lg_var_slopes - grupo los tigres
TRANSCRIPT
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Synopsis
One of the most important design factors in open-pit
mining is determination of the optimal pit Pits may be
redesigned many times during the life of a mine in
response to changes in design parameters as more
information is obtained and to changes in the values of
technical and economic parameters Over the past 35
years the determination of optimum open-pits has been
one of the most active areas of operational research in
the mining industry and many algorithms have been
published The most common optimizing criterion in
these algorithms is maximization of the overall profit
within the designed pit limits subject to mining (access)constraints
Almost all algorithms use a block model of the ore-
body ie a three-dimensional array of identically sized
blocks that covers the entire orebody and sufficient
surrounding waste to allow access to the deepest ore
blocks Of these the LerchsndashGrossmann algorithm
based on graph theory is the only method that is guar-
anteed always to yield the true optimum pit However
the original algorithm assumes fixed slope angles that
are governed by the block dimensions None of the sub-
sequent attempts to incorporate variable slope angles
provides an adequate solution in cases where there are
variable slopes controlled by complex structures andgeology
A general method of incorporating variable slope
angles in the LerchsndashGrossman algorithm is presented
It is assumed that the orebody and the surrounding
waste are divided into regions or domain sectors within
which the rock characteristics are the same and that
each region is specified by four principal slope anglesmdash
north south east and west face slope angles Slope
angles can vary throughout the deposit to follow the
rock characteristics and are independent of the block
dimensions
The size location and final shape of an open-pit are impor-
tant in the planning of the location of waste dumps
stockpiles processing plant access roads and other surface
facilities and for development of a production programme
The pit design also defines minable reserves and the associ-
ated amount of waste to be removed during the life of the
operation The pit design which is a function of numerous
variables may be re-evaluated many times during the life
of the mine as design technical and economic parameters
change or more information is obtained during operation
The use of computer methods is necessary to redesign the pit
as rapidly as possible and to implement complex algorithmson large block models
With the advent and widespread use of computers a num-
ber of algorithms have been developed to determine optimum
open-pits The main objective of these algorithms almost all
of which are based on block models is to find groups of
blocks that should be removed to yield the maximum overall
profit under specified economic conditions and technological
constraints The most common methods are graph theory
LerchsndashGrossmann method)1 network or maximal flow tech-
niques23 various versions of the floating or moving cone4
the Korobov algorithm5 the corrected form of the Korobov
algorithm6 dynamic programming78 and parameterization
techniques910 Of these the algorithm developed by Lerchs
and Grossmann1 based on graph theory is the only algorithmthat can be proved rigorously always to generate the true
optimum pit limit The original algorithm was however
limited to only one slope angle defined by the block dimen-
sions and was incapable of taking into account variable slope
angles
Many factors govern the size and shape of an open-pit
The pit slope is one of the key factors that govern the amount
of waste to be removed to gain access to ore and it is not
restricted to a constant gradient Small changes in slope angle
can change the amount of waste to be removed and have a
significant effect on the degree of selectivity in mining opera-
tions It is often important to change slope angles either for
geotechnical reasons or to follow different structures and rocktypes in the deposit apart from the overriding need to keep
the total amount of waste as small as possible Any truly opti-
mal pit design algorithm must therefore take into account
variable slope angles Incorporation of variable slope con-
straints into the LerchsndashGrossmann algorithm makes it much
more flexible practical and reliable
Many attempts611ndash13 have been made to overcome the
difficulties of incorporating variable slope angles within
the LerchsndashGrossmann algorithm but none provides an
adequate solution for cases in which variable slopes are
controlled by complex structures and geology Alford and
Whittle14 reported the incorporation of variable pit slopes
into the algorithm but gave no details Whittle also reported
a solution elsewhere15 but again the details were not suffi-
cient to enable objective assessment Lipkewich and
Borgman12 proposed a lsquoknightrsquos moversquo pattern to approxi-
mate a conical expansion to the surface Zhao and Kim13
defined a method based on cone templates Dowd and Onur6
used the idea of cone templates to derive a general technique
to deal with the problem but the algorithm does not always
give the correct solution
LerchsndashGrossmann algorithm
Despite the rigorously optimal nature of the Lerchsndash Grossmann algorithm it suffers from the disadvantages of
complexity of the method long computing times and diffi-
culty in incorporating variable pit slopes The method
converts the revenue block model of a deposit into a directed
graph which is a simple diagram consisting of a set of nodes
or vertices and a set of connecting arcs (lines with direction)
used to indicate the relationship between the vertices Each
A77
LerchsndashGrossmann algorithm with variable slope angles
R Khalokakaie P A Dowd and R J Fowell
Manuscript received by the Institution of Mining and Metallurgy on
10 May 2000 Paper published in Trans Instn Min Metall (Sect A
Min technol) 109 MayndashAugust 2000 copy The Institution of Mining
and Metallurgy 2000
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block is represented by a vertex each vertex is assigned a
mass equal to the net value of the corresponding block
Vertices are connected by arcs in such way as to represent the
mining or access constraints These arcs indicate whichblocks should be removed before a particular block can
be mined Fig 1 shows a directed graph for a simple two-
dimensional example in which the pit slope angle is 45deg and
the blocks are squares In this example to mine block 10 it is
first necessary to remove blocks 2 3 and 4
In graph theory notation the vertices are denoted as xI and
the arc connecting vertices xi and x j is denoted (xi x j ) the
order defining the direction of the arc If the set of all vertices
is denoted X and the set of all arcs is denoted A a graph G =
( X A) is defined as the set of all vertices X together with the
set of all arcs A Vertex x j is said to be the successor of vertex
xi if there exists an arc with its initial extremity in xi and its
terminal extremity in x j The set of all successors of a vertex xi
is denoted G xi For example in Fig 1 the set of all successors
of vertex number 18 is G x18 = x10 x11 x12 x2 x3 x4 x5 x6
A set of vertices constitutes a closure of the graph if the
successors of all vertices in the set also belong to the set ie if
the set of blocks represented by the vertices satisfies access
constraints for all blocks in the set Thus the vertices 2 3 4
and 10 constitute a closure A closure is defined as a subset of
vertices Y Igrave X such that if x Icirc Y G x Icirc Y The value of a clo-
sure is the sum of the masses of the vertices within it The
optimal open-pit is defined by the closure with the maximum
value The algorithm thus involves finding the maximum
closure of the graph that represents the block model of the
orebody
Mining and access constraints
For a deposit represented as a grade or revenue block model
pit slopes are specified in terms of blocks that must be
removed to provide access to each block within the block
model In the LerchsndashGrossmann algorithm directed arcs
impose these restrictions They indicate which blocks should
be removed before a particular block can be mined Consider
for example Fig 1 in which each block has three immediate
successors The immediate successor blocks (vertices) of any
specified block (vertex) must be removed before that blockcan be mined The various procedures used to specify mining
and access constraints for block models can be classified into
(1) non-cone-based methods and (2) cone-based methods
The first category non-cone-based methods involves the
use of a pattern or a set of blocks to define mining slopesmdasha
1 5 block configuration a 1 9 block configuration or a com-
bination of these known as a 1 5 9 pattern
In the original formulation of the LerchsndashGrossmann algo-
rithm the 1 5 block pattern is used to specify mining slopes
In this pattern to gain access to one block five overlying
blocksmdashone up and one over as illustrated in Fig 2(a)mdash
must first be removed This pattern requires the use of
five arcs pointing away from each vertex (block) to satisfy
the mining constraints As indicated by Lipkewich and
Borgman12 if this pattern is carried up over several levels an
undesirable wall slope will be obtained For example in a
cubic block model the average slope angle would approxi-
mate 45ndash55deg The second configuration is a 1 9 block
pattern in which nine overlying blocks must be removed to
mine one block (Fig 2(b)) This approximation to slopes pro-
duces a cone with slopes ranging from 35 to 45deg in a cubic
block model A close approximation to a 45deg slope in the
cubic block model is obtained by combining a 1 5 block pat-
tern for the first level above the base block with a 1 9 pattern
for the second level The use of this 1 5 9 pattern in the
LerchsndashGrossmann algorithm is exemplified in a previously
published program16
One of the main disadvantages of use of the first category is
the difficulty of establishing optimum pit outlines with vari-
able slope angles The slope angles are assumed to be defined
by the dimensions of the blocks For example if a 1 5 9
pattern is used in the general rectangular revenue blockmodel of an orebody with 10 m acute 10 m acute 5 m blocks slope
angles of 25deg would be obtained Thus when this procedure
is used different slope angles will require different sizes for
the blocks in the orebody block model but these may not
correspond to the required bench heights The grades of
blocks of different sizes estimated from a given configuration
of data would have different estimation errors and thus cre-
A78
Fig 1 Directed graph representing vertical section
Fig 2 Non-cone-based patterns (a) 1 5 five overlying blocks
must be removed to mine one (b) 1 9 nine overlying blocks must
be removed to mine one
(a)
(b)
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ate difficulties in assessment of the reliability and confidence
levels associated with the final pit values (the optimal pit is
commonly used to define minable reserves with stated levels
of confidence) In addition different parts of the orebody
may require different slope angles It is impossible in this
method to have different angles for different parts of the pit
In cone-based methods a cone is used in a variety of ways
to define the mining slope1213 Dowd and Onur6 used the
idea of cone templates to derive a technique to establish the
optimum pit with variable slope angles This method involvesconstruction of a cone or extraction volume from the block
on a given level to the surface by joining rings or envelopes of
blocks corresponding to the pit slope angles If the mid-point
of any block l ies within the extraction cone it is assumed that
it must be removed before removal of the base block
However the algorithm that was developed does not give the
correct solution in all cases In the work presented here the
method of Dowd and Onur6 has been modified to derive a
general technique for variable slope angles This procedure is
incorporated in the LerchsndashGrossmann algorithm given later
LerchsndashGrossmann algorithm with variable
slope angles
To incorporate variable slope angles in the Lerchsndash
Grossmann algorithm it is assumed that the orebody and sur-
rounding waste have been divided into regions or domains on
the basis of the geotechnical information It is further
assumed that within each region or domain the rock charac-
teristics are the same and can be characterized by a set of
slopes and that each region can be approximated by a poly-
gon Depending on the number of regions the problem is
treated in one of two ways (1) variable slope angles in which
only one region or domain sector is specified to define the pit
slopes and (2) multiple variable slope angles in which more
than one region or domain sector is specified to define the pitslopes For each region or domain sector pit slopes are
assumed to be defined by four principal slope angles in four
principal directions a north face slope east face slope south
face slope and west face slope
Two types of coordinate system as illustrated in Fig 3 are
used The first is the X Y Z Cartesian system in which the X -axis runs westndasheast the Y -axis runs southndashnorth and the
Z -axis is vertical The origin of the system is located in the
southwest of the uppermost level the shaded block shown in
Fig 3 The second system is an i j k coordinate index system
The i j and k coordinates increase along the line of increasing
X Y and Z coordinates respectively In addition the follow-
ing parameters are used to define the block model for the
deposit x dim block dimension in the x direction (westndash
east) y dim block dimension in the y direction (southndash
north) z dim block dimension in the z direction (vertical)
num x number of blocks in the x direction (westndasheast)
num y number of blocks in the y direction (southndashnorth)
and num z number of blocks in the z direction or number of
levels
Variable slope angles
Pit slopes can be approximated by constructing a cone thatrepresents an extraction volume This can be done by creat-
