excavated and processed to recover a salable product. All mineralized material
containing sufficient economic value to cover the costs of mining and processing are
18
classified as economic ore, while material with less than enough mineral concentration
is considered waste. Ore is sent to the processing facility while waste is sent to a waste
pile or left in place. The level of mineral concentration that dictates whether material is
classified ore or waste is the cut-off grade (1).
Cut-off grades can be used for several different situations at a mine as a means for
classification. Most commonly, it is the lowest grade material that “should” be mined
and/or used to calculate total reserves. Most importantly, cut-off grades are used to
identify the optimum exploitation strategy for maximizing the present value of an
operation (1).
In order to determine the optimum cut-off grades for a deposit two fundamental
expressions must be considered. The first determines the optimum exploitation strategy
for maximizing the present value of an operation based upon a finite resource. Lane
suggested that value is a function of the size of the remaining resource Q and the rate
of extraction q (1). These two variables also define the life of the project T . Therefore,
the present value V of the mining operation at any time is a function of the life of mine,
the size of the remaining resource and the chosen rate of extraction (18).
V T + t,Q − q( ) [1]
Differentiating V with respect to Q and T ,
19
dVdQ
= p − tq
δV − dVdT
⎛⎝⎜
⎞⎠⎟
[2]
where, p is the cash flow arising from one unit of resource and t is the time it takes to
process that unit. The tonnage (τ ) is given by t / q , which represents the time required
to process one unit of mineralized material.
The opportunity cost of mining at a particular extraction rate is the term in brackets.
Here, the discount rate δ is multiplied by V to reflect the decrease in value of the
resource resulting from extraction, which is subtracted by the first derivative of the value
( V ) with respect to time ( T ). The opportunity cost can be rewritten as F , and the
equation that maximizes the present value of a mining operation becomes,
dVdQ
= p −τF [3]
The second necessary expression directly relates cash flows to cut-off grades. The
formula for the cash flow arising from one unit of mineralized material is:
p = S − r( )xyg − xc −m − ft [4]
20
where x is the proportion of mineralized material classified as ore, g is the weighted
average grade of the ore and y is the percentage recovery of mineral from the
treatment process. The remaining economic variables are:
S = unit mineral price
r = unit market cost (refining cost)
c = unit processing cost
m = unit mining cost
f = fixed costs (annual)
Combining equations [1] and [2] we derive the objective function that must be
maximized at all times during the life of mine in order to maximize NPV.
Maxg S − r( )xyg − xc −m − f + F( )t{ } [5]
In this expression x and g are directly dependent on the cut-off grade, g . The time t
is also dependent on g but indirectly, which gives rise to three separate cases for
analysis based on which capacity is limiting output.
3.2.2 LIMITING ECONOMIC CUT-OFF GRADES
Lane proposed there are three economic capacities in the mining system that limit
throughput and the exploitation of the deposit. They are the mine, treatment facility and
market (1). The mine represents mining and development rates that govern throughput.
21
The treatment facility consists of the concentrator(s) and ore handling facilities. The
market is limited by any restriction imposed by sales contracts or by a refinery or
smelter. At any given time during the life of an operation one or more of these capacities
will be the limiting factor for the system. Because each of these capacities dictates the
supply of salable product they are deemed limiting economic capacities.
Each limiting capacity has its own calculation taking into account the unit costs and the
specified capacity for that system. Each calculation also contains the opportunity cost of
not mining the remainder of the deposit due to the limiting capacities of the mine, the
processing facilities, and the market. Therefore, depending on which area of the total
system is limiting, the time ‘ t ’ becomes Qm / M , Qc / C or Qr / R if the mine, the
processing plant, or the refinery are limiting, respectively (2), where Qm represents the
quantity of tons mined, Qc is the quantity of tons milled and Qr is the quantity of
ounces refined in a given period. The maximum annual capacity for the mine, the mill
and the market are denoted as M , C and R respectively. The opportunity cost must
then be distributed per ton of material mined, per ton of ore processed, or per ounce of
metal refined, depending on which component is limiting.
