using dynamic programming for aggregating cuts in a single drillhole

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International Journal of Surface Mining, Reclamation

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Using dynamic programming for aggregating cuts in a

single drillholeMark E. Gershon

a & Frederic H. Murphy

a

a School of Business, Temple University , Philadelphia, Pa., 19122, USA

Published online: 27 Apr 2007.

cite this article: Mark E. Gershon & Frederic H. Murphy (1987) Using dynamic programming for aggregating cuts in a single

llhole, International Journal of Surface Mining, Reclamation and Environment, 1:1, 35-40, DOI: 10.1080/09208118708944100

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International Journal

urface Mining1 (1987): 35 40

Usingdynamicprogramming foraggregating cuts in a singledrillhole

MarkE.Gershon Frederic H.Murphy

School

Business

Temple University

Philadelphia Pa.19122 USA

ABSTRACT

In planning a mine

with

layered deposits one must

decide

what to mine as ore or waste, Because

ther-e ar e minimum widths

fo r

waste and or e cuts, th e

dec is io n f or

a depth interval depends on the ore

quality

of

neighboring inte rva ls .

present

a

dynamic programming

approach that o pt im iz es t he cuts

fo r

a

s ingle dr i l. lhole from wh1ch the ore

qual.ities

a re eva luated .

implement a t r i a l

version

using

Lotus

123 to

i l lus t ra te th e

properties

of th e

solution.

INTROUUCTlON

35

Complex

layered

mineral

deposits

consist

of

many

layers of ore interspread

with

layers

of waste

( interburden). These

layers vary

in

both

thickness

and

ore quali ty. To mine

this

type of

deposit ,

one

attempts to mine th e and th e waste

layers

separately, bllt

a

problem ar ises when the layers are

relatively

thin,

smalJer than a minimum

mining

thickness. In

this

case. one needs a decision model

for agg re eaUng th e smaI .le r

layers into

ones

large

enough fo r

mining.

This means that some

waste

may

be mined as

ore

and. some or e may be mined as

waste.

There ar e

deposits

where th e

layers

ar e not pure

miner al o r pure waste,

that i s

there is a con

tinuous

range of quality. In this case i t is no t so

clear whether any

given layer

is to be defined as

or e

or waste. Examples would include o il shale. tar

siinds

or

certain

uranium

deposi ts.

Aggregation

of

lay ers in these deposi t s cannot be accomp.lished by

considering seam thickness only. have to consider

a combination of thickness and

or e

quali ty.

A

large majority of mining operations use a

fixed

cutoff grade without regard to th e

adjacent

Laye r s .

That is any ore

above th is

grade

is pr occs

se. and

any ore below 1s sent to a waste dump. Even

in

the

case

when marginal material

is

sent to a leach dump

wher-e

t he margi na l

o

re

can

be

concentrated,

a mine

uses a doub l e-ct i er-ed cutoff where the fixed dec f s Lnn

t-u l e is:

I f

grade <

Co'

thell

waste;

f Co

<

grade

<

CI then leach;

and f C1 <

grade,

then process.

But Lane (1964) and others have challenged this

practice, advocating th e

use

of a f lexible cutoff

grade

instead, Although

a f lexible cutoff grade is

more di ff j cu l t t.o implement and manage

t does pro-

vide

a production manager th e addit ional f lexibi l i ty

to vary th e

cutoff grade

from time t.o time to keep

th e whole operation

running

smoothly.

The r-esu j r;s of th e appJ.ication described here sup-

port the

argument

of those who

advocate

a flexi.ble

uut.o r

f

grade (Marek and We1henne r . 1985 . The op t t >

mal

aggregation

provided is based on a minimum

average

grade over th e ac-gregated SP-tim. Th.is v r g ~

sometimes con t a

in s

segments that ar-e ightly below

© 1987 A.A.Balkema, P.O.

Box

1675,3000 DR Rotterdam Netherlands

cutoff

and.

i f

t he sur round ing

high grade

segments

ar e

high

enough, may even contain segments

that are

fa r below th e

assumed cutoff. Thus, fo r

the

problem

def ined here, th e ident i f icat ion of one par t icular

segment as ore or

waste depends no t

only on i ts

grade,

bu t on

the

grade

of

th e

surrounding

segments

as

well.

In a larger

context, t 1

so depends on

production schedules, customer contract

requirements

and plant processing settings.