ing rings or envelopes from the mid-point of the base block
and extending them to the surface (Fig 4) in such a way that
the side angles of the cone are equal to the four principal
slope angles
If the pit wall slopes in the four principal directions are not
the same the upper area of the cone on each level (intersec-
tion of the cone with the level) will consist of four quadrants
of different ellipses If the pit wall angles are the same the
upper area of the cone will be a circle Fig 5 shows the
extraction cone and the blocks within it on the first level and
on the two cross-sections On each level the values of the two
semi-major axes and two semi-minor axes depend on the four
principal slope angles and the vertical distance of the mid-
point of the base block from the overlying blocks These
parameters can be found by use of trigonometric functions
The number of blocks in the principal directions on any level
above the base block can be calculated by dividing these para-
meters by the corresponding block dimensions Consider a
block X i j k on level k the parameters and the numbers of
blocks in the principal directions as illustrated in Fig 5 can
be calculated from the equations
(1)
(2)dyk t z
1 = -( ) acute dim
tan (south face angle)
dx
k t z1 =
-( ) acute dim
tan (west face angle)
A79
Fig 3 Block model of deposit and coordinate systems
Fig 4 Construction of cone from base block
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(3)
(4)
(5)
(6)
(7)
(8)
where t is level above the base block and varies from 1 to k ndash1
m1 is number of blocks from the base block to the east n1 is
number of blocks from the base block to the north m2 is
number of blocks from the base block to the west and n2 is
number of blocks from the base block to the south
When the numbers of blocks within the upper area of the
cone on any levelmdashsay the t th level above the base blockmdash
are calculated in the four principal directions all the blocks
X mnk ndash t where m = i ndash m2 i + m1 and n = j ndash n2 j + n1 must
be examined to determine whether they are within the extrac-tion volume This can be done by use of the ellipse equation
a = xdim acute (indashm) (9)
b = ydim acute ( jndashn) (10)
If m is equal to or greater than i and n is equal to or greater
than j
(11)
If m is equal to or greater than i and n is less than or equal to j
(12)
If m is less than or equal to i and n is less than or equal to j
(13)
If m is less than or equal to i and n is equal to or greater than j
(14)
where a and b are the horizontal distances from the mid-point
of the block under consideration to the base block measured
in the westndasheast and southndashnorth directions respectively as
illustrated in Fig 6 If the lsquovaluersquo is less than or equal to 1 it
is assumed that the block is within the extraction cone and it
must be removed before the base block Otherwise it is
assumed that the block is outside the extraction cone Blocks
that lie within the extraction cone are submitted to the graph
algorithm The program was written in such a way that
extraction cones are established only for ore blocks This pre-
vents unnecessary increases in computing time and prevents
waste blocks from being considered several times
With this procedure pit slopes are no longer fixed and are
not limited to one-up and one-over patterns They can vary in
Value =
( )+
( )
a
dx
b
dy
2
22
2
12
Value =
( )+
( )
a
dx
b
dy
2
22
2
22
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
2
2
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
1
2
ndy
y2
2=dim
m
dx
x2
2=dim
n
dy
y1
1=dim
m
dx
x1
1=dim
dy
k t z2 =
-( ) acute dim
tan (north face angle)
dx
k t z2 =
-( ) acute dim
tan(east face angle)
A80
Fig 5 Extraction cone of base blocks showing all blocks within
cone (a) upper area of cone on first level (b) northndashsouth section
A ndash A (c) eastndashwest section B ndash B
(b)
(c)
(a)
Fig 6 Value of parameters a and b
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the principal directions and are independent of block dimen-
sions Fig 7 illustrates a directed graph that represents a
northndashsouth section in a cubic block model in which the east
face angle and the west face angle are assumed to be 60deg and
45deg respectively In this graph vertices 4 5 6 7 14 and 15
are in the extraction cone of block 23
Multiple variable slope angles
In complex cases in which the pit slopes vary in different parts
of the orebody on account of slope stability requirements it is
necessary to divide the orebody into regions or domain sec-
tors within which the rock characteristics are the same and to
use different slope angles for each region In these cases slope
angles are assigned to each block in the four principal direc-
tions within each region this is discussed later
In the case of multiple variable slopes an extraction volume
is constructed level by level by creating rings or envelopes
from the base block and extending them to the surface with
regard for pit slopes that have already been assigned toblocks The extraction volume is constructed from the base
block to the next overlying block and is then constructed
from the point of intersection of the cone with this level to the
second level above the base block This procedure is conti-
nued to the surface (Fig 8)
Consider the construction of an extraction cone in the two-
dimensional case (northndashsouth section) shown in Fig 8 Fora block on level k (row) and column j lines are drawn from
the mid-point of the base block to the left and right with
slopes equal to the east and west face angles of the base block
respectively and the lines are then extended to the level
above The values of parameters dx11 dx2
1 and the number of
blocks to the east m11 and to the west m2
1 on the first level
above the base block are determined by the equations
(15)
(16)
(17)
(18)
where z dim and x dim are the block dimensions in the verti-
cal and horizontal directions respectively On the first level
above the blocks X k ndash1m where m = j ndash m21 j + m1
1 are consid-
ered as part of the extraction cone
There are two intersection points of the extraction cone
with the level above the base block (lines drawn from the
mid-point of the base block to the next overlying level) Theextraction cone is extended from these two points to the next
overlying block (second level above the base block) by using
the slope angles of the blocks that contain the points of inter-
section To determine the slope of the block to be used (in
other words to find the block in which the intersection lies)
the values of the parameters dx11 and dx2
1 are divided by the
block dimension and the result is rounded up This means
that a value of 05 is added to the result of division and then
the integer part is taken ie
(19)
(20)
The values of the parameters dx11 and dx2
1 and the number of
blocks in both directions m12 and m2
2 on the second level
above the base block are determined as
(21)
(22)
(23)
(24)
Again blocks X k ndash2m where m = j ndash m22 j + m1
2 are consid-
ered as part of the extraction cone on the second level above
the base block This procedure is continued to the surface
The procedure presented for multiple variable slopes in
two dimensions can be applied to the three-dimensional case
As with the procedure used for variable slope angles the pit
shape is assumed to be defined by an irregular elliptical out-
m
dx
x22 2
2
=dim
m
dx
x12 1
2
=dim
dx dxz
k j ml 22
21
21
1= +
[ ]dim
tan east face angle of block ( - - )
dx dxz
k j ml 12
11
11
1= +
[ ]dim
tan west face angle of block ( - + )
ml
dx
x21 2
1
0 5= +dim
ml
dx
x11 1
1
0 5= +dim
m
dx
x21 2
1
-dim
m
dx
x11 1
1
= dim
dxz
k j 21 =
[ ]dim
tan east face angle of block ( )
dxz
k j 11 =
[ ]dim
tan west face angle of block ( )
A81
Fig 7 Directed graph representing northndashsouth cross-section in
cubic block model with east face angle of 60deg and west face angle of
45deg
Fig 8 Extraction cone of block for two-dimensional example with
three different regions
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line on each level The outline on each level consists of four
quadrants of different ellipses defined by the pit slope angles
in the four directions The values of the two semi-major axes
two semi-minor axes and the number of blocks in the prin-
cipal directions on any level above the base block should be
calculated in both sections in the same way as described
previously for one section When these parameters are deter-
mined again by use of the ellipse formula any block whose
mid-point lies inside the ellipse is considered to be part of the
cone The values of the four axes and the number of blocks inthe four directions for the t th level above the base block
(block X i j k) can be found from the equations
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
When the numbers of blocks in the four principal direc-
tions (m11 n1
1 m21 and n2
1) have been calculated the blocks are
examined according to the procedure described for variable
slope angles to determine whether they are within the extrac-
tion volume
Assigning slope angles to blocks
If more than one region or domain sector is specified to
define mining slopes it is necessary to assign slope angles to
each block To assign slope angles to the blocks the first step
is to determine which blocks are inside the particular region
A block is deemed to be inside a region if its mid-point lies
within that region Blocks deemed to be within a given region
have the slopes of that region assigned to them Different
methods can be used to determine whether a point is inside
outside or on the boundary of a polygon The approach
adopted here is the angle sum method based on coding origi-
nally written by Dowd17 In this method lines are drawn from
the point in question to each of the vertices that define the
boundary of the polygon and the angles between each succes-sive pair of lines are then summed Angles measured in the
clockwise direction are positive and those in the anticlockwise
direction are negative The point is inside the polygon if the
sum of the angles is 2p radians and outside if the sum is zero
The basic requirement for this method is the signed angle
between pairs of lines from the point to successive pairs of
vertices that define the boundary of the polygon These
angles can be determined by use of either the dot product of
two vectors or the equation of a triangle The signs of angles
can also be determined from the cross-product of two
vectors
Triangle equation
a2 = b2 + c2 ndash 2bc cos q
Dot product of two vectors
v1v2 = ccedilv1ccedilccedilv2ccedil cos q
Cross-product of two vectors
v1acutev2 = ccedilv1ccedilccedilv2ccedil sin q
A block whose mid-point lies within a slope region isassigned the slopes of that region The method is imple-
mented by first imposing a bounding box around the
regionmdashthis is the smallest rectangle that contains the region
or polygon Then the mid-points of all blocks in the first level
of the region that are inside the bounding box are examined
to see whether or not they are inside the region If the mid-
point of any block lies inside the region slope angles are
n
dy
y
t t
22=
dim
mdx
x
t t
22=
dim
n
dy
y
t t
11=
dim
m
dx
x
t t
11=
dim
dy dy
z
i j nl k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan north face angle of block ( - )2-1
dx dx
z
i ml j k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan east face angle of block ( - )2-1
dy dy
z
i j nl k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan south face angle of block ( + )1
-1
dx dx
z
i ml j k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan west face angle of block ( + )1
-1
nl
dy
y
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
nl
dy
y
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
A82
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assigned to all blocks at this location from the minimum to
the maximum depth of the region
Case study
The method of variable slope angles has been incorporated
into the LerchsndashGrossmann algorithm and has been coded
into an interactive Windows software package18 Data from a
real orebody were used to illustrate and test the application of
the software in the determination of optimum open-pit limitsThe data are from the Bjoumlrkdal low-grade gold deposit
located approximately 35 km northwest of Skelleftearing in the
north of Sweden19
Gold mineralization in the Bjoumlrkdal area occurs within
a network of steeply dipping quartz veins in the contact
between older granodiorite and limestoneacid volcanic rocks
The gold is erratically distributed but is mainly concentrated
in and around high-grade quartz veins It occurs as both fine
and coarse grains and is free-milling
The block grade model of the deposit contains 101 acute 82 acute36 blocks in the eastndashwest northndashsouth and vertical direc-
tions respectively Each block is assigned the estimated
(kriged) recoverable tonnage of ore above a cutoff grade andthe estimated (kriged) average grade of this tonnage The
method of estimation has been detailed elsewhere19 The
deposit is divided into 15 m (eastndashwest) acute 10 m (northndashsouth)
acute 5 m (vertical) blocks and the recoverable tonnage is based
on a selective mining unit of 5 m (eastndashwest) acute 4 m
(northndashsouth) acute 5 m (vertical) The physical and economic
parameters for this case are specific gravity of ore and waste
271 tm3 cost of mining of ore and wastemdashas given Table 1
processing cost SEK 52tonne of ore price of gold
SEK 90g and recovery 91
The slope regions and associated principal slope angles are
shown in Fig 9 and Table 2 The slopes used here are solely
for the sake of example and do not necessarily correspond to
actual slopes The overall results of pit optimization are
shown in Table 3 Two cross-sections through the optimal pit
are shown in Fig 10 The application of the software to the
case study has enabled a much more realistic pit design that is
able to accommodate real slope angles within a traditional
block model for a complex low-grade gold orebody
A83
Fig 9 Deposit and surrounding waste subdivided into four geo-
technical regions
