A System Dynamics Approach to Mine Modelling and Cut-Off
Grade Management
K Sinding 1 and E R Larsen2
ABSTRACT
We discuss the concept of optimal cut-off grade policies and their
advantages in tenns of maximisation of net present value of mining
operations. There are indications, however, that such optimal policies are
not widely used in the mining industry. In order to explore why this is the
case we identify a number of stakeholders who may have diverging
interests in the operating policies of a given mine. These various interests
are mapped and a number of important relationships are explored in a
large qualitative systems dyanrnics model.
INTRODUCTION
The cut-off grade is defined as the grade, or relative content of
valuable material, which is used to distinguish between ore and
waste in a specific mine or orebody (Dagdelen, 1992). The
cut-off value is highly site specific'and is defined by the unique
characteristics of a particular mining operation.
The modem concept of how an optimal cut-off grade policy
can be calculated was formulated by Kenneth Lane in 1964 and
later extended 'and generalised to a range of different mining
situations (Lane, 1964; Lane, 1988). This approach is based on
the idea that calculation of the cut-off grade must take into
account the fixed (opportunity) cost of not receiving future cash
flows more rapidly as a result of the cut-off grade decision made
now (Dagdelen, 1992).
The theory is well established and has not been challenged.
However, there are signs that the methods advocated by Lane are
not widely used. Indeed, the introduction to Lane's 1988 book is
not sparing in its criticism of arguments used in defence of cut-off
determination not based on optimal cut-off determination taking
the opportunity cost element into account.
The other major problem influencing determination of cut-off
grade is proper accounting for costs in the various stages of the
mining process. This comes back to the very site specifi'c nature
of many cost elements. For example, there will be significant
differences in the cost of ore located close to or far from the shaft
in an underground mine. If these cost differences are accounted
for there will be a diff~rence in cut-off grade between the two
locations. However, if the costs are averaged out in the
accounting process, then ore close to the shaft will be subsidising
ore of the same grade far away from the shaft.
Indications from numerous mine visits, and comments from
colleagues in the mining industry, indicate that little or no weight
is given to the opportunity cost element in determining cut-off
grade. On the other hand, the need for exact and detailed cost
accounting is more widely appreciated, perhaps because the
,concept is easier to understand than opportunity cost.
Nevertheless, from the point of view taken by Lane, the use of
sub-optimal cut-off grades is unsatisfactory and inefficient,
especially from a shareholder point of view. It also begs a number
of questions about what factors really determine observed cut-off
grade policies.
'
In this paper we employ a management science technique
whereby mental models of reality (eg a mining operation) are
transformed first into maps (causal loop diagrams) which show
how various factors interact, and second into quantitative
computer simulation models.
The work we report here is of an exploratory nature in that we
concentrate on formulating a qualitative model of mining
operations. We begin by outlining the principles involved in
determining optimal cut-off grade policies based on precise
accounting of cost and inclusion of the opportunity cost element
related to cut-off decisions made now.
The apparent lack of enthusiasm for this approach indicate that
other interests and stakeholders may be able to influence the
cut-off grade decision. To analyse this possibility we first
introduce the concept of causal loop diagrams as a tool for
mapping the interrelationships of the possible stakeholders in a
given mine. The stakeholders are then m!lpped and the possible
avenues of influence on the cut-off grade decision are identified.
In conclusion we discuss what areas are most fruitful for further
quantitative modelling.
OPT~ALCU~OFFGRADES
The seminal work on cut-off grade selection by Lane (1964,
1988) shows that any determination of an optimal cut-off grade
must take into account the fact that there is an opportunity cost
associated with not receiving cash-flows more rapidly in the
future as a result of cut-off policies adopted now. While
emphasising this as the key requirement, Lane also stressed the
importance of limiting ·factors in a mining system, for example
the capacity of a mill to treat ore, or the capacity of a customer (ie
a smelter specified in a long tei.1ll contract) to receive concentrate.
Using the simplified case presented by Dagdelen (1992), where
the milling capacity is the limiting factor, the optimal cut-off
grade in a year i between 1 and the end of mine life N is
calculated as:
gMil1 =
2. Research Associate, The Learning Center, London Business
School, Sussex Place, Regents Park, London NW1, UK.