Mine Limiting gm = c
S − r( ) * y [6]
The equation for a mine limiting economic cut-off grade indicates that the mineralized
material should be classified as ore for as long as its implicit value, S − r( ) * y * g ,
22
exceeds the costs of further processing, c . It is important to recognize that time costs
and mine costs are not relevant. This is because the formula is based on the
assumption that the decision to mine beyond the present time has already been made
(1). The mine limiting equation also does not make reference to present values due to
the fact that there is no trade-off of future losses against present gains to modify the
current policy.
Process Limiting gc =
c +f + F( )
CS − r( ) * y
[7]
For the process limiting equation, the opportunity cost F represents an additional time
cost associated with processing the ore. This creates higher cut-offs when F is large in
the early years of production and lower cut-offs as F declines along with the resource.
This realization represents dynamic cut-off theory, whereby cut-off grades change
throughout the life of a mine to maximize the value of the resource being mined.
Market Limiting gr =c
S − r −f + F( )
R
⎡
⎣⎢⎢
⎤
⎦⎥⎥
* y
[8]
Similar to the process limiting cut-off grade formula, the market limiting equation
includes the present value term in the form of an opportunity cost, which along with the
fixed costs is distributed according to the limiting capacity. This results in declining cut-
off grades as F declines due to the resource being depleted.
23
3.2.3 BALANCING CUT-OFF GRADES
Often a mine is constrained by more than one of the limiting factors mentioned in the
previous section. If this is the case, the optimum cut-off grade is calculated by balancing
the limiting cut-off grades and the maximum capacity for each of the limiting factors. As
previously mentioned the time ‘ t ’ becomes Qm / M , Qc / C , or Qr / R depending if the
mine, the processing plant, or the refinery are limiting output, respectively (2). Setting
these ratios equal to each other gives rise to three new cut-off grades called balancing
cut-off grades.
If both the mine and processing facility are limiting than cut-off grade gmc is the grade
that satisfies the equation
QmM
= QcC
[9]
Similarly, if the processing plant and the refinery are the limiting factors than gcr is the
grade that satisfies,
QcC
= QrR
[10]
And if the mine and the refinery are both limiting than gmr must satisfy,
QmM
= QrR
[11]
Applying these ratios to the cumulative grade distribution curve, a single point is
observed where the proportion of; mineralized material, recoverable mineral per unit of
24
mineralized material, and the recoverable mineral per unit of ore above the
corresponding grade equals the balancing ratio C / M , R / M , or R / C respectively. A
graphical representation of gmc is shown in Figure 3.2.
Figure 3.2 Graphical representation of balancing cut-off when the mine and the mill are limiting.
Therefore, six possible cut-off grades must be examined to determine the effective
optimum cut-off; three break-even cut-off grades based on the limiting capacity and
three balancing cut-offs that are dependent on which capacity(s) are limiting the mining
system.
3.2.4 EFFECTIVE OPTIMUM CUT-OFF
The optimum cut-off grade is selected from the 6 possible cut-offs discussed thus far.
Again, this will be dependent on which areas are limiting the output of the mining
Prop
ortio
n of
Min
eral
ized
M
ater
ial
Grade
Cumulative Grade Distribution for a Mine Planning Increment
Ratio C/M =Processing cap./Mining cap. Balancing cut-off
grade
gmc
25
system. Because the cut-off grade g corresponds to the present value V of the mine,
the optimum cut-off grade can be determined by maximizing the rate of change of V ,
with respect to resource usage ( dV / dQ ) (1,2,18). Setting equation [5] equal to the
variable v , which represents the increment in present value per unit of resource utilized,
we get
v = S − r( )xyg − xc −m − f + F( )t. [12]
As with the limiting and balancing cut-offs, v takes on three forms depending on which
area(s) of the mine are limiting.