Tradit ionally,

th e

decis ions

of what to process and

how deep to mine have been treated snparately. The

ultimate pit

l imit

problem (Meyer,

1969; Johnson,

1973)

defines

th e deepest

economical

mining

penetra

tion

at any

point in

an open pit mine. In

pract ice,

th e

ultimate

p it is found f i r s t and then product ion

schedUling conducted

Within th is p it .

Ollr resul ts

show

that

th e decis ions

must

be made

simultaneously.

Many l inear programming applicat ions

in

mining

(Albach,

1969 Barbaro and Raman , 1983) develop

product ion schedul es without in troducing the loca

tion of th e wast.e and ore to

the

problem. Various

quantit ies of ore at

different

grndes ar e assumed to

be

available,

a lmos t a s i f

they

harl

already

been

mined and were

s i t t ing in stockpiles. These applica

t ions ar e us efu l fo r higher level planning,

bu t

in

l ight

of

th e diff icul t

sequencing

problems (Gershon,

1983) between

production (m.ining

ore)

and overburden

r emoval (mining was te ), th e posi t ion of th e

or e

must

carry

equal

importance

with i t s

grade in

order

to

be

able to

actually

implement a production

schedule.

But

i t should

now

be

apparent that th e true ul timate

pit cannot be found unti l product ion schedul ing has

been

completed.

The

analysis

herein

also

considers

both

position and gt-ade . since i f either one were

ignored,

th e

results would have no

meaning.

The goal

of

this paper is to optimize th e aggrega

t ions into

m.ineable

layers of are

and interbul den.

The model developed and

presented

here also

yIelds

results

that have a

bearing

on t ~ r important

aspects of mine planning and mine production sche

duling, namely

the

role of

cutoff

grades

- ultimate pit Ijmits

- posJtion and grade in production scheduling



36

The next

section deHcribes

a dynam ic progr am ming

model

developed

fo r

this

purpose. The

remainder

of

the

paper

provides an e xa mp le , app ly ing the methodo

logy

to an oil sha le depos it . The example is imple

mented in OTUS 123, which we use as a shell

to test

various

formulations.

a lso d iscuss ext ensions

to

othe r a spect s of

mine planning.

Model Statement

The data for the problem consists of the assay

results

token

at regular

intervals

of depth using a

core

drawn from a single dril lhole. For each sample

we know

th e

depth

at

which

was taken and

the ore

QUAlity.

This

m ea su re of

quality is represented

by a

single

value. In th e case

of oil

shale. th e measure

of quality can be simply th e oil content pe r unit

volume of rock. As an alternative, we can define an

economJ c

measure that combines var-Ious or e

par-aeev

tel s. In the c as e

of

coal,

th e

measure

includes

sulfur, Btu and ash cumbined

Into

8n estimate

of

prof i tabi l i ty,

In add t ion to

th e

g eo lo gical d ata, th er e a re

operating

parameters that must be

defined fo r each

particular mining operation, These include

th e

mini

mum thickness

a llowab le for

a minable ore

layer,

th e

minimum thickness

fo r

a waste layer, and

th e

costs

and prices that determine

prof i tabi l i ty.

The

thickness

of an indiv.1dual

cu t

cannot be deter

mined independently of the other

cuts.

For

example,

expanding

th e

size

of an

ar e

cu t may

reduce

th e

size

of an lnterburden cu t to below th e minimum width,

necessitating that i t be mined as an ore

layer.

We

t he re fo re , r eso rt to dynam ic progr am ming

techniques.

In many mining opp.rations, a

fixed

cutoff grade is

used to separate th e or-e fl om th e In t erburden .

Figure 1

i l lustr tes

th e characterization of are

and

waste

layers

resulting

from th e st r i t use of

grade cutuffs with no

minimum size

cut. For example,

a re la ye rs C, D and E may each be too narrow to mine

individually, bu t

th e

average grade

fo r

t he se t hr ee

layers. including

th e

two waste layers that

separate

them. may be above the cutoff . The r e su I t wou.ld be

to

mine this segment

as

one seam.

Similarly,

i f

th e

waste

layer between seams F and G

is

too narrow,

then

part of the use fu l ma te ri al from

either F

or

G

needs

to be included in

th e

lnterhur

den.

Thus, this good material is lost. Dynamic

programming allows us

to take

these interactions

i n to cons ide ra ti on.