Table 1 Cost of mining of ore and waste
Level m Cost of mining SEKt
From To Waste Ore
0 120 110 110
120 130 1130 1130
130 140 1160 1160
140 150 1190 1190
150 160 1220 1220
160 170 1250 1250
170 180 1280 1280
180 200 1320 1320
Fig 10 Cross-sections at (a) easting 150 m and (b) northing 120 m
Table 2 Slope angles applied to example shown in Fig 9
Region North face East face South face West face
1 30deg 40deg 42deg 38deg
2 41deg 37deg 50deg 46deg
3 35deg 35deg 35deg 35deg
4 39deg 39deg 46deg 46deg
7262019 Lg_var_slopes - Grupo Los Tigres
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Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
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13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
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block is represented by a vertex each vertex is assigned a
mass equal to the net value of the corresponding block
Vertices are connected by arcs in such way as to represent the
mining or access constraints These arcs indicate whichblocks should be removed before a particular block can
be mined Fig 1 shows a directed graph for a simple two-
dimensional example in which the pit slope angle is 45deg and
the blocks are squares In this example to mine block 10 it is
first necessary to remove blocks 2 3 and 4
In graph theory notation the vertices are denoted as xI and
the arc connecting vertices xi and x j is denoted (xi x j ) the
order defining the direction of the arc If the set of all vertices
is denoted X and the set of all arcs is denoted A a graph G =
( X A) is defined as the set of all vertices X together with the
set of all arcs A Vertex x j is said to be the successor of vertex
xi if there exists an arc with its initial extremity in xi and its
terminal extremity in x j The set of all successors of a vertex xi
is denoted G xi For example in Fig 1 the set of all successors
of vertex number 18 is G x18 = x10 x11 x12 x2 x3 x4 x5 x6
A set of vertices constitutes a closure of the graph if the
successors of all vertices in the set also belong to the set ie if
the set of blocks represented by the vertices satisfies access
constraints for all blocks in the set Thus the vertices 2 3 4
and 10 constitute a closure A closure is defined as a subset of
vertices Y Igrave X such that if x Icirc Y G x Icirc Y The value of a clo-
sure is the sum of the masses of the vertices within it The
optimal open-pit is defined by the closure with the maximum
value The algorithm thus involves finding the maximum
closure of the graph that represents the block model of the
orebody
Mining and access constraints
For a deposit represented as a grade or revenue block model
pit slopes are specified in terms of blocks that must be
removed to provide access to each block within the block
model In the LerchsndashGrossmann algorithm directed arcs
impose these restrictions They indicate which blocks should
be removed before a particular block can be mined Consider
for example Fig 1 in which each block has three immediate
successors The immediate successor blocks (vertices) of any
specified block (vertex) must be removed before that blockcan be mined The various procedures used to specify mining
and access constraints for block models can be classified into
(1) non-cone-based methods and (2) cone-based methods
The first category non-cone-based methods involves the
use of a pattern or a set of blocks to define mining slopesmdasha
1 5 block configuration a 1 9 block configuration or a com-
bination of these known as a 1 5 9 pattern
In the original formulation of the LerchsndashGrossmann algo-
rithm the 1 5 block pattern is used to specify mining slopes
In this pattern to gain access to one block five overlying
blocksmdashone up and one over as illustrated in Fig 2(a)mdash
must first be removed This pattern requires the use of
five arcs pointing away from each vertex (block) to satisfy
the mining constraints As indicated by Lipkewich and
Borgman12 if this pattern is carried up over several levels an
undesirable wall slope will be obtained For example in a
cubic block model the average slope angle would approxi-
mate 45ndash55deg The second configuration is a 1 9 block
pattern in which nine overlying blocks must be removed to
mine one block (Fig 2(b)) This approximation to slopes pro-
duces a cone with slopes ranging from 35 to 45deg in a cubic
block model A close approximation to a 45deg slope in the
cubic block model is obtained by combining a 1 5 block pat-
tern for the first level above the base block with a 1 9 pattern
for the second level The use of this 1 5 9 pattern in the
LerchsndashGrossmann algorithm is exemplified in a previously
published program16
One of the main disadvantages of use of the first category is
the difficulty of establishing optimum pit outlines with vari-
able slope angles The slope angles are assumed to be defined
by the dimensions of the blocks For example if a 1 5 9
pattern is used in the general rectangular revenue blockmodel of an orebody with 10 m acute 10 m acute 5 m blocks slope
angles of 25deg would be obtained Thus when this procedure
is used different slope angles will require different sizes for
the blocks in the orebody block model but these may not
correspond to the required bench heights The grades of
blocks of different sizes estimated from a given configuration
of data would have different estimation errors and thus cre-
A78
Fig 1 Directed graph representing vertical section
Fig 2 Non-cone-based patterns (a) 1 5 five overlying blocks
must be removed to mine one (b) 1 9 nine overlying blocks must
be removed to mine one
(a)
(b)
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ate difficulties in assessment of the reliability and confidence
levels associated with the final pit values (the optimal pit is
commonly used to define minable reserves with stated levels
of confidence) In addition different parts of the orebody
may require different slope angles It is impossible in this
method to have different angles for different parts of the pit
In cone-based methods a cone is used in a variety of ways
to define the mining slope1213 Dowd and Onur6 used the
idea of cone templates to derive a technique to establish the
optimum pit with variable slope angles This method involvesconstruction of a cone or extraction volume from the block
on a given level to the surface by joining rings or envelopes of
blocks corresponding to the pit slope angles If the mid-point
of any block l ies within the extraction cone it is assumed that
it must be removed before removal of the base block
However the algorithm that was developed does not give the
correct solution in all cases In the work presented here the
method of Dowd and Onur6 has been modified to derive a
general technique for variable slope angles This procedure is
incorporated in the LerchsndashGrossmann algorithm given later
LerchsndashGrossmann algorithm with variable
slope angles
To incorporate variable slope angles in the Lerchsndash
Grossmann algorithm it is assumed that the orebody and sur-
rounding waste have been divided into regions or domains on
the basis of the geotechnical information It is further
assumed that within each region or domain the rock charac-
teristics are the same and can be characterized by a set of
slopes and that each region can be approximated by a poly-
gon Depending on the number of regions the problem is
treated in one of two ways (1) variable slope angles in which
only one region or domain sector is specified to define the pit
slopes and (2) multiple variable slope angles in which more
than one region or domain sector is specified to define the pitslopes For each region or domain sector pit slopes are
assumed to be defined by four principal slope angles in four
principal directions a north face slope east face slope south
face slope and west face slope
Two types of coordinate system as illustrated in Fig 3 are
used The first is the X Y Z Cartesian system in which the X -axis runs westndasheast the Y -axis runs southndashnorth and the
Z -axis is vertical The origin of the system is located in the
southwest of the uppermost level the shaded block shown in
Fig 3 The second system is an i j k coordinate index system
The i j and k coordinates increase along the line of increasing
X Y and Z coordinates respectively In addition the follow-
ing parameters are used to define the block model for the
deposit x dim block dimension in the x direction (westndash
east) y dim block dimension in the y direction (southndash
north) z dim block dimension in the z direction (vertical)
num x number of blocks in the x direction (westndasheast)
num y number of blocks in the y direction (southndashnorth)
and num z number of blocks in the z direction or number of
levels
Variable slope angles
Pit slopes can be approximated by constructing a cone thatrepresents an extraction volume This can be done by creat-
ing rings or envelopes from the mid-point of the base block
and extending them to the surface (Fig 4) in such a way that
the side angles of the cone are equal to the four principal
slope angles
If the pit wall slopes in the four principal directions are not
the same the upper area of the cone on each level (intersec-
tion of the cone with the level) will consist of four quadrants
of different ellipses If the pit wall angles are the same the
upper area of the cone will be a circle Fig 5 shows the
extraction cone and the blocks within it on the first level and
on the two cross-sections On each level the values of the two
semi-major axes and two semi-minor axes depend on the four
principal slope angles and the vertical distance of the mid-
point of the base block from the overlying blocks These
parameters can be found by use of trigonometric functions
The number of blocks in the principal directions on any level
above the base block can be calculated by dividing these para-
meters by the corresponding block dimensions Consider a
block X i j k on level k the parameters and the numbers of
blocks in the principal directions as illustrated in Fig 5 can
be calculated from the equations
(1)
(2)dyk t z
1 = -( ) acute dim
tan (south face angle)
dx
k t z1 =
-( ) acute dim
tan (west face angle)
A79
Fig 3 Block model of deposit and coordinate systems
Fig 4 Construction of cone from base block
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(3)
(4)
(5)
(6)
(7)
(8)
where t is level above the base block and varies from 1 to k ndash1
m1 is number of blocks from the base block to the east n1 is
number of blocks from the base block to the north m2 is
number of blocks from the base block to the west and n2 is
number of blocks from the base block to the south
When the numbers of blocks within the upper area of the
cone on any levelmdashsay the t th level above the base blockmdash
are calculated in the four principal directions all the blocks
X mnk ndash t where m = i ndash m2 i + m1 and n = j ndash n2 j + n1 must
be examined to determine whether they are within the extrac-tion volume This can be done by use of the ellipse equation
a = xdim acute (indashm) (9)
b = ydim acute ( jndashn) (10)
If m is equal to or greater than i and n is equal to or greater
than j
(11)
If m is equal to or greater than i and n is less than or equal to j
(12)
If m is less than or equal to i and n is less than or equal to j
(13)
If m is less than or equal to i and n is equal to or greater than j
(14)
where a and b are the horizontal distances from the mid-point
of the block under consideration to the base block measured
in the westndasheast and southndashnorth directions respectively as
illustrated in Fig 6 If the lsquovaluersquo is less than or equal to 1 it
is assumed that the block is within the extraction cone and it
must be removed before the base block Otherwise it is
assumed that the block is outside the extraction cone Blocks
that lie within the extraction cone are submitted to the graph
algorithm The program was written in such a way that
extraction cones are established only for ore blocks This pre-
vents unnecessary increases in computing time and prevents
waste blocks from being considered several times
With this procedure pit slopes are no longer fixed and are
not limited to one-up and one-over patterns They can vary in
Value =
( )+
( )
a
dx
b
dy
2
22
2
12
Value =
( )+
( )
a
dx
b
dy
2
22
2
22
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
2
2
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
1
2
ndy
y2
2=dim
m
dx
x2
2=dim
n
dy
y1
1=dim
m
dx
x1
1=dim
dy
k t z2 =
-( ) acute dim
tan (north face angle)
dx
k t z2 =
-( ) acute dim
tan(east face angle)
A80
Fig 5 Extraction cone of base blocks showing all blocks within
cone (a) upper area of cone on first level (b) northndashsouth section
A ndash A (c) eastndashwest section B ndash B
(b)
(c)
(a)
Fig 6 Value of parameters a and b
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the principal directions and are independent of block dimen-
sions Fig 7 illustrates a directed graph that represents a
northndashsouth section in a cubic block model in which the east
face angle and the west face angle are assumed to be 60deg and
45deg respectively In this graph vertices 4 5 6 7 14 and 15
are in the extraction cone of block 23
Multiple variable slope angles
In complex cases in which the pit slopes vary in different parts
of the orebody on account of slope stability requirements it is
necessary to divide the orebody into regions or domain sec-
tors within which the rock characteristics are the same and to
use different slope angles for each region In these cases slope
angles are assigned to each block in the four principal direc-
tions within each region this is discussed later
In the case of multiple variable slopes an extraction volume
is constructed level by level by creating rings or envelopes
from the base block and extending them to the surface with
regard for pit slopes that have already been assigned toblocks The extraction volume is constructed from the base
block to the next overlying block and is then constructed
from the point of intersection of the cone with this level to the
second level above the base block This procedure is conti-
nued to the surface (Fig 8)