Mine vm = S − r( )xyg − xc −m −
f + F( )M
[13]
Mill vc = S − r( )xyg − x c +
f + F( )C
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪−m [14]
Refinery vr = S − r −
f + F( )R
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪xyg − xc −m [15]
By plotting v as a function cut-off grade, it is observed that the graph is concave with a
single maximum. This holds true for all forms of v . The maximum corresponds to the
limiting economic grade for the component being analyzed (1).
26
Figure 3.3 Increment in present value versus cut-off grade with the mine and mill components in balance.
When two forms of v are plotted on the same graph as in Figure 3.3, the feasible region
(shaded in purple) for the optimum form of v called ve is always the lower of the two
curves. The effective optimum cut-off becomes the maximum point along the feasible
ve curve. In the example above, the maximum value for ve occurs at gmc when the
mine and the mill are in balance. However, in Figure 3.4 the balancing cut-off is above
both the mine and mill limiting cut-offs and the maximum point along the feasible ve
curve must be located.
In Figure 3.4 the effective optimum grade occurs at the process limiting cut-off gc , or
the median value between the limiting and balancing cut-offs. Lane devised a set of
conditions that must be applied to determine the effective optimum cut-off grade at a
Incr
emen
t in
Pres
ent V
alue
(v)
Grade
Effective Optimum Cut-Off Grade
v mine
v mill
gm gc
gmc
Process limiting cut-off Mine limiting cut-off
Maximum ve
27
single point in time.
Figure 3.4 Increment in present value versus cut-off highlighting the maximum value in the feasible region of ve when the mine and the mill are in balance.
When the mine and the processing facility are limiting the effective optimum cut-off
grade is,
Gmc = gm if gmc < gm
= gc if gmc > gc
= gmc otherwise
When the mine and market are limiting,
Gmr = gm if gmr < gm
= gr if gmr > gr
= gmr otherwise
Incr
emen
t in
Pres
ent V
alue
(v)
Grade
Effective Optimum Cut-Off Grade
v mine
v mill
Maximum ve
gm gc= Gmc gmc
28
When the mill and market are limiting,
Gcr = gr if gcr < gr
= gc if gcr > gr
= gcr otherwise
Identifying the feasible region of ve is not always as clear as the examples above. Often
all three forms of v must be analyzed making it difficult to identify the true maximum.
Lane admits that the peaks for the various incremental present value curves are easily
identified, however, the exact intersections of the feasible region can be difficult to
determine graphically and a more robust method such as the golden search method
(19) must be applied.
3.2.5 CUT-OFF POLICY
A cut-off grade policy is a sequence of optimum cut-off grades over a specified period of
time (1). A cut-off policy serves as a long-term strategy for how much material to mine,
process and refine as a function of grade that will maximize the value of a mining
project. Similar to determining a single effective optimum cut-off grade for a single point
in time, a complete cut-off policy will follow an exploitation track that maximizes the
present value of the resource being mined at all times, while adhering to the capacity
constraints associated to the mine, the mill and the market. It is therefore necessary to
have a mine design including all the major operational and economic parameters in
order to determine a complete life of mine (LOM) cut-off policy.
29
A cut-off policy calculation begins with identifying a terminal value for V . Most often the
terminal value will be zero if the policy is for the life of the resource. Next, initial values
for V are estimated to use in the opportunity cost term
F = δV − dV
dT⎛⎝⎜
⎞⎠⎟
[16]
and a policy is calculated. Finally, the present value at termination is compared to the
specified terminal value. Based on the results, the initial estimates for V are adjusted
and a new policy is calculated. This iterative approach ultimately returns a solution
where the present value at termination is within some tolerance of the terminal value.
