lg_var_slopes - grupo los tigres

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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



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)



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



(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



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



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



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






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



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



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)























slope angles






























levels






slope angles


















(1)

(2)dyk t z

1 = -( ) acute dim


dx

k t z1 =

-( ) acute dim


A79





(3)

(4)

(5)

(6)

(7)

(8)














than j

(11)


(12)


(13)


(14)















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


dx

k t z2 =

-( ) acute dim


A80




(b)

(c)

(a)
































1 and the number of




(15)

(16)

(17)

(18)




1 are consid-














(19)

(20)


1 and the number of




(21)

(22)

(23)

(24)


2 are consid-







m

dx

x22 2

2

=dim

m

dx

x12 1

2

=dim

dx dxz

k j ml 22

21

21

1= +

[ ]dim


dx dxz

k j ml 12

11

11

1= +

[ ]dim


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


dxz

k j 11 =

[ ]dim


A81



45deg
















(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)


tions (m11 n1

1 m21 and n2




tion volume
























vectors

Triangle equation













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


dx dx

z

i ml j k t

t i

i

t

t

2 2

1

1

1

=

+- +[ ]

=

-

aringdim


dy dy

z

i j nl k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-1

dx dx

z

i ml j k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-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





Case study







north of Sweden19




























A83


technical regions



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










Conclusion













the slope angles



















A84

Table 3 Optimum pit



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











Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries

























slope angles






























levels






slope angles


















(1)

(2)dyk t z

1 = -( ) acute dim


dx

k t z1 =

-( ) acute dim


A79





(3)

(4)

(5)

(6)

(7)

(8)














than j

(11)


(12)


(13)


(14)















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


dx

k t z2 =

-( ) acute dim


A80




(b)

(c)

(a)
































1 and the number of




(15)

(16)

(17)

(18)




1 are consid-














(19)

(20)


1 and the number of




(21)

(22)

(23)

(24)


2 are consid-







m

dx

x22 2

2

=dim

m

dx

x12 1

2

=dim

dx dxz

k j ml 22

21

21

1= +

[ ]dim


dx dxz

k j ml 12

11

11

1= +

[ ]dim


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


dxz

k j 11 =

[ ]dim


A81



45deg
















(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)


tions (m11 n1

1 m21 and n2




tion volume
























vectors

Triangle equation













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


dx dx

z

i ml j k t

t i

i

t

t

2 2

1

1

1

=

+- +[ ]

=

-

aringdim


dy dy

z

i j nl k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-1

dx dx

z

i ml j k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-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





Case study







north of Sweden19




























A83


technical regions



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










Conclusion













the slope angles



















A84

Table 3 Optimum pit



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











Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries





(3)

(4)

(5)

(6)

(7)

(8)














than j

(11)


(12)


(13)


(14)















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


dx

k t z2 =

-( ) acute dim


A80




(b)

(c)

(a)
































1 and the number of




(15)

(16)

(17)

(18)




1 are consid-














(19)

(20)


1 and the number of




(21)

(22)

(23)

(24)


2 are consid-







m

dx

x22 2

2

=dim

m

dx

x12 1

2

=dim

dx dxz

k j ml 22

21

21

1= +

[ ]dim


dx dxz

k j ml 12

11

11

1= +

[ ]dim


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


dxz

k j 11 =

[ ]dim


A81



45deg
















(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)


tions (m11 n1

1 m21 and n2




tion volume
























vectors

Triangle equation













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


dx dx

z

i ml j k t

t i

i

t

t

2 2

1

1

1

=

+- +[ ]

=

-

aringdim


dy dy

z

i j nl k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-1

dx dx

z

i ml j k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-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





Case study







north of Sweden19




























A83


technical regions



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










Conclusion













the slope angles



















A84

Table 3 Optimum pit



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











Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries

































1 and the number of




(15)

(16)

(17)

(18)




1 are consid-














(19)

(20)


1 and the number of




(21)

(22)

(23)

(24)


2 are consid-







m

dx

x22 2

2

=dim

m

dx

x12 1

2

=dim

dx dxz

k j ml 22

21

21

1= +

[ ]dim


dx dxz

k j ml 12

11

11

1= +

[ ]dim


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


dxz

k j 11 =

[ ]dim


A81



45deg
















(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)


tions (m11 n1

1 m21 and n2




tion volume
























vectors

Triangle equation













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


dx dx

z

i ml j k t

t i

i

t

t

2 2

1

1

1

=

+- +[ ]

=

-

aringdim


dy dy

z

i j nl k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-1

dx dx

z

i ml j k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-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





Case study







north of Sweden19




























A83


technical regions



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










Conclusion













the slope angles



















A84

Table 3 Optimum pit



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











Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries
















(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)


tions (m11 n1

1 m21 and n2




tion volume
























vectors

Triangle equation













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


dx dx

z

i ml j k t

t i

i

t

t

2 2

1

1

1

=

+- +[ ]

=

-

aringdim


dy dy

z

i j nl k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-1

dx dx

z

i ml j k t

t i

i

t

t

1 1

1

1

1

=

+- +[ ]

=

-

aring

dim


-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





Case study







north of Sweden19




























A83


technical regions



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










Conclusion













the slope angles



















A84

Table 3 Optimum pit



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











Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries







Case study







north of Sweden19




























A83


technical regions



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










Conclusion













the slope angles



















A84

Table 3 Optimum pit



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











Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries





Conclusion













the slope angles



















A84

Table 3 Optimum pit



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











Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries













Authors


















NUGGET 2001







Submissions



1 December 2000





Enquiries



lg_var_slopes - grupo los tigres

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