Dynamic progr am ming, a tool fo r solving problems

that can be v iew ed as a

sequence

of

related

deci

sions,

has been

used

in

th e

mining industry

fo r

a

variety of problems. One example

is th e

ultimate pit

p ro bl em Le

rchs

and G ro ss man n. 1 96 5: Konigsberg,

1982),

Another is t he p roduct ion schedul ing

problem

(Davis

and

Wil liams, 1979) .

Dynamic progr am s have th e

following

structure.

Decisions

can be

ordered

:in a sequence,

such

as

time

or

depth.

At each step,

or

stage, in

th e sequence

th er e a re states

that

describe th e

potential

situations

in which one has to make decisions.

Associated with each state and decision is a rule

that describes th e

resulting

state in th e next step

in the sequence. Dynamic programming yields,the

optimal set

of

decisions for each state and step,

tak.tng into account

current

benefits and future

benefits from subsequent stages. The benefits ar e

expressed in

terms

uf an equation known as a d yn ami c

programm.tng

recurs

ion. See Wagner,

1975.)

With

a

Quality utof f

0.06

0.05

0.04

a

:J

0.03

0

u

0

0.02

0.01

o

V

V

A

B

D E F

G

I

o

Figure

1

Ore

20

epth

Qua lity Versus

40

epth



37

The Dynamic Program

Solution

Acceleration

The decisi.ons made ar e to mine as are or

waste

or to

abandon

th e

mine.

There ar e

two st at es to the

system in

which

d ec is io ns a re

made.

mining as ore

or

waste.

Once th e decision is to abandon there ar e

no

further deci

sions.

Le t

This is a r el at i vel y simple model, ye t we ca n

improve on th e

computational ef f iciency. This i s

w or th wh il e b ec au se e ac h

dr i l l hol e can be par ti t ioned

Into

as many as 1500

one-foot sections,

and

i f

one

wanted to use

th is

model

as par t

of

a lar ger

model

fo r planning mining

benches,

one would want to solve

th e dynamic program repeatedly.

st at e

no t

st at e

o indicates waste

indicates

ore.

The stag e o f th e

system,

t , represents t he d ep th .

The

depth is

divided

into uniform segments, indexed

by

t

t=I, . . . ,D.

The

return depends

on

whether

we

decide

to

remain in

th e same st at e or whether we dec ide to change s t a t e s

from one

interval

to th e ne xt.

I f

we are in s t a t e

and

tile

action

is

to

remain in

i

then

th e

return is

th e

value cost) of

including

one uni t

depth in

action i i e . d is ca rd o r p ro ce ss ). Denote th is

return

by

Rt {i , i ) . I f

we

ar e

in s t a t e i and th e

action

i s

to

change

to

s t a t e

- i , then the

return

is

th e

value cost)

o f f ol lo wi ng action -1

fo r

th e

minimum thickness of

a

cut,

m -i ). The

d ec is io n t o

abandon a t stage

t ha s

th e value a t negative

i f

a

c os t fo r closing th e

mine).

Le t

ft i

be th e future

return

from being

in

st at e i

a t depth inter val

t

through

t he p la nn in g h or iz on .

The dynamic programming

recursion

can

then

be

stated

a s f ol lo ws :

There is a property of th e model that allows us to

improve th e

ef f iciency

of the

solution

within

th e

context of the

dynamic program.

Again, th is

property

i s

based

on th e existence of

th e

minimum

cu t

i n t e r

va.l,

m i .

For

e xam pl e, s ay

we

ar e in

th e

situ atio n

th at,

fo r some stage t , we mine no matter what s t a t e

we

h av e b een in

and th e

ar e being

mined in

th is cu t

i s of a uniformly h ig h g ra de .

then

will continue

to mine as ore the succession of

e a r l i e r s tag es in

th e

dynamic program

u n til

th e

g ra de d ro ps

below a

certain cutof f

level of quality.

Conversely, i f

we

ar e

in a

situ atio n

where we

discard th e material

because

th e ar e quality is uniformly in ferio r no

matter what

has been th e

s ta r ti n g s t at e , we

discard

all stages u n t i l th e ar e quality r ea ch es t he

same

cutof f . This situ atio n is guaranteed

to

occur

whe

never

there is

a

c ut l en gt h gr eater than

or

equal

to

m l)

+ m O

as e i t h e r waste or ore.