Consider the construction of an extraction cone in the two-
dimensional case (northndashsouth section) shown in Fig 8 Fora block on level k (row) and column j lines are drawn from
the mid-point of the base block to the left and right with
slopes equal to the east and west face angles of the base block
respectively and the lines are then extended to the level
above The values of parameters dx11 dx2
1 and the number of
blocks to the east m11 and to the west m2
1 on the first level
above the base block are determined by the equations
(15)
(16)
(17)
(18)
where z dim and x dim are the block dimensions in the verti-
cal and horizontal directions respectively On the first level
above the blocks X k ndash1m where m = j ndash m21 j + m1
1 are consid-
ered as part of the extraction cone
There are two intersection points of the extraction cone
with the level above the base block (lines drawn from the
mid-point of the base block to the next overlying level) Theextraction cone is extended from these two points to the next
overlying block (second level above the base block) by using
the slope angles of the blocks that contain the points of inter-
section To determine the slope of the block to be used (in
other words to find the block in which the intersection lies)
the values of the parameters dx11 and dx2
1 are divided by the
block dimension and the result is rounded up This means
that a value of 05 is added to the result of division and then
the integer part is taken ie
(19)
(20)
The values of the parameters dx11 and dx2
1 and the number of
blocks in both directions m12 and m2
2 on the second level
above the base block are determined as
(21)
(22)
(23)
(24)
Again blocks X k ndash2m where m = j ndash m22 j + m1
2 are consid-
ered as part of the extraction cone on the second level above
the base block This procedure is continued to the surface
The procedure presented for multiple variable slopes in
two dimensions can be applied to the three-dimensional case
As with the procedure used for variable slope angles the pit
shape is assumed to be defined by an irregular elliptical out-
m
dx
x22 2
2
=dim
m
dx
x12 1
2
=dim
dx dxz
k j ml 22
21
21
1= +
[ ]dim
tan east face angle of block ( - - )
dx dxz
k j ml 12
11
11
1= +
[ ]dim
tan west face angle of block ( - + )
ml
dx
x21 2
1
0 5= +dim
ml
dx
x11 1
1
0 5= +dim
m
dx
x21 2
1
-dim
m
dx
x11 1
1
= dim
dxz
k j 21 =
[ ]dim
tan east face angle of block ( )
dxz
k j 11 =
[ ]dim
tan west face angle of block ( )
A81
Fig 7 Directed graph representing northndashsouth cross-section in
cubic block model with east face angle of 60deg and west face angle of
45deg
Fig 8 Extraction cone of block for two-dimensional example with
three different regions
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line on each level The outline on each level consists of four
quadrants of different ellipses defined by the pit slope angles
in the four directions The values of the two semi-major axes
two semi-minor axes and the number of blocks in the prin-
cipal directions on any level above the base block should be
calculated in both sections in the same way as described
previously for one section When these parameters are deter-
mined again by use of the ellipse formula any block whose
mid-point lies inside the ellipse is considered to be part of the
cone The values of the four axes and the number of blocks inthe four directions for the t th level above the base block
(block X i j k) can be found from the equations
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
When the numbers of blocks in the four principal direc-
tions (m11 n1
1 m21 and n2
1) have been calculated the blocks are
examined according to the procedure described for variable
slope angles to determine whether they are within the extrac-
tion volume
Assigning slope angles to blocks
If more than one region or domain sector is specified to
define mining slopes it is necessary to assign slope angles to
each block To assign slope angles to the blocks the first step
is to determine which blocks are inside the particular region
A block is deemed to be inside a region if its mid-point lies
within that region Blocks deemed to be within a given region
have the slopes of that region assigned to them Different
methods can be used to determine whether a point is inside
outside or on the boundary of a polygon The approach
adopted here is the angle sum method based on coding origi-
nally written by Dowd17 In this method lines are drawn from
the point in question to each of the vertices that define the
boundary of the polygon and the angles between each succes-sive pair of lines are then summed Angles measured in the
clockwise direction are positive and those in the anticlockwise
direction are negative The point is inside the polygon if the
sum of the angles is 2p radians and outside if the sum is zero
The basic requirement for this method is the signed angle
between pairs of lines from the point to successive pairs of
vertices that define the boundary of the polygon These
angles can be determined by use of either the dot product of
two vectors or the equation of a triangle The signs of angles
can also be determined from the cross-product of two
vectors
Triangle equation
a2 = b2 + c2 ndash 2bc cos q
Dot product of two vectors
v1v2 = ccedilv1ccedilccedilv2ccedil cos q
Cross-product of two vectors
v1acutev2 = ccedilv1ccedilccedilv2ccedil sin q
A block whose mid-point lies within a slope region isassigned the slopes of that region The method is imple-
mented by first imposing a bounding box around the
regionmdashthis is the smallest rectangle that contains the region
or polygon Then the mid-points of all blocks in the first level
of the region that are inside the bounding box are examined
to see whether or not they are inside the region If the mid-
point of any block lies inside the region slope angles are
n
dy
y
t t
22=
dim
mdx
x
t t
22=
dim
n
dy
y
t t
11=
dim
m
dx
x
t t
11=
dim
dy dy
z
i j nl k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan north face angle of block ( - )2-1
dx dx
z
i ml j k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan east face angle of block ( - )2-1
dy dy
z
i j nl k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan south face angle of block ( + )1
-1
dx dx
z
i ml j k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan west face angle of block ( + )1
-1
nl
dy
y
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
nl
dy
y
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
A82
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assigned to all blocks at this location from the minimum to
the maximum depth of the region
Case study
The method of variable slope angles has been incorporated
into the LerchsndashGrossmann algorithm and has been coded
into an interactive Windows software package18 Data from a
real orebody were used to illustrate and test the application of
the software in the determination of optimum open-pit limitsThe data are from the Bjoumlrkdal low-grade gold deposit
located approximately 35 km northwest of Skelleftearing in the
north of Sweden19
Gold mineralization in the Bjoumlrkdal area occurs within
a network of steeply dipping quartz veins in the contact
between older granodiorite and limestoneacid volcanic rocks
The gold is erratically distributed but is mainly concentrated
in and around high-grade quartz veins It occurs as both fine
and coarse grains and is free-milling
The block grade model of the deposit contains 101 acute 82 acute36 blocks in the eastndashwest northndashsouth and vertical direc-
tions respectively Each block is assigned the estimated
(kriged) recoverable tonnage of ore above a cutoff grade andthe estimated (kriged) average grade of this tonnage The
method of estimation has been detailed elsewhere19 The
deposit is divided into 15 m (eastndashwest) acute 10 m (northndashsouth)
acute 5 m (vertical) blocks and the recoverable tonnage is based
on a selective mining unit of 5 m (eastndashwest) acute 4 m
(northndashsouth) acute 5 m (vertical) The physical and economic
parameters for this case are specific gravity of ore and waste
271 tm3 cost of mining of ore and wastemdashas given Table 1
processing cost SEK 52tonne of ore price of gold
SEK 90g and recovery 91
The slope regions and associated principal slope angles are
shown in Fig 9 and Table 2 The slopes used here are solely
for the sake of example and do not necessarily correspond to
actual slopes The overall results of pit optimization are
shown in Table 3 Two cross-sections through the optimal pit
are shown in Fig 10 The application of the software to the
case study has enabled a much more realistic pit design that is
able to accommodate real slope angles within a traditional
block model for a complex low-grade gold orebody
A83
Fig 9 Deposit and surrounding waste subdivided into four geo-
technical regions
Table 1 Cost of mining of ore and waste
Level m Cost of mining SEKt
From To Waste Ore
0 120 110 110
120 130 1130 1130
130 140 1160 1160
140 150 1190 1190
150 160 1220 1220
160 170 1250 1250
170 180 1280 1280
180 200 1320 1320
Fig 10 Cross-sections at (a) easting 150 m and (b) northing 120 m
Table 2 Slope angles applied to example shown in Fig 9
Region North face East face South face West face
1 30deg 40deg 42deg 38deg
2 41deg 37deg 50deg 46deg
3 35deg 35deg 35deg 35deg
4 39deg 39deg 46deg 46deg
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Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
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13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
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ate difficulties in assessment of the reliability and confidence
levels associated with the final pit values (the optimal pit is
commonly used to define minable reserves with stated levels
of confidence) In addition different parts of the orebody
may require different slope angles It is impossible in this
method to have different angles for different parts of the pit
In cone-based methods a cone is used in a variety of ways
to define the mining slope1213 Dowd and Onur6 used the
idea of cone templates to derive a technique to establish the
optimum pit with variable slope angles This method involvesconstruction of a cone or extraction volume from the block
on a given level to the surface by joining rings or envelopes of
blocks corresponding to the pit slope angles If the mid-point
of any block l ies within the extraction cone it is assumed that
it must be removed before removal of the base block
However the algorithm that was developed does not give the
correct solution in all cases In the work presented here the
method of Dowd and Onur6 has been modified to derive a
general technique for variable slope angles This procedure is
incorporated in the LerchsndashGrossmann algorithm given later
LerchsndashGrossmann algorithm with variable
slope angles
To incorporate variable slope angles in the Lerchsndash
Grossmann algorithm it is assumed that the orebody and sur-
rounding waste have been divided into regions or domains on
the basis of the geotechnical information It is further
assumed that within each region or domain the rock charac-
teristics are the same and can be characterized by a set of
slopes and that each region can be approximated by a poly-
gon Depending on the number of regions the problem is
treated in one of two ways (1) variable slope angles in which
only one region or domain sector is specified to define the pit
slopes and (2) multiple variable slope angles in which more
than one region or domain sector is specified to define the pitslopes For each region or domain sector pit slopes are
assumed to be defined by four principal slope angles in four
principal directions a north face slope east face slope south
face slope and west face slope
Two types of coordinate system as illustrated in Fig 3 are
used The first is the X Y Z Cartesian system in which the X -axis runs westndasheast the Y -axis runs southndashnorth and the
Z -axis is vertical The origin of the system is located in the
southwest of the uppermost level the shaded block shown in
Fig 3 The second system is an i j k coordinate index system
The i j and k coordinates increase along the line of increasing
X Y and Z coordinates respectively In addition the follow-
ing parameters are used to define the block model for the
deposit x dim block dimension in the x direction (westndash
east) y dim block dimension in the y direction (southndash
north) z dim block dimension in the z direction (vertical)
num x number of blocks in the x direction (westndasheast)
num y number of blocks in the y direction (southndashnorth)
and num z number of blocks in the z direction or number of
levels
Variable slope angles
Pit slopes can be approximated by constructing a cone thatrepresents an extraction volume This can be done by creat-
ing rings or envelopes from the mid-point of the base block
and extending them to the surface (Fig 4) in such a way that
the side angles of the cone are equal to the four principal
slope angles
If the pit wall slopes in the four principal directions are not
the same the upper area of the cone on each level (intersec-
tion of the cone with the level) will consist of four quadrants
of different ellipses If the pit wall angles are the same the
upper area of the cone will be a circle Fig 5 shows the
extraction cone and the blocks within it on the first level and
on the two cross-sections On each level the values of the two
semi-major axes and two semi-minor axes depend on the four
principal slope angles and the vertical distance of the mid-
point of the base block from the overlying blocks These
parameters can be found by use of trigonometric functions
The number of blocks in the principal directions on any level
above the base block can be calculated by dividing these para-
meters by the corresponding block dimensions Consider a
block X i j k on level k the parameters and the numbers of
blocks in the principal directions as illustrated in Fig 5 can
be calculated from the equations
(1)
(2)dyk t z
1 = -( ) acute dim
tan (south face angle)
dx
k t z1 =
-( ) acute dim
tan (west face angle)
A79