As stated earlier, the opportunity cost is the rate of change of present value with respect
to time. Therefore, an estimate of this term is the difference between the present value
at time t = 0 , ( V ) and the present value at time t = t +1, ( W ) for the same amount of
remaining resource (1). The F term then be rewritten as
F = δV +V −W [17]
The mathematical iterative process proposed by Lane can be quite complex when
fluctuations in prices, costs and other variations in economic parameters are introduced.
Consequently, robust cut-off policy calculations are most efficiently performed with a
computer.
30
3.2.6 SHORTCOMINGS OF LANE’S METHOD
Lane’s method for determining optimal cut-off grades is based on maximizing the
present value of cash flows generated from a mine. Lane’s model assumes that a single
mine producing a single stream of material is processed by one facility and refined at
one facility. Therefore mining complexes with multiple sources and multiple processing
streams are difficult to model using Lane’s methods. Another major assumption is that
the resource has been defined and a mine schedule has already been determined.
However, in practice the mine schedule is based on cut-off grades and Lane’s algorithm
thus becomes iterative. Also Lane’s methods fail to capture blending requirements
related to processing ore. (20)
In an attempt to resolve the shortcomings associated to Lane’s methods many authors
have applied extensions or modifications to the original algorithm (2,5,9,19,21).
3.3 GRADE TONNAGE CURVES
The use of grade tonnage data is imperative when analyzing cut-off grade strategy. The
grade tonnage curve is a frequency distribution of the amount of mineralized material
above a calculated cut-off in a particular deposit. The data often comes directly from a
geological block model of the mineralized body of rock created from exploration and
definition drilling. A block model is a three dimensional array of minable blocks each
containing specific attributes such as density, metal grade and lithology. This data is
then used to create a histogram of the tonnages belonging to each grade category. An
example of a tonnage histogram is shown in Figure 3.5.
31
Figure 3.5 Simulated tonnage histogram of gold deposit.
Next, the cumulative frequency of tons is calculated where, n refers to the individual
grade categories and T represents the tons of material within those grade categories.
Q = Tn
n=0∑
The weighted average grade of those tons is then determined by
g =Tngn
n=0∑
Tnn=0∑
0
200
400
600
800
1000
1200 To
nnag
e ('0
00s)
Grade Categories (g/t)
Tonnage Histogram
32
where gn represents the average grade of tones with in each grade category n . Plotting
the two functions for Q and g we obtain a grade tonnage curve, as illustrated in Figure
3.6.
Here, the blue line represents the cumulative tonnage of mineralized material and the
red line is the average grade of that material plotted on the primary and secondary y-
axis, respectively.
Consider a calculated effective optimum cut-off grade of 4.0 g/t. By inspection of the
grade tonnage curve it can be seen that there are 1.7 million tons of ore averaging a
grade of ~6 g/t.
Figure 3.6 Sample grade tonnage curve with cut-off grade of 4.0 g/t.
Grade tonnage curves serve as a visual aid in evaluating the exploitation potential of a
deposit at several different cut-off grade scenarios. The curve displays the average
grade and quantity of ore above a specified cut-off.
0.00
2.00
4.00
6.00
8.00
10.00
12.00
- 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7 8 9 10 Avg.
Gra
de A
bove
Cut
-off
(g/t)
Tons
Abo
ve C
ut-o
ff (M
illio
ns)
Cut-off Grade (g/t)
Grade Tonnage Curve
Tons
Avg. Grade
33
4 PROCESSING
At an open pit mine, mineralized material is excavated and sent either to a waste
stockpile or a processing facility to recover the minerals of interest. As previously stated,
the determination of ore and waste is based on a calculated cut-off grade. A dynamic
cut-off strategy based on Lane’s methods must include the capacities for processing in
the derivation of the COG. It is therefore imperative to understand how processing and
cut-off grades are related. The following chapter is divided in to two categories. The first
section defines modular processing and provides a brief overview of the economic and
operating parameters. The second section examines how cut off grades are determined
when multiple processing streams are utilized.
4.1 MODULAR PROCESSING
Modular mineral processing plants are small, often mobile mineral recovery facilities.