The

cutof f

can

be determined by finding th e or e grade that makes

the user

in d ifferen t between

processing

th e material

or discarding i t .

now make t he se n ot io ns more

precise and more general.

make th e follOWing

assumption fo r

th e

remainder

of th e

paper:

Ore quality ca n be

tran slated into

an economic

value and

th is

value increases

monotonically

with

th e quality.

RI O,I) + fm I)+I I)

f l

min

RI I,O)

+ fm O +I O

take advantage of the

min mum

cu t

width to

elimi

nate

computations

a t t h e b eg in ni ng and

ending

s t a

tes.

For 2

t

min[m O ,m I)), that is a t th e

surface, th e r e

Is

no

o p ti m iz a ti o n s i nc e

we must

take

a cu t a t l e a s t

equal

to the smaller of m O or m l)

a t stage

1.

Restating th e r ecu rsio n fo r s ta ge

1:

Th i s a ss u mp t io n

allows us to

estab lish

a single

cutoff

poi nt ,

distinguishing between

th e

p ro fitab le

and unprofitable ore for

processing.

Note that th e

single cutoff is r e I e v a n t fo r th e stretch where

mining continues.

The

d ec is io n o f whether

to

con

tinue mining or

to stop

mining altogether involves

d ifferen t

cutof f s.

There

is no

single cutoff

grade

to

determine

when to

abandon. This decision

depends

no t only

on th e

a ve ra g e g ra de

over

f ut ur e s ta ge s bu t

also on

th e d istrib u tio n

o f g ra de s. To

understand

t hi s, note that if

the ore

quality i s

highly

var iable, much of what is mined is discarded and

processing

costs ar e

saved.

Thus, th e

a ve ra ge g ra de

required

to continue mining i s lower than i f th e ar e

is of uniform

quality

and a ll of

i t

incurs

pro

cessing costs.

I)

a t

f t i = min

where

Rl O,l) is t he v al ue of

mining th e

f i r s t c ut as are

Ri l,O)

i s

th e

c os t

of

removing

th e

f i r s t

cu t

as

overbur

Also,

assume, fo r

example,

that m l)

>

m O . Then,

i f

we

are in st at e

I

with

m O t

m l ,

th e

only

f easible decision i s to

remain in

s t a t e I,

other

wise,

we would

v io late th e

minimum cu t

requirement.

Note

that i f

m O =m I =I,

th at

i s ,

there is

no

mini··

mum cu t size, then

there

is a single

cutoff

level,

C, where

all

m at er ia l of

grade gr eater than

C

will

be

process ed as o re

and

all material

below

th is

grade

will be discarded as waste. By

having

no mini

mum

cu t

size,

each

segment

ca n

be

evaluated indepen

dently.

Since th e value increases monotonically with

th e or e quality, there i s

a

single ar e quality, c,

wheve we ar e indif f er ent

between processing

and

d ~ s c r d i n g

This

c

applies

to

each

segment.

For t

>

D - m l) we face s p e ci a l c o n si d e ra t io n s

in

formulating

th e

dynamic

p ro gr am . T he re

is no minimum

cut for

waste

a t th e bottom of the hole, since th e

minimum cu ts are defined for the

purposes

of

mining

bu t

th e

waste

at th e bottom of the hole

will

no t

be

mined . The o nl y r ea so n fa r mining

waste

is to enable

th e

mining

of allY or e

below.

For

t

>

0 -

m l), if

i

e O

then the decision is

to abandon

w

l th a

t o t a l

re tu rn af

O. and

R O,l)

i s

no t

allowed.

A simlJle consequence of this is tilat

i f

a l l

segments

above th e

cutof f grade are longer t ha n m l)

and

all segments

below th e

cutoff

grade a re l on ge r

than

m O ,

then

t h e a g gr e ga t io n

of

inter vais

into

mining cuts can be done according to

cutoff

grade.

T hi s le ad s

us

to conclude that in

th e

optimal solu

tion, a l l cuts

th at

ar e processed

have

an

average

quality

gr eater than

or

equal

to c and

all cuts

that

ur-e discarded have an

average qual it y

below c.