Fig 3 Block model of deposit and coordinate systems
Fig 4 Construction of cone from base block
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(3)
(4)
(5)
(6)
(7)
(8)
where t is level above the base block and varies from 1 to k ndash1
m1 is number of blocks from the base block to the east n1 is
number of blocks from the base block to the north m2 is
number of blocks from the base block to the west and n2 is
number of blocks from the base block to the south
When the numbers of blocks within the upper area of the
cone on any levelmdashsay the t th level above the base blockmdash
are calculated in the four principal directions all the blocks
X mnk ndash t where m = i ndash m2 i + m1 and n = j ndash n2 j + n1 must
be examined to determine whether they are within the extrac-tion volume This can be done by use of the ellipse equation
a = xdim acute (indashm) (9)
b = ydim acute ( jndashn) (10)
If m is equal to or greater than i and n is equal to or greater
than j
(11)
If m is equal to or greater than i and n is less than or equal to j
(12)
If m is less than or equal to i and n is less than or equal to j
(13)
If m is less than or equal to i and n is equal to or greater than j
(14)
where a and b are the horizontal distances from the mid-point
of the block under consideration to the base block measured
in the westndasheast and southndashnorth directions respectively as
illustrated in Fig 6 If the lsquovaluersquo is less than or equal to 1 it
is assumed that the block is within the extraction cone and it
must be removed before the base block Otherwise it is
assumed that the block is outside the extraction cone Blocks
that lie within the extraction cone are submitted to the graph
algorithm The program was written in such a way that
extraction cones are established only for ore blocks This pre-
vents unnecessary increases in computing time and prevents
waste blocks from being considered several times
With this procedure pit slopes are no longer fixed and are
not limited to one-up and one-over patterns They can vary in
Value =
( )+
( )
a
dx
b
dy
2
22
2
12
Value =
( )+
( )
a
dx
b
dy
2
22
2
22
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
2
2
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
1
2
ndy
y2
2=dim
m
dx
x2
2=dim
n
dy
y1
1=dim
m
dx
x1
1=dim
dy
k t z2 =
-( ) acute dim
tan (north face angle)
dx
k t z2 =
-( ) acute dim
tan(east face angle)
A80
Fig 5 Extraction cone of base blocks showing all blocks within
cone (a) upper area of cone on first level (b) northndashsouth section
A ndash A (c) eastndashwest section B ndash B
(b)
(c)
(a)
Fig 6 Value of parameters a and b
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the principal directions and are independent of block dimen-
sions Fig 7 illustrates a directed graph that represents a
northndashsouth section in a cubic block model in which the east
face angle and the west face angle are assumed to be 60deg and
45deg respectively In this graph vertices 4 5 6 7 14 and 15
are in the extraction cone of block 23
Multiple variable slope angles
In complex cases in which the pit slopes vary in different parts
of the orebody on account of slope stability requirements it is
necessary to divide the orebody into regions or domain sec-
tors within which the rock characteristics are the same and to
use different slope angles for each region In these cases slope
angles are assigned to each block in the four principal direc-
tions within each region this is discussed later
In the case of multiple variable slopes an extraction volume
is constructed level by level by creating rings or envelopes
from the base block and extending them to the surface with
regard for pit slopes that have already been assigned toblocks The extraction volume is constructed from the base
block to the next overlying block and is then constructed
from the point of intersection of the cone with this level to the
second level above the base block This procedure is conti-
nued to the surface (Fig 8)
Consider the construction of an extraction cone in the two-
dimensional case (northndashsouth section) shown in Fig 8 Fora block on level k (row) and column j lines are drawn from
the mid-point of the base block to the left and right with
slopes equal to the east and west face angles of the base block
respectively and the lines are then extended to the level
above The values of parameters dx11 dx2
1 and the number of
blocks to the east m11 and to the west m2
1 on the first level
above the base block are determined by the equations
(15)
(16)
(17)
(18)
where z dim and x dim are the block dimensions in the verti-
cal and horizontal directions respectively On the first level
above the blocks X k ndash1m where m = j ndash m21 j + m1
1 are consid-
ered as part of the extraction cone
There are two intersection points of the extraction cone
with the level above the base block (lines drawn from the
mid-point of the base block to the next overlying level) Theextraction cone is extended from these two points to the next
overlying block (second level above the base block) by using
the slope angles of the blocks that contain the points of inter-
section To determine the slope of the block to be used (in
other words to find the block in which the intersection lies)
the values of the parameters dx11 and dx2
1 are divided by the
block dimension and the result is rounded up This means
that a value of 05 is added to the result of division and then
the integer part is taken ie
(19)
(20)
The values of the parameters dx11 and dx2
1 and the number of
blocks in both directions m12 and m2
2 on the second level
above the base block are determined as
(21)
(22)
(23)
(24)
Again blocks X k ndash2m where m = j ndash m22 j + m1
2 are consid-
ered as part of the extraction cone on the second level above
the base block This procedure is continued to the surface
The procedure presented for multiple variable slopes in
two dimensions can be applied to the three-dimensional case
As with the procedure used for variable slope angles the pit
shape is assumed to be defined by an irregular elliptical out-
m
dx
x22 2
2
=dim
m
dx
x12 1
2
=dim
dx dxz
k j ml 22
21
21
1= +
[ ]dim
tan east face angle of block ( - - )
dx dxz
k j ml 12
11
11
1= +
[ ]dim
tan west face angle of block ( - + )
ml
dx
x21 2
1
0 5= +dim
ml
dx
x11 1
1
0 5= +dim
m
dx
x21 2
1
-dim
m
dx
x11 1
1
= dim
dxz
k j 21 =
[ ]dim
tan east face angle of block ( )
dxz
k j 11 =
[ ]dim
tan west face angle of block ( )
A81
Fig 7 Directed graph representing northndashsouth cross-section in
cubic block model with east face angle of 60deg and west face angle of
45deg
Fig 8 Extraction cone of block for two-dimensional example with
three different regions
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line on each level The outline on each level consists of four
quadrants of different ellipses defined by the pit slope angles
in the four directions The values of the two semi-major axes
two semi-minor axes and the number of blocks in the prin-
cipal directions on any level above the base block should be
calculated in both sections in the same way as described
previously for one section When these parameters are deter-
mined again by use of the ellipse formula any block whose
mid-point lies inside the ellipse is considered to be part of the
cone The values of the four axes and the number of blocks inthe four directions for the t th level above the base block
(block X i j k) can be found from the equations
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
When the numbers of blocks in the four principal direc-
tions (m11 n1
1 m21 and n2
1) have been calculated the blocks are
examined according to the procedure described for variable
slope angles to determine whether they are within the extrac-
tion volume
Assigning slope angles to blocks
If more than one region or domain sector is specified to
define mining slopes it is necessary to assign slope angles to
each block To assign slope angles to the blocks the first step
is to determine which blocks are inside the particular region
A block is deemed to be inside a region if its mid-point lies
within that region Blocks deemed to be within a given region
have the slopes of that region assigned to them Different
methods can be used to determine whether a point is inside
outside or on the boundary of a polygon The approach
adopted here is the angle sum method based on coding origi-
nally written by Dowd17 In this method lines are drawn from
the point in question to each of the vertices that define the
boundary of the polygon and the angles between each succes-sive pair of lines are then summed Angles measured in the
clockwise direction are positive and those in the anticlockwise
direction are negative The point is inside the polygon if the
sum of the angles is 2p radians and outside if the sum is zero
The basic requirement for this method is the signed angle
between pairs of lines from the point to successive pairs of
vertices that define the boundary of the polygon These
angles can be determined by use of either the dot product of
two vectors or the equation of a triangle The signs of angles
can also be determined from the cross-product of two
vectors
Triangle equation
a2 = b2 + c2 ndash 2bc cos q
Dot product of two vectors
v1v2 = ccedilv1ccedilccedilv2ccedil cos q
Cross-product of two vectors
v1acutev2 = ccedilv1ccedilccedilv2ccedil sin q
A block whose mid-point lies within a slope region isassigned the slopes of that region The method is imple-
mented by first imposing a bounding box around the
regionmdashthis is the smallest rectangle that contains the region
or polygon Then the mid-points of all blocks in the first level
of the region that are inside the bounding box are examined
to see whether or not they are inside the region If the mid-
point of any block lies inside the region slope angles are
n
dy
y
t t
22=
dim
mdx
x
t t
22=
dim
n
dy
y
t t
11=
dim
m
dx
x
t t
11=
dim
dy dy
z
i j nl k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan north face angle of block ( - )2-1
dx dx
z
i ml j k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan east face angle of block ( - )2-1
dy dy
z
i j nl k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan south face angle of block ( + )1
-1
dx dx
z
i ml j k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan west face angle of block ( + )1
-1
nl
dy
y
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
nl
dy
y
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
A82
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 79
assigned to all blocks at this location from the minimum to
the maximum depth of the region
Case study
The method of variable slope angles has been incorporated
into the LerchsndashGrossmann algorithm and has been coded
into an interactive Windows software package18 Data from a
real orebody were used to illustrate and test the application of
the software in the determination of optimum open-pit limitsThe data are from the Bjoumlrkdal low-grade gold deposit
located approximately 35 km northwest of Skelleftearing in the
north of Sweden19
Gold mineralization in the Bjoumlrkdal area occurs within
a network of steeply dipping quartz veins in the contact
between older granodiorite and limestoneacid volcanic rocks
The gold is erratically distributed but is mainly concentrated
in and around high-grade quartz veins It occurs as both fine
and coarse grains and is free-milling
The block grade model of the deposit contains 101 acute 82 acute36 blocks in the eastndashwest northndashsouth and vertical direc-
tions respectively Each block is assigned the estimated
(kriged) recoverable tonnage of ore above a cutoff grade andthe estimated (kriged) average grade of this tonnage The
method of estimation has been detailed elsewhere19 The
deposit is divided into 15 m (eastndashwest) acute 10 m (northndashsouth)
acute 5 m (vertical) blocks and the recoverable tonnage is based
on a selective mining unit of 5 m (eastndashwest) acute 4 m
(northndashsouth) acute 5 m (vertical) The physical and economic
parameters for this case are specific gravity of ore and waste
271 tm3 cost of mining of ore and wastemdashas given Table 1
processing cost SEK 52tonne of ore price of gold
SEK 90g and recovery 91
The slope regions and associated principal slope angles are
shown in Fig 9 and Table 2 The slopes used here are solely
for the sake of example and do not necessarily correspond to
actual slopes The overall results of pit optimization are
shown in Table 3 Two cross-sections through the optimal pit
are shown in Fig 10 The application of the software to the
case study has enabled a much more realistic pit design that is
able to accommodate real slope angles within a traditional
block model for a complex low-grade gold orebody
A83
Fig 9 Deposit and surrounding waste subdivided into four geo-
technical regions
Table 1 Cost of mining of ore and waste
Level m Cost of mining SEKt
From To Waste Ore
0 120 110 110
120 130 1130 1130
130 140 1160 1160
140 150 1190 1190
150 160 1220 1220
160 170 1250 1250
170 180 1280 1280
180 200 1320 1320
Fig 10 Cross-sections at (a) easting 150 m and (b) northing 120 m
Table 2 Slope angles applied to example shown in Fig 9
Region North face East face South face West face
1 30deg 40deg 42deg 38deg
2 41deg 37deg 50deg 46deg
3 35deg 35deg 35deg 35deg
4 39deg 39deg 46deg 46deg
7262019 Lg_var_slopes - Grupo Los Tigres
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Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
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httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 99
13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
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(3)
(4)
(5)
(6)
(7)
(8)
where t is level above the base block and varies from 1 to k ndash1
m1 is number of blocks from the base block to the east n1 is
number of blocks from the base block to the north m2 is
number of blocks from the base block to the west and n2 is
number of blocks from the base block to the south
When the numbers of blocks within the upper area of the
cone on any levelmdashsay the t th level above the base blockmdash
are calculated in the four principal directions all the blocks