They concentrate ore through the use of gravity or flotation. The term “modular” refers
to the flexible arrangement offered by the design of the milling unit. Traditional
processing plants use large, high-energy consuming comminution circuits housed in a
permanently constructed building or facility. These facilities require significant amounts
of space and depending on the location of the mine and can have very high construction
costs. Modular processing plants were originally designed to process ore underground
in an effort to reduce the haulage and energy costs associated with traditional milling
(22). The smaller size of the mill also reduces the surface footprint when compared to a
conventional mill.
34
Modular processing plants are typically designed to process lower tonnages (10-20 tph)
than larger fixed facilities (50-125 tph). However, they have the flexibility of being mobile
and the option of adding or removing elements of the circuit as necessary. The system
itself is designed in such a way that it can be modified based on the type of ore being
processed. For example, a newly developed gold mine with considerable amounts of
refractory gold at the surface may use flotation to concentrate the ore. Later on, the ore
contains coarser gold that benefits from gravity concentration. These units can be
added and/or removed at very little capital cost when compared to a fixed infrastructure.
The flow sheet for a modular processing plant is much the same as a conventional
facility using the same recovery method except for the size. An example flow sheet
diagram for the Gekko Python underground gravity/flotation plant is presented below.
The Python uses coarse and fine crushing, wet screening, continuous gravity
concentration and flash flotation to concentrate gold (22).
Figure 4.1 Process flow diagram for the Gecko Python Plant, from (22).
35
This type of mill design can benefit any project that is limited in available space but also
benefits projects with multiple ore types and/or with varying grades. The modular
processing plant can be used in conjunction with a larger conventional plant when it is
required that specific ore be processed using a separate recovery method. Consider an
open pit deposit containing predominantly low-grade oxide ore with lesser amounts of
higher-grade sulfide rich ore. The mine currently blends and processes all the ore using
a 3000 t/d heap leach pad with a carbon in column recovery circuit. Lab testing has
determined that the HL recovery method is ideal for the oxide ore resulting in a recovery
of ~90%. However, test work on the sulfide ore when run as separate batch, has a
recovery of ~30%. With exploratory drilling suggesting more high-grade sulfide rich ore,
a new processing stream should be incorporated to maximize the recovery of mineral in
the sulfide ore. A modular flotation unit is one such solution.
By incorporating a modular flotation plant the need for blending and stockpiling the
different ore types is removed. This reduces operating expenses related to transporting
and re-handling the previously mined ore.
4.1.1 CAPITAL AND OPERATING COSTS
Although the operating and capital expenses of a modular processing plant are far less
than a conventional plant, the exact capital and operating costs are difficult to estimate
due to the variability in project location and ore type. Each project requires a different
set of modules to obtain maximum recovery from the ore being processed resulting in
project specific costs. Sepro Systems of Vancouver reports their 30 tph skid mounted
36
gravity/flotation plants cost approximately $1.2-$1.5 million USD (*Personal
Communication, Sepro Systems). Table 1 lists the estimated total operating costs for
two sizes of the Gekko Python plant, the P200 and P500, installed in an underground
South African gold mine. The P200 and the P500 have annual throughput capacities of
146,000 tons and 360,000 tons, respectively. Unit operating costs range from $8.80-
$12.50 USD per ton, depending on the throughput rate (22). The variable operating
costs for both the Sepro and Gekko modular processing plants are most sensitive to
local labor costs followed by consumables and power consumption.
Table 1 Estimated unit operating costs for the Gekko Python modular processing plant, from (22).
It is important to note that due to economies of scale, as process capacity is increased
the unit costs will decrease. Kappes (25) points out that operating costs are not very
sensitive to the size of a HL operation. In an article on heap leach design and practice
he reviewed the cash operating costs of 27 HL operations. Including mining costs, a
3000 t/d operation has a unit cost of $10.12 per ton, a 15,000 t/d has a unit cost of
$7.70 per ton and a 30,000 t/d (typical of Nevada) has an operating cost of $5.20 per
ton.