38

DYN MI PROGR MMING

SOLUTION FOR DETERMINING MINE

CUTS

imme d ia te

r e turn

s t age

data

s t a t e

s t a t e

s t a t e

1 s t a t e

1 abandon

d e c

d ec 1 d ec de c

1

30

0 . 0 0 0

S 9 99 9 9 9 1 0

29

0 . 0 0 1 S 9 99

1 0

9

28

0 . 0 0 0

S 9 99

1 0

1 0

27

0 . 0 0 1

S

3 8

1 0

9

26

0 .0 0 9 S

2 9

1 0

2S

0 . 0 0 8

s

2 2 1 0

2

24

0 .0 1 3 S 9 1 3

23

0 . 0 1 6

s 6

1 0

6

22

0 .0 1 8

s IS

1 0

8

21

0 .0 3 9 S

46

1 0

29

2

0 .0 6 1 S

94

1 0

S l

19 0 .0 2 7

S

lOS

1 0

17

18

0 .0 1 7

S 10 4

1 0

7

17

0 . 0 0 1

S

66

1 0 9

16

0 .0 0 2

S

7 1 0

8

IS

0 . 0 0 1

S 1 9 1 0 9

14

0 .0 0 2

S 3 4 1 0

8

13 O.OlS

S

2 0

1 0

S

12

0 .0 0 3

S

1 9

1 0

7

11

0 . 0 0 1

S 1 9 1 0

9

10

0 .0 0 4

S

1 7

1 0

6

9

0 .0 0 3

S

2 9 1 0

7

8 O OOS

S 2 7 1 0 S

7

0 .0 0 4 S

2 4

1 0

6

6

0 . 0 0 6

S

2 2 1 0 4

S

0 .0 0 9

S

1 6

1 0

1

4

0 . 0 1 1 S

1 0 1 0

1

3

0 .0 1 2

S

2

1 0

2

2

0 . 0 1 0

S

2

1 0

1

0 .0 0 7

999 0

1 0

999

t o t a l

re turn

o p tim a l

so lu t ion

s t a t e

s t a t e

s ta te

1

s t a t e 1

abandon

s t a t e

s t a t e 1

de c dec 1

d e c

d ec 1

S 9 99 9 99

1 0

S

999 1 0

9

S 999 1 0

1 0

S 3 8

1 0

9

S 2 9

1 0 1

S 2 2

1 0

2

S

9

1 0

3

3

S

6

1 0

9

6

9

1 IS

1 0

17

IS

17

10 46

4

46

46

46

41

97 S

97

97 97

92 11 4

36

11 4

11 4

11 4

109 121

87

12 1

121 121

11 6

112 104

11 2

11 6

11 2

11 1

10 4

11 1 10 4

11 1

11 1

10 6

9S 10 6

10 2

10 6

10 6

10 1

87 10 1

98

10 1

1 1

96

92

96

10 6

96

10 6

91

92 91

99

92

99

87

87

86

9

87

9

82 84

82

84

84

84

79

77

79

74 72

74

72

74

74

69 66

69

68

69

69

64

62 64

6S

64 6S

S9 61

S9

64

61

64

S6 64

S4

6S

64

6S

S9

67

S l

67

67 67

62

67

54

67

67

67

932

64 57

932

64



39

0 . 0 7

0 . 0 6

0 . 0 5

:>

0 . 0 4

0

;

0

u

0 . 0 3

0

0 . 0 2

0.01

o

as

a

Function of Depth

o

4 8

12

6

Depth

2 0

2 4

28

Figure

2

Ore

Quali ty

rhcae r e s u l t s

a r e

th e most s implis t ic

c a se s o f th is

pr ob l em ,

c a s e s where

th e solut ions

a r e o b v i o u s .

Our

main r e s u l t , which follows. while no t l ea di ng t o a

complete s ol ut i on

of

th e

problem,

p ro vid es t he b a s is

fo r

ou r

simplif ication.

Assume that th e

decision

a t stage

t

is to process

the ore both when we enter stage t p ro ce s si ng t he

o r-e

an d

when

n o t . Then fo r s t a g e

t - l th e o p t i m a l

decision

is to

p r ~ s s

th e segment

ha s

an

or e

qual i ty

above

c .

LJk ewise, i f

th e d e c i s i o n is

to

d is c a r d no matter what th e

i n i t i a l

s ta te then the

d e c i s i o n

fo r

s t a g e s

t - l

e t c

. . Is to

d is c a r d a t

l e a s t u n t i l a segment having qualjty

gre a t e r

than C

is r-eache

d .

The b e n e f it

o f

t hi s r es ul t i s th a t

e v a lu a tio n s

of

th e f u nc t io n nl e q ua t io n ar e n ec es sa ry o nl y for the

a p pr op r ia t e d e ci s io n

and

no t

both

altern ativ es

when

there is a long s t r e t c h o f ar e that i s mineable o r a

long

stretch

of waste to be

discarded.