APCOM XXV 1995 Conference
dNPV
i
(P-s)y
where
c
is milling cost;
fa
is annual fixed cost;
C
is annual mill capacity;
d
1. Research Associate, Department of Earth Resources
Engineering, Imperial College of Science, Technology and
Medicine, Royal School of Mines, Prince Consort Road,
London SW? 2BP, UK.
fa
c+C+-C-
i
NPV
is the discount rate applied;
i
is the net present value of future cash flows from i to the
end of mining;
P
is the mineral sales price;
s
is the marketing cost; and
y
is the fraction of minerals recovered.
Brisbane. 9 - .14 July 1995
241
K SINDING and ER LARSEN
With the exception of NPyi, all of these variables are
exogenously given in the sense that they change only in response
to external events or if action is taken to change them. If none of
the factors are altered, th~ cut-off grade depends only on the
gradual decline in NPY' as mining progresses. From the
expression for optimal cut-off grade it is then clear that cut-off
will decline as the end of the mine life approaches.
In practice, the determination of the optimal cut-off grade
depends on the develop1TIent of an iterative model with which the
initially unknown NPY' is estimated. An algorithm which solves
this problem, and generates an optimal cut-off grade policy has
been developed by Dagdelen (1992). Using this algorithm on a
hypothetical mine, and comparing the resulting cut-off grade to a
traditionally determined cut-off grade policy, Dagdelen obtained
radically different results. Most importantly, the optimal cut-off
grade policy generated a much higher net present value (a 90 per
cent improvement) while undiscounted cash flows were
somewhat lower (by 35 per cent) and mine life cut to less than a
third (ten years instead of 36). This simplified example did not
allow for stockpiling of low grade ore. Had this been possible, an
even larger improvement in NPY might have been possible. This
brief outline of how optimal cut-off grade poilcies are determined
omits many of the refinements described by Lane (1988), and is
based on the simplified case where processing capacity is the
only limiting factor.
There are no published indications of how widely the
prescriptions for setting optimal cut-off grade policies are used in
the mining industry. A few papers have mentioned the method in
passing (Real and Torres Lopes, 1994; Taylor, 1985), but seem to
indicate that other considerations are equally or more important
than optimisation when the cut-off grade is determined. On the
one hand this is consistent with our observations at minesites and
with the descriptions from colleagues in the mining industry. On
the other hand, this use of non-optimal cut-off grades is, to some
extent, predicted by Lane (1988), who sees this as the result of
special interest groups motivated by other considerations than
maximisation of overall NPY.
The example cited by Lane in support of this contention is that
staff at a given mine will be interested in setting a lower than
optimal cut-off grade so as to extend their period of employment.
However, while this may be a valid argument for managers in
mines close to the end of their operating lives, it is not necessarily
true that managers are in a position to do so when reserves can
support a long mine life. For example, if management is replaced
by natural turnover every five years and the remaining mine life
is twenty years, extending mine life is unlikely to be a primary
concern, until late in a mines life.
This means that we must look for other causes to explain
non-optimal cut-off grade policies. To analyse the possibilities we
have encountered we must first digress in order to introduce the
concept of causal loop diagrams and their use.
increased supply of the metal. The linear cause and effect
diagram is shown in Figure 1.
But when supply expands, a situation of excess supply may
develop, pushing down prices. By taking this effect into account,
beginning and end of the linear cause and effect chain is
connected to form a closed loop, as shown in Figure 2.
The other type of loop is reinforcing, indicating that change
leads to more change. As an example, consider the standard
prescription for how a mine should adjust its cut-off grade in
response to a price increase, ie by lowering the cut-off grade. If
each mine follows the same rule, all will still put the same
quantity of ore through the mill, but less metal concentrate will be
produced, both locally and overall. This restricts supply, and
prices are pushed upwards even further. A causal loop with this
reinforcing effect is shown in Figure 3.
Price '~E & D Investment'~Supply,
FIG 1 - Linear cause and effect diagram.
Supply
FIG 2 - Balancing casual loop diagram for a metals market.
242
Cutoff
grade
(
Metal Price
~
CAUSAL LOOP MODELLING
Causal loop diagrams are a part of what management scientists
refer to as 'systems thinking'. This is a way of seeing structures
in complex systems. The approach emphasises interrelationships
rather than linear cause-and-effect chains, and processes of
change rather than snapshots (Senge, 1990). This involves
mapping feedback processes, where causal loops are used to
represent 'cause' and 'effect' relationships. There are two distinct
types of feedback loops: balancing and reinforcing. The
following examples serve to illustrate how they work.
Consider first the relationship between metal price, exploration
and development, and metal supply. A price increase indicates a
situation of excess demand. This is an incentive for mining
companies to increase investment in exploration and mining
capacity development. In turn, and after a long lag, this leads to
Exploration and
development investment
Metal Price
FIG
Supply
\
Concentrate
output
~
3 - A reinforcing feed back loop.