X mnk ndash t where m = i ndash m2 i + m1 and n = j ndash n2 j + n1 must
be examined to determine whether they are within the extrac-tion volume This can be done by use of the ellipse equation
a = xdim acute (indashm) (9)
b = ydim acute ( jndashn) (10)
If m is equal to or greater than i and n is equal to or greater
than j
(11)
If m is equal to or greater than i and n is less than or equal to j
(12)
If m is less than or equal to i and n is less than or equal to j
(13)
If m is less than or equal to i and n is equal to or greater than j
(14)
where a and b are the horizontal distances from the mid-point
of the block under consideration to the base block measured
in the westndasheast and southndashnorth directions respectively as
illustrated in Fig 6 If the lsquovaluersquo is less than or equal to 1 it
is assumed that the block is within the extraction cone and it
must be removed before the base block Otherwise it is
assumed that the block is outside the extraction cone Blocks
that lie within the extraction cone are submitted to the graph
algorithm The program was written in such a way that
extraction cones are established only for ore blocks This pre-
vents unnecessary increases in computing time and prevents
waste blocks from being considered several times
With this procedure pit slopes are no longer fixed and are
not limited to one-up and one-over patterns They can vary in
Value =
( )+
( )
a
dx
b
dy
2
22
2
12
Value =
( )+
( )
a
dx
b
dy
2
22
2
22
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
2
2
Value =
( )+
( )
a
dx
b
dy
2
1
2
2
1
2
ndy
y2
2=dim
m
dx
x2
2=dim
n
dy
y1
1=dim
m
dx
x1
1=dim
dy
k t z2 =
-( ) acute dim
tan (north face angle)
dx
k t z2 =
-( ) acute dim
tan(east face angle)
A80
Fig 5 Extraction cone of base blocks showing all blocks within
cone (a) upper area of cone on first level (b) northndashsouth section
A ndash A (c) eastndashwest section B ndash B
(b)
(c)
(a)
Fig 6 Value of parameters a and b
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the principal directions and are independent of block dimen-
sions Fig 7 illustrates a directed graph that represents a
northndashsouth section in a cubic block model in which the east
face angle and the west face angle are assumed to be 60deg and
45deg respectively In this graph vertices 4 5 6 7 14 and 15
are in the extraction cone of block 23
Multiple variable slope angles
In complex cases in which the pit slopes vary in different parts
of the orebody on account of slope stability requirements it is
necessary to divide the orebody into regions or domain sec-
tors within which the rock characteristics are the same and to
use different slope angles for each region In these cases slope
angles are assigned to each block in the four principal direc-
tions within each region this is discussed later
In the case of multiple variable slopes an extraction volume
is constructed level by level by creating rings or envelopes
from the base block and extending them to the surface with
regard for pit slopes that have already been assigned toblocks The extraction volume is constructed from the base
block to the next overlying block and is then constructed
from the point of intersection of the cone with this level to the
second level above the base block This procedure is conti-
nued to the surface (Fig 8)
Consider the construction of an extraction cone in the two-
dimensional case (northndashsouth section) shown in Fig 8 Fora block on level k (row) and column j lines are drawn from
the mid-point of the base block to the left and right with
slopes equal to the east and west face angles of the base block
respectively and the lines are then extended to the level
above The values of parameters dx11 dx2
1 and the number of
blocks to the east m11 and to the west m2
1 on the first level
above the base block are determined by the equations
(15)
(16)
(17)
(18)
where z dim and x dim are the block dimensions in the verti-
cal and horizontal directions respectively On the first level
above the blocks X k ndash1m where m = j ndash m21 j + m1
1 are consid-
ered as part of the extraction cone
There are two intersection points of the extraction cone
with the level above the base block (lines drawn from the
mid-point of the base block to the next overlying level) Theextraction cone is extended from these two points to the next
overlying block (second level above the base block) by using
the slope angles of the blocks that contain the points of inter-
section To determine the slope of the block to be used (in
other words to find the block in which the intersection lies)
the values of the parameters dx11 and dx2
1 are divided by the
block dimension and the result is rounded up This means
that a value of 05 is added to the result of division and then
the integer part is taken ie
(19)
(20)
The values of the parameters dx11 and dx2
1 and the number of
blocks in both directions m12 and m2
2 on the second level
above the base block are determined as
(21)
(22)
(23)
(24)
Again blocks X k ndash2m where m = j ndash m22 j + m1
2 are consid-
ered as part of the extraction cone on the second level above
the base block This procedure is continued to the surface
The procedure presented for multiple variable slopes in
two dimensions can be applied to the three-dimensional case
As with the procedure used for variable slope angles the pit
shape is assumed to be defined by an irregular elliptical out-
m
dx
x22 2
2
=dim
m
dx
x12 1
2
=dim
dx dxz
k j ml 22
21
21
1= +
[ ]dim
tan east face angle of block ( - - )
dx dxz
k j ml 12
11
11
1= +
[ ]dim
tan west face angle of block ( - + )
ml
dx
x21 2
1
0 5= +dim
ml
dx
x11 1
1
0 5= +dim
m
dx
x21 2
1
-dim
m
dx
x11 1
1
= dim
dxz
k j 21 =
[ ]dim
tan east face angle of block ( )
dxz
k j 11 =
[ ]dim
tan west face angle of block ( )
A81
Fig 7 Directed graph representing northndashsouth cross-section in
cubic block model with east face angle of 60deg and west face angle of
45deg
Fig 8 Extraction cone of block for two-dimensional example with
three different regions
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line on each level The outline on each level consists of four
quadrants of different ellipses defined by the pit slope angles
in the four directions The values of the two semi-major axes
two semi-minor axes and the number of blocks in the prin-
cipal directions on any level above the base block should be
calculated in both sections in the same way as described
previously for one section When these parameters are deter-
mined again by use of the ellipse formula any block whose
mid-point lies inside the ellipse is considered to be part of the
cone The values of the four axes and the number of blocks inthe four directions for the t th level above the base block
(block X i j k) can be found from the equations
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
When the numbers of blocks in the four principal direc-
tions (m11 n1
1 m21 and n2
1) have been calculated the blocks are
examined according to the procedure described for variable
slope angles to determine whether they are within the extrac-
tion volume
Assigning slope angles to blocks
If more than one region or domain sector is specified to
define mining slopes it is necessary to assign slope angles to
each block To assign slope angles to the blocks the first step
is to determine which blocks are inside the particular region
A block is deemed to be inside a region if its mid-point lies
within that region Blocks deemed to be within a given region
have the slopes of that region assigned to them Different
methods can be used to determine whether a point is inside
outside or on the boundary of a polygon The approach
adopted here is the angle sum method based on coding origi-
nally written by Dowd17 In this method lines are drawn from
the point in question to each of the vertices that define the
boundary of the polygon and the angles between each succes-sive pair of lines are then summed Angles measured in the
clockwise direction are positive and those in the anticlockwise
direction are negative The point is inside the polygon if the
sum of the angles is 2p radians and outside if the sum is zero
The basic requirement for this method is the signed angle
between pairs of lines from the point to successive pairs of
vertices that define the boundary of the polygon These
angles can be determined by use of either the dot product of
two vectors or the equation of a triangle The signs of angles
can also be determined from the cross-product of two
vectors
Triangle equation
a2 = b2 + c2 ndash 2bc cos q
Dot product of two vectors
v1v2 = ccedilv1ccedilccedilv2ccedil cos q
Cross-product of two vectors
v1acutev2 = ccedilv1ccedilccedilv2ccedil sin q
A block whose mid-point lies within a slope region isassigned the slopes of that region The method is imple-
mented by first imposing a bounding box around the
regionmdashthis is the smallest rectangle that contains the region
or polygon Then the mid-points of all blocks in the first level
of the region that are inside the bounding box are examined
to see whether or not they are inside the region If the mid-
point of any block lies inside the region slope angles are
n
dy
y
t t
22=
dim
mdx
x
t t
22=
dim
n
dy
y
t t
11=
dim
m
dx
x
t t
11=
dim
dy dy
z
i j nl k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan north face angle of block ( - )2-1
dx dx
z
i ml j k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan east face angle of block ( - )2-1
dy dy
z
i j nl k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan south face angle of block ( + )1
-1
dx dx
z
i ml j k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan west face angle of block ( + )1
-1
nl
dy
y
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
nl
dy
y
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
A82
7262019 Lg_var_slopes - Grupo Los Tigres
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assigned to all blocks at this location from the minimum to
the maximum depth of the region
Case study
The method of variable slope angles has been incorporated
into the LerchsndashGrossmann algorithm and has been coded
into an interactive Windows software package18 Data from a
real orebody were used to illustrate and test the application of
the software in the determination of optimum open-pit limitsThe data are from the Bjoumlrkdal low-grade gold deposit
located approximately 35 km northwest of Skelleftearing in the
north of Sweden19
Gold mineralization in the Bjoumlrkdal area occurs within
a network of steeply dipping quartz veins in the contact
between older granodiorite and limestoneacid volcanic rocks
The gold is erratically distributed but is mainly concentrated
in and around high-grade quartz veins It occurs as both fine
and coarse grains and is free-milling
The block grade model of the deposit contains 101 acute 82 acute36 blocks in the eastndashwest northndashsouth and vertical direc-
tions respectively Each block is assigned the estimated
(kriged) recoverable tonnage of ore above a cutoff grade andthe estimated (kriged) average grade of this tonnage The
method of estimation has been detailed elsewhere19 The
deposit is divided into 15 m (eastndashwest) acute 10 m (northndashsouth)
acute 5 m (vertical) blocks and the recoverable tonnage is based
on a selective mining unit of 5 m (eastndashwest) acute 4 m
(northndashsouth) acute 5 m (vertical) The physical and economic
parameters for this case are specific gravity of ore and waste
271 tm3 cost of mining of ore and wastemdashas given Table 1
processing cost SEK 52tonne of ore price of gold
SEK 90g and recovery 91
The slope regions and associated principal slope angles are
shown in Fig 9 and Table 2 The slopes used here are solely
for the sake of example and do not necessarily correspond to
actual slopes The overall results of pit optimization are
shown in Table 3 Two cross-sections through the optimal pit
are shown in Fig 10 The application of the software to the
case study has enabled a much more realistic pit design that is
able to accommodate real slope angles within a traditional
block model for a complex low-grade gold orebody
A83
Fig 9 Deposit and surrounding waste subdivided into four geo-
technical regions
Table 1 Cost of mining of ore and waste
Level m Cost of mining SEKt
From To Waste Ore
0 120 110 110
120 130 1130 1130
130 140 1160 1160
140 150 1190 1190
150 160 1220 1220
160 170 1250 1250
170 180 1280 1280
180 200 1320 1320
Fig 10 Cross-sections at (a) easting 150 m and (b) northing 120 m
Table 2 Slope angles applied to example shown in Fig 9
Region North face East face South face West face
1 30deg 40deg 42deg 38deg
2 41deg 37deg 50deg 46deg
3 35deg 35deg 35deg 35deg
4 39deg 39deg 46deg 46deg
7262019 Lg_var_slopes - Grupo Los Tigres
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Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 99
13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 59
the principal directions and are independent of block dimen-
sions Fig 7 illustrates a directed graph that represents a
northndashsouth section in a cubic block model in which the east
face angle and the west face angle are assumed to be 60deg and
45deg respectively In this graph vertices 4 5 6 7 14 and 15
are in the extraction cone of block 23
Multiple variable slope angles
In complex cases in which the pit slopes vary in different parts
of the orebody on account of slope stability requirements it is
necessary to divide the orebody into regions or domain sec-
tors within which the rock characteristics are the same and to
use different slope angles for each region In these cases slope
angles are assigned to each block in the four principal direc-
tions within each region this is discussed later
In the case of multiple variable slopes an extraction volume
is constructed level by level by creating rings or envelopes
from the base block and extending them to the surface with
regard for pit slopes that have already been assigned toblocks The extraction volume is constructed from the base
block to the next overlying block and is then constructed
from the point of intersection of the cone with this level to the
second level above the base block This procedure is conti-