EX MPLE PROBLEM

Table 1 and Figure 2 i l lustrate th e resu lts from

using th e dynamic program to determine when to mine

as

or e or waste an d when to abandon. The data on

ar e

q ua lity is adapted from

measurements

of

o i l - s h a l e

concentrat.ions. The re t urn function from mining and

processing

is

IODO g -

.01),

th e cost o f m in in g w as te is

5

and th e cost of aban

dunment Is zero, The

mjniml m

or e cu t

is

4 and th e

minimum

waste

cu t

is 2

The dynamic

program i s

l

ap t cmenteu

in

Lotus

123

w it .h ou t a ny o f

t.he

acce

ler ation

rnsu

I

ts

be

i

ug

used.

Lotus

tur ns

o ut

to

be

an uxnet leut. sh e

lJ

fo r ao l viug t.ns

t

ve rs io u s of

dynamic

programs

since th e recursive str uctur e of

Lotus adapts to th e

r ec ur si on i n

dynamic programming

s ee; for example Ho,

1 9 8 6 ).

The

columns

in

Table

1

ar e th e stage, da ta, th e

immediate

re t urns

fo r al l

combinations

of

decisions

and

states

th e

v alu es o f

th e

f un ct io na l e qu at io ns f or

combinations

of

decisions and s t a t e s

with

th e

f u n c tio n a l

equation

r eplicated fo r e ac h L otu s cell using a stngle

Lotus

command), and th e v alu e o f th e optimal s ol ut i on

d et er m in ed w it h

th e

Lotus X

o p e r a to r .

The

-999

eliminates inf easible decisions. Figure 2

pl ot s

th e

ar e

q u alities

and th e solid b ar s i nd ic at e

th e

or e to

be processed.

A

weakness of Lotus is that

i t

cannot

display

th e maximizing argument).

The s ol ut i on i l lus t ra tes th e prope rt i e s of th e model

shown

above.

F i r s t , th e lowest leveJs of th e

resource ar e

no t

mined because o f th e low qua l i t y of

th e

o re g ra de . The or e grade ha s

to

reach

.0 1

before

mining is pr of itable.

Notice

t h a t

in

th e inter val

from stages 14 to 30 th e inter vals a bo ve and below

th e

minimum

cutof f

gr ade . 05 )

ar e l a rge r

than th e

minimum cu t

widths

so t h a t whether to mine or

no t

depends

only

on th e e-rade quality. Stages

10-13

ar e

mined

because

o f

the high grade a t

stage 13 , Note

that th e

stages

comh.tned with 13 to cr eate th e mini-

mum size

are cut

ar e

th e

adjacent

stages

t ha t

pro

.duce

th e highest average

or e quality.

This

wi.lJ

always be th e case when

th is

ar e cu t is surrounded

by s t ~ v s w it h g ra de s below

th e

cutof f

fo r

a minimum

waste

c ut

and minimum

cu t

w id ths o f ad-jacent

cuts

arp. unaffected.

Extensions

We can

Lden t

l fy 3

a re as fo r pnt

en

t t

a 1 x ensIons or

nppljcations of the mn del b ey on d th e

p r e s e n t

con

t e xt :



40

1.

2.

3.

Add

capabili ty t o i ncorpora te grades

from

multiple commodities,

Use for mine planning with more than one

drlllhole,

Use basis fo r three dimens ional aggrega

t ion.

March 6-10.

Davis, R.E. and C.E.

Williams,

1979, Optimization

Procedures for Open Pit Mine Scheduling ,

ProceeQings,

16th Symposium on t he App li ca ti on o f

Computers and Operations Research in the Minerals

Industry, Tucson Arizona,

October.

The f irst of

these

f i ts into the same approach

pre

s en ted here,

bu t

th e

profit

function

would need

to

ref lect jo in t ly

th e g rade s o f t he v ar io us

com

modities that would be refined from th e are.

The

second

item, mine planning with multiple

drillholes, i l lustr t s a

higher level

use

of the

model. In

this

paper

the

mining levels

are

designed

fo r one dril lhole only,

bu t

the

practical

use

of

such an approach is

limited

to the com-

modities

mentioned

earlier

such as uranium

a ll

shale and ta r sands. To apply th e same approach in

more common seam deposits such as coal or phospha-

tes, i t is

necessary to

apply this

a lgor ithm to

each

drill hole and then to develop an approach to corre

lating

the

results

between

dri l l

holes. Such an.

approach toward mine planning could greatly enhance

the product ion

planning process. Work to

date

in

this area (Rendu, 1982) has focused on geological

correlation

between

the

dril lholes

without

the

bene

f it

of

the economic framework provided

here.