Similarities are observed when comparing operating costs for modular CIL units as well.
Hughes and Gray (22) determine that barring any changes to power, water and reagent
consumption, increased capacity means lower unit operating costs. The Gekko Python
P200, with an annual throughput capacity of 146,000 tons, has an estimated operating
cost of $12.10 per ton. While the Python P500, with an annual capacity of 360,000 tons
has an operating cost of $8.80 per ton.
For both the HL and modular CIL processing streams the unit labor and management
costs become the dominating variable. When capacity is low the utilization of man-hour
to processed ton is also low. As capacity is increased, and the demand for labor stays
the same overall unit costs will decrease. In practice this may not be the case. For
example, in a HL operation if the capacity was doubled certain variable costs would
indeed increase such as the unit costs for reagents and haulage costs to transport the
additional ore. Other areas of the mine will also be affected including waste disposal
and tailings management to handle the extra material.
68
8 CONCLUSIONS
The objective of this research was to maximize the value of small-scale, low-grade,
open pit homogeneous gold deposits through cut-off grade strategy and modular
processing. The NPV for the modular case was on average 11.43% higher than that of
the base case. This indicates that by adding process capacity and dividing the ore into
separate streams, project value will increase.
The hypothetical case in this research does not consider capital costs. However, this
study suggests that an additional modular processing plant to process high grade ore
should be introduced if the capital cost is less than the difference between the average
NPV calculated for both the base and modular case over the set of 15 simulated grade
tonnage curves.
It was demonstrated through a sensitivity analysis that gold price has the greatest
influence on NPV and therefore ore/waste classification. When gold price was low, cut-
off grades are higher and less material is classified as ore. When gold price is high,
optimum cut-off grades are low and more material is classified as ore. The methods
used in this research based on Lane’s algorithm, determine the optimal ore/waste
classification scheme for any point in time during the life of the mine. This confirms the
tactic of altering cut-off grade when commodity prices rise or fall.
The results also demonstrate that an optimum cut-off strategy needs to consider the
capacities for the limiting components outlined by Lane, and the opportunity costs of
69
mining at a determined cut-off grade level. A comparison of the NPV obtained via the
break-even method and dynamic optimization methods presented here, suggest that by
overlooking the capacities and opportunity costs, break-even calculations may lead to
sub optimal cut-off grades resulting in underutilized resources and revenue loss. This
substantiates that cut-off grades determined by the break-even method are inadequate
for maximizing the value of a resource.
Results from the hypothetical case study reveal that a low-grade open pit gold mine will
benefit from the use of multiple processing streams when a dynamic cut-off policy is
used, particularly, when incorporating a “high grade” milling stream to maximize the
potential revenue of the mineralized material. Therefore, a mine with increased
processing flexibility has more value than a mine that does not. This means that for a
given grade tonnage curve and a set of design and production parameters, the
classification of ore and waste is what ultimately determines the NPV of a mining
project.
70
9 RECOMMENDATIONS
The strategy of utilizing multiple processing streams in which ore and waste are more
finely classified must be considered when processing low-grade ore in remote locations.
The flexibility offered by modular processing streams allows for a wider range of
feasible ore grades and commodity prices than that offered by traditional constructed
processing plants. Further development of modular processing technology that allows
for designs that can be scaled up to throughput rates similar to conventional facilities
will require a more robust cut-off analysis.
Work done in this thesis on optimal cut-off policies for mines with multiple processing
plants including a modular stream can be further developed with case studies which
closer reflect real world situations. For example, projects that do not have the ability to
process material using a HLF can be modeled with multiple modular processing
streams. A cut-off policy of this kind will benefit a mining complex with multiple ore types
and/or multiple phases each containing a separate grade tonnage curve. In theory, the
model will provide the best sequence of extraction based on the determined optimum
cut-off grades.