nued to the surface (Fig 8)
Consider the construction of an extraction cone in the two-
dimensional case (northndashsouth section) shown in Fig 8 Fora block on level k (row) and column j lines are drawn from
the mid-point of the base block to the left and right with
slopes equal to the east and west face angles of the base block
respectively and the lines are then extended to the level
above The values of parameters dx11 dx2
1 and the number of
blocks to the east m11 and to the west m2
1 on the first level
above the base block are determined by the equations
(15)
(16)
(17)
(18)
where z dim and x dim are the block dimensions in the verti-
cal and horizontal directions respectively On the first level
above the blocks X k ndash1m where m = j ndash m21 j + m1
1 are consid-
ered as part of the extraction cone
There are two intersection points of the extraction cone
with the level above the base block (lines drawn from the
mid-point of the base block to the next overlying level) Theextraction cone is extended from these two points to the next
overlying block (second level above the base block) by using
the slope angles of the blocks that contain the points of inter-
section To determine the slope of the block to be used (in
other words to find the block in which the intersection lies)
the values of the parameters dx11 and dx2
1 are divided by the
block dimension and the result is rounded up This means
that a value of 05 is added to the result of division and then
the integer part is taken ie
(19)
(20)
The values of the parameters dx11 and dx2
1 and the number of
blocks in both directions m12 and m2
2 on the second level
above the base block are determined as
(21)
(22)
(23)
(24)
Again blocks X k ndash2m where m = j ndash m22 j + m1
2 are consid-
ered as part of the extraction cone on the second level above
the base block This procedure is continued to the surface
The procedure presented for multiple variable slopes in
two dimensions can be applied to the three-dimensional case
As with the procedure used for variable slope angles the pit
shape is assumed to be defined by an irregular elliptical out-
m
dx
x22 2
2
=dim
m
dx
x12 1
2
=dim
dx dxz
k j ml 22
21
21
1= +
[ ]dim
tan east face angle of block ( - - )
dx dxz
k j ml 12
11
11
1= +
[ ]dim
tan west face angle of block ( - + )
ml
dx
x21 2
1
0 5= +dim
ml
dx
x11 1
1
0 5= +dim
m
dx
x21 2
1
-dim
m
dx
x11 1
1
= dim
dxz
k j 21 =
[ ]dim
tan east face angle of block ( )
dxz
k j 11 =
[ ]dim
tan west face angle of block ( )
A81
Fig 7 Directed graph representing northndashsouth cross-section in
cubic block model with east face angle of 60deg and west face angle of
45deg
Fig 8 Extraction cone of block for two-dimensional example with
three different regions
7262019 Lg_var_slopes - Grupo Los Tigres
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line on each level The outline on each level consists of four
quadrants of different ellipses defined by the pit slope angles
in the four directions The values of the two semi-major axes
two semi-minor axes and the number of blocks in the prin-
cipal directions on any level above the base block should be
calculated in both sections in the same way as described
previously for one section When these parameters are deter-
mined again by use of the ellipse formula any block whose
mid-point lies inside the ellipse is considered to be part of the
cone The values of the four axes and the number of blocks inthe four directions for the t th level above the base block
(block X i j k) can be found from the equations
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
When the numbers of blocks in the four principal direc-
tions (m11 n1
1 m21 and n2
1) have been calculated the blocks are
examined according to the procedure described for variable
slope angles to determine whether they are within the extrac-
tion volume
Assigning slope angles to blocks
If more than one region or domain sector is specified to
define mining slopes it is necessary to assign slope angles to
each block To assign slope angles to the blocks the first step
is to determine which blocks are inside the particular region
A block is deemed to be inside a region if its mid-point lies
within that region Blocks deemed to be within a given region
have the slopes of that region assigned to them Different
methods can be used to determine whether a point is inside
outside or on the boundary of a polygon The approach
adopted here is the angle sum method based on coding origi-
nally written by Dowd17 In this method lines are drawn from
the point in question to each of the vertices that define the
boundary of the polygon and the angles between each succes-sive pair of lines are then summed Angles measured in the
clockwise direction are positive and those in the anticlockwise
direction are negative The point is inside the polygon if the
sum of the angles is 2p radians and outside if the sum is zero
The basic requirement for this method is the signed angle
between pairs of lines from the point to successive pairs of
vertices that define the boundary of the polygon These
angles can be determined by use of either the dot product of
two vectors or the equation of a triangle The signs of angles
can also be determined from the cross-product of two
vectors
Triangle equation
a2 = b2 + c2 ndash 2bc cos q
Dot product of two vectors
v1v2 = ccedilv1ccedilccedilv2ccedil cos q
Cross-product of two vectors
v1acutev2 = ccedilv1ccedilccedilv2ccedil sin q
A block whose mid-point lies within a slope region isassigned the slopes of that region The method is imple-
mented by first imposing a bounding box around the
regionmdashthis is the smallest rectangle that contains the region
or polygon Then the mid-points of all blocks in the first level
of the region that are inside the bounding box are examined
to see whether or not they are inside the region If the mid-
point of any block lies inside the region slope angles are
n
dy
y
t t
22=
dim
mdx
x
t t
22=
dim
n
dy
y
t t
11=
dim
m
dx
x
t t
11=
dim
dy dy
z
i j nl k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan north face angle of block ( - )2-1
dx dx
z
i ml j k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan east face angle of block ( - )2-1
dy dy
z
i j nl k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan south face angle of block ( + )1
-1
dx dx
z
i ml j k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan west face angle of block ( + )1
-1
nl
dy
y
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
nl
dy
y
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
A82
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 79
assigned to all blocks at this location from the minimum to
the maximum depth of the region
Case study
The method of variable slope angles has been incorporated
into the LerchsndashGrossmann algorithm and has been coded
into an interactive Windows software package18 Data from a
real orebody were used to illustrate and test the application of
the software in the determination of optimum open-pit limitsThe data are from the Bjoumlrkdal low-grade gold deposit
located approximately 35 km northwest of Skelleftearing in the
north of Sweden19
Gold mineralization in the Bjoumlrkdal area occurs within
a network of steeply dipping quartz veins in the contact
between older granodiorite and limestoneacid volcanic rocks
The gold is erratically distributed but is mainly concentrated
in and around high-grade quartz veins It occurs as both fine
and coarse grains and is free-milling
The block grade model of the deposit contains 101 acute 82 acute36 blocks in the eastndashwest northndashsouth and vertical direc-
tions respectively Each block is assigned the estimated
(kriged) recoverable tonnage of ore above a cutoff grade andthe estimated (kriged) average grade of this tonnage The
method of estimation has been detailed elsewhere19 The
deposit is divided into 15 m (eastndashwest) acute 10 m (northndashsouth)
acute 5 m (vertical) blocks and the recoverable tonnage is based
on a selective mining unit of 5 m (eastndashwest) acute 4 m
(northndashsouth) acute 5 m (vertical) The physical and economic
parameters for this case are specific gravity of ore and waste
271 tm3 cost of mining of ore and wastemdashas given Table 1
processing cost SEK 52tonne of ore price of gold
SEK 90g and recovery 91
The slope regions and associated principal slope angles are
shown in Fig 9 and Table 2 The slopes used here are solely
for the sake of example and do not necessarily correspond to
actual slopes The overall results of pit optimization are
shown in Table 3 Two cross-sections through the optimal pit
are shown in Fig 10 The application of the software to the
case study has enabled a much more realistic pit design that is
able to accommodate real slope angles within a traditional
block model for a complex low-grade gold orebody
A83
Fig 9 Deposit and surrounding waste subdivided into four geo-
technical regions
Table 1 Cost of mining of ore and waste
Level m Cost of mining SEKt
From To Waste Ore
0 120 110 110
120 130 1130 1130
130 140 1160 1160
140 150 1190 1190
150 160 1220 1220
160 170 1250 1250
170 180 1280 1280
180 200 1320 1320
Fig 10 Cross-sections at (a) easting 150 m and (b) northing 120 m
Table 2 Slope angles applied to example shown in Fig 9
Region North face East face South face West face
1 30deg 40deg 42deg 38deg
2 41deg 37deg 50deg 46deg
3 35deg 35deg 35deg 35deg
4 39deg 39deg 46deg 46deg
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 89
Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 99
13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 69
line on each level The outline on each level consists of four
quadrants of different ellipses defined by the pit slope angles
in the four directions The values of the two semi-major axes
two semi-minor axes and the number of blocks in the prin-
cipal directions on any level above the base block should be
calculated in both sections in the same way as described
previously for one section When these parameters are deter-
mined again by use of the ellipse formula any block whose
mid-point lies inside the ellipse is considered to be part of the
cone The values of the four axes and the number of blocks inthe four directions for the t th level above the base block
(block X i j k) can be found from the equations
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
When the numbers of blocks in the four principal direc-
tions (m11 n1
1 m21 and n2
1) have been calculated the blocks are
examined according to the procedure described for variable
slope angles to determine whether they are within the extrac-
tion volume
Assigning slope angles to blocks
If more than one region or domain sector is specified to
define mining slopes it is necessary to assign slope angles to
each block To assign slope angles to the blocks the first step
is to determine which blocks are inside the particular region
A block is deemed to be inside a region if its mid-point lies
within that region Blocks deemed to be within a given region
have the slopes of that region assigned to them Different
methods can be used to determine whether a point is inside
outside or on the boundary of a polygon The approach
adopted here is the angle sum method based on coding origi-
nally written by Dowd17 In this method lines are drawn from
the point in question to each of the vertices that define the
boundary of the polygon and the angles between each succes-sive pair of lines are then summed Angles measured in the
clockwise direction are positive and those in the anticlockwise
direction are negative The point is inside the polygon if the
sum of the angles is 2p radians and outside if the sum is zero
The basic requirement for this method is the signed angle
between pairs of lines from the point to successive pairs of
vertices that define the boundary of the polygon These
angles can be determined by use of either the dot product of
two vectors or the equation of a triangle The signs of angles
can also be determined from the cross-product of two
vectors
Triangle equation
a2 = b2 + c2 ndash 2bc cos q
Dot product of two vectors
v1v2 = ccedilv1ccedilccedilv2ccedil cos q
Cross-product of two vectors
v1acutev2 = ccedilv1ccedilccedilv2ccedil sin q
A block whose mid-point lies within a slope region isassigned the slopes of that region The method is imple-
mented by first imposing a bounding box around the
regionmdashthis is the smallest rectangle that contains the region
or polygon Then the mid-points of all blocks in the first level
of the region that are inside the bounding box are examined
to see whether or not they are inside the region If the mid-
point of any block lies inside the region slope angles are
n
dy
y
t t
22=
dim
mdx
x
t t
22=
dim
n
dy
y
t t
11=
dim
m
dx
x
t t
11=
dim
dy dy
z
i j nl k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan north face angle of block ( - )2-1
dx dx
z
i ml j k t
t i
i
t
t
2 2
1
1
1
=
+- +[ ]
=
-
aringdim
tan east face angle of block ( - )2-1
dy dy
z
i j nl k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan south face angle of block ( + )1
-1
dx dx
z
i ml j k t
t i
i
t
t
1 1
1
1
1
=
+- +[ ]
=
-
aring
dim
tan west face angle of block ( + )1
-1
nl
dy
y
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
21
2
1
1
0 5- =
-
= +
aringdim
nl
dy
y
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
ml
dx
x
t
i
i
t
11
1
1
1
0 5- =
-
= +
aringdim
A82
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 79
assigned to all blocks at this location from the minimum to
the maximum depth of the region
Case study
The method of variable slope angles has been incorporated
into the LerchsndashGrossmann algorithm and has been coded
into an interactive Windows software package18 Data from a
real orebody were used to illustrate and test the application of
the software in the determination of optimum open-pit limitsThe data are from the Bjoumlrkdal low-grade gold deposit
located approximately 35 km northwest of Skelleftearing in the
north of Sweden19
Gold mineralization in the Bjoumlrkdal area occurs within
a network of steeply dipping quartz veins in the contact
between older granodiorite and limestoneacid volcanic rocks