The third item a pp lie s to

both

mine planning and

production

s hedu

ling. In ei

ther

case, planning

decisions are often based

on

sequencing the

removal

of

three dimensional blocks of

are.

Smaller b locks

yield

better

~ e t j l but larger blocks

are

easier to

schedule.

Too

often, th e

tradeoffs involved

lead to

the use of larger aggregated

blocks.

The algorithm

developed

here

performs this type

of aggregat ion ,

but In

only

one dimension. The manner in which the

aggregation takes

place can have a strong

effect

on

th e

production

schedules

produced.

At this time, no

methods of

block

(3-0) aggregation other than

guessing have been developed, but an

extension of

the approach

described here

could

provide

a

basis

fo r

doing

so .

A model has been developed that

solves

the problem

of

aggregating

mine

cuts

into mineable seams in a

s ing le dr li lho l e. Dynamic programming solves

this

problem readily due to the

small

dimension of

the

state

space.

We have also

provided

the means to

make th e dynamic programming

algorithm

more ffi-

cient

by taking advantage of the special structure

of

the problem. These efficiencies become more

important

when this procedure

needs

to be imple

mented

within

the context of more complex are

depo

si ts requiring more dri l lholes . In

addition

to the

problem of

aggregation

In a single

hole. the

approach has

implications

for

related

mining

problems. Among these are cutoff grade strategies,

mIne planning and

production

scheduling. The next

direction

of t he res earch

involves

extensions o f

the

model to analyzing more complex ore

deposits

having

more dril lholes.

This

problem

focuses

on how to

correlate the resulting

seams at

each dril lhole

in a

practical fashion fo r mining.

References

Albach,

H 1967, Long Range

Planning

in Open

Pit

Mining , Management Science, 13, B549-68.

Barbaro, R.W. and R.V. Rama.ni 1983. General.ized

Multiperiod

MIP

Model for Production Scheduling ,

SM

Preprint

83-123, SM Annual

Meeting,

Atlanta,

Gershon, M., 1983 . Optimal Mine Production

Scheduling: Evaluat ion of

Large

Scale

Mathematical

Programming Approaches International Journal

of

Mining

Engineering,

I,

4,

315-329.

Ho , James, 1986.

Optimacros:

Optimal Decis ions with

Spreadsheet

Macros . ORS TlMS

Miami.

October.

Johnson, T.

B. ,

1973. A Comparative Study of Methods

fo r Determining Ultimate Open-Pit Mining

Limits,

Proceedings , 11th Symposium on t he App li ca ti on o f

Computers and Operations Research

in

the Minerals

Industry, Tucson Arizona, April.

Koenigsberg,

E.,

1982. The Optimum Contours of an

Open Pit Mine: An Application

of

Dynamic

Programming

Proceedings,

17th Symposium on the

Application

of

Computers and Operations Research

in t he Mineral s Indus try, Golden, Colorado, April.

Lane K.

1964. Choos ing

the

Optimum

Cutoff

Grade

QuarterlY

of

the Colorado

School of

Mines,

October.

Lerchs, H. and

I.

F. Grossman

1965.

Optimum Design

of Open Pit Mines, Canadian

Inst i tute of

Mining,

Bulletin, Vol, 8, No, 633, January, p p. 4 7- 54 .

Marek, J.M. and H.E. Welhener, 1985. Cutoff Grade

Strategy-A Baiancing Act , SM Preprint 85-320,

SM Fall Meeting , Albuquerque, October 17-19.

Meyer M.

1969. Apply ing

Linear Programming to the

Design of Ultimate Pit Limits, Management

Science,

16,

BI21-35.

Rendu, J.M., 1982. Computer Est ima tion o f Ore and

Waste Zones

in

Complex

Mineral

Deposits ,

SME

Preprint 82-95, SME Annual

Meeting, Dallas,

February

14-18.

Wagner H.M. 1975. Principles

of

Operations

~ e s e r c h Prentice-Hall, Englewood Cliffs , New

Jersey.

using dynamic programming for aggregating cuts in a single drillhole

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