Based on the abundance of extensions and modifications to Lane’s methods, it is
conceivable that a combination of these extensions could be compiled into one model.
This includes:
1. The ability to analyze underground and open-pit mines containing multiple metals
of interest. Because most underground mining techniques are selective the
71
functional forms of Qm and Qc would need to be defined for a specific
underground mine. Barr (12) suggests the quantity of material mined and
processed could be discrete functions defined by a series of alternative stope
designs.
2. The replacement of deterministic prices and costs with stochastic variables. The
use of stochastic variables to forecast commodity prices can be included directly
in to the limiting cut-off calculation (step 5.3 of algorithm) and the annual profit
calculation (step 5.6 of algorithm). This substitution will allow for improved mine
design planning when analyzing mines with longer life spans. The longer the
mine will be in production the less reliable deterministic prices become
3. The incorporation of rehabilitation costs into the limiting cut-off grade calculation
following the methods proposed by Gholamnejad (6).
72
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APPENDIX A. Derivation of Lane’s Equations
Derivation of the equations used in this thesis for Lane’s method for determining the
maximum present value of a mining operation based on a finite resource, from (18).
The value of the mining operation V , in the first period, is the cash flow ( Pq ) associated
with mining q units of material and the present value (PV) of any facility is given by:
V T,Q( ) = Pq + 11+δ( )t V T + t,Q − q( )⎡⎣ ⎤⎦
= Pq + V T + t( )⎡⎣ ⎤⎦ / 1+δ( )t [XXVIII]
Focusing on the second part of the equation,
V T + t,Q − q( )⎡⎣ ⎤⎦1+δ( )t
Using the Binomial Series expansion,
1+ n( )n ≈ 1+ nn( ) if n ≪1
The equation is rewritten as:
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V T + t,Q − q( )1+δ( )t
=V T + t,Q − q( ) * 1+δ t( )
Next, using the Taylor Series expansion for two variables for this part of the equation,
V T + t( ) =V T( ) + t dV
dT+ t 2
2d 2VdT 2 + ...and [XXIX]
V Q − q( ) =V Q( )− q dV
dQ− q2
2d 2VdQ2 + ...and [XXX]
Combining equations [II] and [III],
V T + t,Q − q( ) = V T,Q( ) + t dV
dT− q dV
dQ⎡
⎣⎢
⎤
⎦⎥ * 1+δ t( ) [XXXI]
Multiplying equation [IV] by −rt t dV
dT⎛⎝⎜
⎞⎠⎟≈ 0 and because −rt is very small, it means −rt 2
and −rtq are extremely small, so they can be ignored and equation [IV] becomes:
V T,Q( )− rtV + t dV
dT− q dV
dQ [XXXII]
Substituting equation [V] into equation [I],
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V T,Q( ) = Pq +V T,Q( )− rtV + t dV
dT− q dV
dQ [XXXIII]
After differentiation and cancelling of the common terms on both sides of equation [VI],
it can be set equal to 0:
Pq − rtV + t dV
dT− q dV
dQ= 0 [XXXIV]
Next, equation [VII] is solved for the variables of interest,
q dVdQ
= Pq − t δV + dVdT
⎛⎝⎜
⎞⎠⎟
dVdQ
= P − tq
δV + dVdT
⎛⎝⎜
⎞⎠⎟
and if,
F = rV + dV
dT⎛⎝⎜
⎞⎠⎟
and τ = tq
the maximum PV of a mining operation based on a finite resource Q determined by
Lane is,
dVdQ
= P − Fτ [XXXV]
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APPENDIX B. Simulated Grade Tonnage Data
Grade tonnage data for the set of 15 simulated grade tonnage curves used in this
research is presented in Tables B1-B5 below. The data set includes the tons of
material, the average grade of the material and the ounces of gold contained in each
grade category.
Table B1 Simulated grade tonnage data for GT1-GT3.