MAPPING THE BENCHMARK MODEL
The principle of including the opportunity cost element
developed by Lane (1988), and the iteration algorithm suggested
by Dagdelen (1992) may be described as the benchmark against
which alternative solutions to the mine optimisation problem
must be measured. The optimal cut-off determination and the
iteration involved can, in its simplest form, be mapped as shown
Brisbane, 9 - 14 July 1995
APCOM XXV 1995 Conference
SYSTEM DYNAMICS APPROACH TO MINE MODELLING
Total reserves ~
~ \
~
Interestrate
~.
(
~ Oremined~
""
Cuwff(ade
~
.
Mme caP'""ty
~ Mill capacity
\
costs)
(
c)oncentrate
NPVofyears
remaininig
~
Total NPV
Mine Cash Flow
.
Metal Pnce
Fro 4 - Causal loop diagram for the benchmark optimal cut-off grade model.
in Figure 4. The main loop in this model represents the iteration
needed for determining the optimal sequence of cut-off grades
over the life of the mine. The total NPV is derived from the NPV
of years remaining as the NPV from now to the end of mining.
The small loop connecting ore mined and total reserves reflects
that total reserves contributes to ore mined, while ore mined
reduces total reserves.
This model reflects the mechanics of the process which
determines an optimal cut-off grade policy. ne only action
needed is to set the process in motion. The model does not, for
example, relate cash flow, or remaining NPV, to costs, although
an attempt to reduce costs is one of the first and most obvious
responses to lower metal prices, or higher supply costs.
Lenders
Loc~ue"'l own",\
/
Parent company
management~
1-----__
Cash Flow
RealizedNP
A COMPLEX MODEL OF CUT·OFF GRADE
DETERMINATION
Non-optimal cut-off grades may be employed for a considerable
number of interrelated reasons, and as a result of the influences of
a range of stakeholders in individual mining operations. The
identity of the principal stakeholders is shown in Figure 5, while
the interrelationships between these stakeholders is mapped in
Figure 6 and also discussed below.
The parent company or major shareholder are concerned not
only with subsidiaries, but also their own financial performance.
While a cut-off grade designed to optimise total NPV at the mine
may be in the best long-term interest of the parent company,
realities such as parent debt service, performance of parent
company shares on the stock market or internal financing needs
may indicate a different cut-off grade policy (typically an even
higher cut-off grade).
Lenders are not concerned with optimal cut-offs, but with debt
service. As a result, they will be in favour of stable cash flows
over the repayment period. However, many factors influellce the
debt required, and the debt is not just incurred at the time when a
mine is first developed.
The mineral owner, most commonly the state in the form of a
regional authority, invariably lays claim to part of the mineral rent
derived from the operation by raising tax revenues. The rents are
captured using various tax instmments (Gamaut, 1983), most of
which are based on either profits, calculated net rent or revenue.
APCOM XXV 1995 Conference
Neighbours
Customers
RegIonal
authority
Fro 5 - Stakeholders in the cut-off grade setting.
For these, government will receive the largest discounted
revenues if an optimal cut-off grade policy is pursued at the mine.
However, some tax instmments are based on the physical mineral
output or on fixed annual payments. For these a cut-off grade
policy which maximises mine life is preferable.
This is not the only problem facing the regional authority. Even
if it is both optimal and rational to favour an optimal cut-off
grade policy, it may serve to exacerbate economic cycles in the
local economy. An optimal cut-off grade policy tends to
concentrate cash flows in a much shorter period and tax revenues
are correspondingly concentrated. This gives large but short-lived
revenue streams to the regional authority, and, since the authority
is likely to be politically unable to resist pressures to spend the
Brisbane. l;l- 14 July 1995
243
K SINDING and E R LARSEN
Concentrate
customers
Local benefits
1
Environmental
pro}
Environmental
problems
Tax revenue
Investment cost
FIG 6 - Complex causal loop diagram offactors influencing cut-off grade policy.
revenue, roughly corresponding bursts in government spending.
This creates two problems in the regional economy. First, boom
and bust cycles associated with the concentrated exploitation of
mines under optimal cut-off grade policies are exacerbated by
high levels of g~vernment spending. Second, once government
expenditures have increased they are much more difficult to
reduce when incomes decline.