The gold is erratically distributed but is mainly concentrated
in and around high-grade quartz veins It occurs as both fine
and coarse grains and is free-milling
The block grade model of the deposit contains 101 acute 82 acute36 blocks in the eastndashwest northndashsouth and vertical direc-
tions respectively Each block is assigned the estimated
(kriged) recoverable tonnage of ore above a cutoff grade andthe estimated (kriged) average grade of this tonnage The
method of estimation has been detailed elsewhere19 The
deposit is divided into 15 m (eastndashwest) acute 10 m (northndashsouth)
acute 5 m (vertical) blocks and the recoverable tonnage is based
on a selective mining unit of 5 m (eastndashwest) acute 4 m
(northndashsouth) acute 5 m (vertical) The physical and economic
parameters for this case are specific gravity of ore and waste
271 tm3 cost of mining of ore and wastemdashas given Table 1
processing cost SEK 52tonne of ore price of gold
SEK 90g and recovery 91
The slope regions and associated principal slope angles are
shown in Fig 9 and Table 2 The slopes used here are solely
for the sake of example and do not necessarily correspond to
actual slopes The overall results of pit optimization are
shown in Table 3 Two cross-sections through the optimal pit
are shown in Fig 10 The application of the software to the
case study has enabled a much more realistic pit design that is
able to accommodate real slope angles within a traditional
block model for a complex low-grade gold orebody
A83
Fig 9 Deposit and surrounding waste subdivided into four geo-
technical regions
Table 1 Cost of mining of ore and waste
Level m Cost of mining SEKt
From To Waste Ore
0 120 110 110
120 130 1130 1130
130 140 1160 1160
140 150 1190 1190
150 160 1220 1220
160 170 1250 1250
170 180 1280 1280
180 200 1320 1320
Fig 10 Cross-sections at (a) easting 150 m and (b) northing 120 m
Table 2 Slope angles applied to example shown in Fig 9
Region North face East face South face West face
1 30deg 40deg 42deg 38deg
2 41deg 37deg 50deg 46deg
3 35deg 35deg 35deg 35deg
4 39deg 39deg 46deg 46deg
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 89
Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 99
13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 79
assigned to all blocks at this location from the minimum to
the maximum depth of the region
Case study
The method of variable slope angles has been incorporated
into the LerchsndashGrossmann algorithm and has been coded
into an interactive Windows software package18 Data from a
real orebody were used to illustrate and test the application of
the software in the determination of optimum open-pit limitsThe data are from the Bjoumlrkdal low-grade gold deposit
located approximately 35 km northwest of Skelleftearing in the
north of Sweden19
Gold mineralization in the Bjoumlrkdal area occurs within
a network of steeply dipping quartz veins in the contact
between older granodiorite and limestoneacid volcanic rocks
The gold is erratically distributed but is mainly concentrated
in and around high-grade quartz veins It occurs as both fine
and coarse grains and is free-milling
The block grade model of the deposit contains 101 acute 82 acute36 blocks in the eastndashwest northndashsouth and vertical direc-
tions respectively Each block is assigned the estimated
(kriged) recoverable tonnage of ore above a cutoff grade andthe estimated (kriged) average grade of this tonnage The
method of estimation has been detailed elsewhere19 The
deposit is divided into 15 m (eastndashwest) acute 10 m (northndashsouth)
acute 5 m (vertical) blocks and the recoverable tonnage is based
on a selective mining unit of 5 m (eastndashwest) acute 4 m
(northndashsouth) acute 5 m (vertical) The physical and economic
parameters for this case are specific gravity of ore and waste
271 tm3 cost of mining of ore and wastemdashas given Table 1
processing cost SEK 52tonne of ore price of gold
SEK 90g and recovery 91
The slope regions and associated principal slope angles are
shown in Fig 9 and Table 2 The slopes used here are solely
for the sake of example and do not necessarily correspond to
actual slopes The overall results of pit optimization are
shown in Table 3 Two cross-sections through the optimal pit
are shown in Fig 10 The application of the software to the
case study has enabled a much more realistic pit design that is
able to accommodate real slope angles within a traditional
block model for a complex low-grade gold orebody
A83
Fig 9 Deposit and surrounding waste subdivided into four geo-
technical regions
Table 1 Cost of mining of ore and waste
Level m Cost of mining SEKt
From To Waste Ore
0 120 110 110
120 130 1130 1130
130 140 1160 1160
140 150 1190 1190
150 160 1220 1220
160 170 1250 1250
170 180 1280 1280
180 200 1320 1320
Fig 10 Cross-sections at (a) easting 150 m and (b) northing 120 m
Table 2 Slope angles applied to example shown in Fig 9
Region North face East face South face West face
1 30deg 40deg 42deg 38deg
2 41deg 37deg 50deg 46deg
3 35deg 35deg 35deg 35deg
4 39deg 39deg 46deg 46deg
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 89
Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 99
13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 89
Conclusion
The LerchsndashGrossmann algorithm is well known for being
the only method that can be proved rigorously always to
yield the true optimum pit However when the algorithm was
first introduced it was based on a fixed slope angle governed
by the block dimensions The methods presented here have
been incorporated into the algorithm to overcome this limi-
tation and to take account of variable slope angles As
demonstrated by a case study the algorithm is able to gener-
ate an optimal open-pit with variable slopes The method can
be used for both cubic and rectangular block models Slope
angles can vary in different parts of the orebody without
change to the block dimensions which are independent of
the slope angles
Methods for the determintion of varying slope angles for
incorporation into pit design algorithms are described in the
accompanying contribution20
References1 Lerchs H and Grossmann I F Optimum design of open pitmines CIM Bull 58 1965 47ndash542 Johnson T B and Barnes R J Application of the maximal flow
algorithm to ultimate pit design In Levary R R ed Engineering design better results through operations research methods (Amsterdam
North Holland 1988) 518ndash313 Yegulalp T M and Arias J A A fast algorithm to solve the ulti-mate pit limit problem In Proc 23rd symposium on the application of
computers and operations research in the mineral industries (APCOM)(Littleton Colorado AIME 1992) 391ndash74 Lemieux M Moving cone optimizing algorithm In Weiss A ed
Computer methods for the 80s in the mineral industry (New York AIME1979) 329ndash45
5 Korobov S Method for determining optimal open pit limits(Montreal Ecole Polytechnique de lrsquoUniversiteacute de Montreacuteal 1974)
24 p Technical report EP74-R-4
6 Dowd P A and Onur A H Open-pit optimizationmdashpart 1optimal open-pit design Trans Instn Min Metall (Sect A Min
industry) 102 1993 A95ndash1047 Wilke F L and Wright E A Determining the optimal ultimatepit design for hard rock open pit mines using dynamic programming
Erzmetall 37 1984 139ndash448 Yamatomi J et al Selective extraction dynamic cone algorithm
for three-dimensional open pit designs In Proc 25th symposium on theapplication of computers and operations research in the mineral industries
(APCOM) (Brisbane Australasian Institute of Mining andMetallurgy 1995) 267ndash749 Matheron G Parameacutetrage des contours optimaux(Fontainebleau Centre de Geacuteostatistique et de Morphologie matheacute-matique 1975) 54 p Internal report N-403 Note geacuteostatistique 12810 Franccedilois-Bongarccedilon D and Guibal D Algorithms for parame-terizing reserves under different geometrical constraints In Proc 17th
symposium on the application of computers and operations research in themineral industries (APCOM) (New York AIME 1982) 297ndash30911 Chen T 3D pit design with variable wall slope capabilities In
Proc 14th symposium on the application of computers and operations
research in the mineral industries (APCOM) (New York AIME 1976)615ndash2512 Lipkewich M P and Borgman L Two- and three-dimensionalpit design optimization techniques In Weiss A ed A decade of digital computing in the mineral industry (New York AIME 1969) 505ndash23
A84
Table 3 Optimum pit
Level Number of blocks Tonnage t Value SEK 10000 Mean
no Pit Ore Waste Ore Waste Ore Waste grade gt
1 0 0 0 0 0 0 0 0
2 7 0 7 0 14 2275 0 ndash157 0
3 20 0 20 1 2612 39 3888 0 ndash442 0
4 47 6 41 8 3210 87 2065 282 ndash721 1534
5 89 19 70 18 9437 161 9488 714 ndash1337 1516
6 98 24 74 23 0709 176 1141 1204 ndash1492 16427 116 23 93 26 6350 209 1350 1728 ndash1807 1787
8 115 22 93 30 3495 203 3880 2237 ndash1834 1831
9 132 31 101 38 9825 229 3075 3137 ndash2081 1890
10 188 70 118 88 4912 293 6188 9866 ndash2431 2241
11 304 109 195 149 6605 468 2195 16389 ndash4063 2195
12 381 160 221 229 5243 544 8582 26614 ndash4627 2258
13 652 186 466 257 2782 1 067 9118 28046 ndash10091 2179
14 833 192 641 274 9017 1 418 1709 29037 ndash13985 2131
15 912 199 713 294 2511 1 559 3890 32596 ndash15438 2193
16 884 213 671 290 3476 1 506 3824 26323 ndash14435 1996
17 899 183 716 230 409 9 1 596 8076 20062 ndash15237 1956
18 853 181 672 216 5247 1 517 1978 21904 ndash14281 2140
19 793 181 612 224 8208 1 386 9518 24532 ndash12953 2227
20 730 182 548 225 1219 1 258 6031 25297 ndash11712 225721 734 212 522 270 8927 1 220 9622 36663 ndash11338 2536
22 723 228 495 286 7627 1 182 7348 35422 ndash10456 2405
23 663 209 454 285 2033 1 062 3442 31186 ndash9869 2217
24 638 233 405 312 8226 983 9124 32982 ndash8620 2173
25 650 281 369 389 3992 931 7258 43271 ndash7662 2244
26 652 344 308 464 9911 860 1989 52621 ndash6190 2268
27 625 324 301 441 3248 828 9878 56523 ndash6066 2460
28 579 342 237 478 6142 698 2033 69848 ndash4569 2667
29 479 296 183 415 8638 557 7037 59377 ndash3819 2623
30 384 241 143 338 9629 441 5171 46425 ndash2907 2554
31 300 223 77 295 2311 314 5189 39207 ndash1538 2516
32 220 181 39 249 7752 197 3748 40369 ndash662 2856
33 150 137 13 195 4743 109 4007 31172 ndash280 2816
34 100 94 6 140 5129 62 7371 26861 ndash121 319235 57 56 1 83 1226 32 7299 19642 ndash27 3741
36 23 23 0 35 8420 10 9055 7001 0 3230
Total 15 030 5405 9625 7 313 6911 23 234 7841 89 8537 ndash20 3249 2380
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 99
13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk
7262019 Lg_var_slopes - Grupo Los Tigres
httpslidepdfcomreaderfulllgvarslopes-grupo-los-tigres 99
13 Zhao Y and Kim Y C A new optimum pit limit design algo-rithm Reference 3 423ndash3414 Alford C G and Whittle J Application of LerchsndashGrossmannpit optimization to the design of open pit mines In Large open pit
mining conference AusIMMndashIE Aust Newman Combined Group1986 201ndash715 Whittle J The facts and fallacies of open pit optimization (NorthBalwyn Victoria Australia Whittle Programming Pty Ltd 1989)
16 Dowd P A The optimal design of quarries Mineral resourceevaluation II methods and case histories Spec Publ geol Soc Londno 79 1994 141ndash5517 Dowd P A Un ensemble de sous-programmes pour reacutealiser le
krigeage drsquoun bloc irreacutegulier (Montreal Ecole Polytechnique delrsquoUniversiteacute de Montreacuteal 1973) 18 p Rapport Technique EP-73-R 18 Khalokakaie R Computer-aided optimal open pit design withvariable slope angles PhD thesis University of Leeds 1999
19 Dowd P A Bjoumlrkdal gold-mining project northern Sweden
Trans Instn Min Metall (Sect A Min industry) 104 1995 A149ndash63
20 Khalokakaie R Dowd P A and Fowell R J Incorporation of slope design into optimal pit design algorithms Trans Instn Min
Metall (Sect A Min technol) 109 2000 A70ndash6
Authors
R Khalokakaie graduated from the University of Tehran with a
master of science degree in mining engineering He completed a
PhD in optimal open-pit design at the University of Leeds in 1999
and has recently taken up a post as lecturer in mining engineering at
the University of Shahroud Iran
P A Dowd Fellow is Professor of Mining Engineering and head of
the School of Process Environmental and Materials Engineering at
the University of Leeds He was President of the Institution of
Mining and Metallurgy for 1998ndash99
Address Department of Mining and Minerals Engineering
University of Leeds Leeds LS2 9JT England
R J Fowell Fellow was formerly a reader in the University of
Newcastle upon Tyne where he gained his PhD and is now
Reader in Mining Engineering at the University of Leeds
CALL FOR PAPERS amp EXPRESSION OF INTEREST
Third Cardiff Mineral Resource Evaluation Conference
DEALING WITH THE NUGGET EFFECT IN PRACTICEmdash
NUGGET 2001
3ndash4 May 2001 Cardiff Wales
Following the successful Cardiff MRE21 meeting an international two-day symposium on the theme lsquoDealing with the
nugget effect in practice NUGGET 2001rsquo is being planned in association with the Institution of Mining and
Metallurgy The nugget effect poses a major challenge to precious-metal mining companies during exploration evalua-
tion and exploitation This meeting will cover all aspects of dealing with the nugget effect from deposit geology throughgrade control The first dayrsquos programme will comprise presentation of invited and offered papers and the second will
consist of workshops of case studies etc and a panel-led discussion session
Submissions
Synopses of approximately 500 words are invited both as conference papers and for workshop sessions Synopses may
refer to case histories research results andor experiences from actual projects The deadline for receipt of synopses is
1 December 2000
Expression of interest
Anyone interested in attending the meeting should contact the convenor to be placed on the mailing list Please provide
e-mail address if possible
SponsorshipThe organizers are also seeking corporate sponsors for the meeting
Enquiries
Dr Simon Dominy Department of Earth Sciences Cardiff University PO Box 914 Cardiff CF10 3YE Wales UK
Tel +44 (0)29 2087 4924 Fax +44 (0)29 2087 4326 e-mail dominyscardiffacuk