Local management of the mining operation is the real target of
the comments by Lane (1988) noted above. However, adoption of
sub-optimal cut-off grades is only a likely scenario when such
managers have a rational expectation of extending their
employment beyond the closure date indicated by an optimal
policy. If the turnover rate for local management is such that
management is replaced over a period shorter than the remaining
mine life then only the last manager has a real incentive to extend
mine life. A better understanding of what motivates local
management can be gained by understanding what their pay-off
from various courses of action will be. As managers they can
influence most or all types of decisions in the operation (and for
that reason management is not represented in the 'causal loop
diagram) in order to maximise their own pay-off in terms of
financial remuneration, corporate advancement opportunities, and
external career possibilities.
Employees do not figure in the causal loop diagram but they
derive benefits from production in the form of local benefits.
More than management, employees have an interest in extending
mine life. The principal way' of doing this is by reducing the
cut-off grade. But the most likely avenue of influence is not
management, but rather the regional authority (which must in that
case balance its preference for maximum discounted tax revenues
with the pressure to retain jobs and with them indirect personal
income tax revenues).
244
A further complication is caused by the relationship between
employees as workers and employees as neighbours of an activity
creating environmental externality effects. Thus, environmental
benefits may be moderated by the level of benefits the local
residents derive from the mining activity (directly due to
employment in production, as well as indirectly through transfers
from the regional authority).
The customers of a mine are the smelters and refiners which
carry out the downstream processing of their output. Relations
between mines and their principal customers are frequently
governed by long-term contracts (Gentry, 1984), even where the
smelting operation is a downstream part of the same company or
group of companies. When external events result in a change in
price, the optimal response may be to increase cut-off grade, and
produce more concentrate. But this extra quantity cannot be sold
within existing contracts, leading either to sales at less attractive
terms (since smelter payments are not only a function of metal
prices) or to sales outside the corporate family.
The neighbours as employees or dependents have already been
noted as having an interest in the way a mine operates. From an
environmental perspective the extent of environmental
disturbance clearly depends both on how long a time the mining
operations go on, and on how extensive they are. An optimal
cut-off grade policy results in a shorter and more concentrated
mining operation, followed by closure. The amount of ore mined
is lower and as a result the quantity of waste generated is also
smaller. While local environmental concerns favour optimal
cut-off grade policies, and local employment favours
maximisation of mine life through lower cut-off gades, another
environmental consideration favours lower than optimal cut.off
grade policies. This is what may be called the materials efficiency
perspective. By extracting more material from each mine fewer
sites will have to be disturbed to produce the same quantity of
metal.
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APCOM XXV 1995 Conference
SYSTEM DYNAMICS APPR0ACH TO MINE MODELLING
CONCLUSIONS
REFERENCES
In this paper we have discussed the concept of optimal cut-off
grade policies and indicated their advantages in terms of
maximi&ation of net present value of mining operations.
Observations have suggested, however, that such optimal policies
are not widely used in practice. In order to explain why this is the
case we have identified a number of stakeholders who have
diverging interests in the operating policies of a given mine.
These various interests have been mapped and a number of
important relationships have been explored in a large qualitative
systems dynamics model.
The complex model presented here opens a number of avenues
for computer based modelling. The initial challenge is to use the
algorithm presented by Dagdelen to create a tool which can be
used as a point of departure for extensions to the model. These
are not only those which involve stockpiling of intermediate
material Jor later processing, or for optimisation with several
minerals, but also the extensions which takes behavioural or
discretionary decisions into account.
Dagdelen, K, 1992. Cutoff grade maximization, in Proceedings 23th
APCOM Symposium (Ed: Y C Kim) pp 157-165 (SME Inc).
Gamaut, R and Ross, A C, 1983. Taxation of Mineral Rents (Oxford:
Clarendon Press).
Gentry, D W and O'Neil, T. 1984. Mine Financial Analysis (SME:
Littleton, CO).
Lane, K F, 1964. Choosing the optimum cuttoff grade, Colorado School
ofMines Quarterly, 59:811-824.
Lane, K F, 1988. The Economic Definition of Ore (Mining Journal
Books: London).
Real, F and Torres Lopes, A, 1994. Setting of planning objectives for
mine management in sUlphides mines, in Mining Planning and
Equipment Selection, June, pp 263-267 (Balkema).
Senge, PM, 1990. The Fifth Dicipline. The Art and Practice of the
Learning OrfJanization (New York: Doubleday).
Taylor, H, 1985. Cutoff grades - s,ome further reflections, Trans Inst Min
Metall (Section A: Mining industry) October:A204-A216.
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