7/18/2019 163653 L G Alogorithm

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Transactions, C.I.M., Volume LXV1I I, 1965,  pp . 17-24

ABSTRACT Open-Pit Model

n o p e n - p it m i n i n g o p e r a t i on c a n ! e " i e # e $ a s a pr oc es s !% #& ic & t& e o pe n sur fa c e o f a mi ne i s c o nt i nu-o u sl y $e'orme$. T&e p lann in( o' a min in( pro(ram in-"ol"es t&e $esi(n o' t&e ' inal s&ape o' t&is open sur'ace.T & e a p p r oa c & $ e " e lo p e$ i n t & i s p a p er i s ! a s e $ o n t & e'ollo#in( assumptions) 1. t&e t%pe o' material , i ts mine"alue an$ i ts e*tract ion coat is (i"en 'or eac& point) 2.restr ict ions on t&e (eometr% o' t&e pi t are speci' ie$ +sur-' a c e ! o u n $ a r i e s a n $ m a * i m u m a l l o # a ! le # a l l s l o p e s . t&e o!/ecti"e is to ma*imi0e total pro'i t total mine"alue o' material e*tracte$ minus total e*tract ion cost .

T#o numeric met&o$s are propose$) simple $%namic p ro( rammi n( a l(o r i t&m f o r t & e t w o - d i m e n si o n a l p i t ( o ra s ingle vertica l section of a mine), and a more e laborategrap a lgori tm for t&e (eneral t&ree-$imensional  pi t.

=mow

Intro$ uction

A 3C m i n i n ( p r o ( r a m i s a c o m p l e * o p e r a -

tion t&at ma% e*ten$ o"er man% %ears, an$ in-"ol"e &u(e capital e*pen$itures an$ ris, 8e'ore !n-

d e r t a " i ng s u c & a n o p e r a t io n , i t m u s t ! e n o #n #&atare t&ere is to !e mine$ +t%pes, (ra$es, uanti t iesan$ spat ial $ is tri!ut ion an$ &o# muc& o' t&e ores&oul$ !e mine$ to mae t&e operation pro'ita!le.

T&e reser"es o' ore an$ i t s spatial $istri!ution aree s t i m at e $ ! % ( e o l o( i c a l i n t e r p r e ta t i o n o ' t & e i n ' o r m a -t i o n o!taine$ 'rom $ r il l cores . T&e o ! /e c t o '  pi t $e -s i(n t&en is to $etermine t&e amount o' ore to !emine$.

ssumin( t&at t&e concentration o' ores an$ im- puri ti es in no#n at ea c& poi nt , t&e  pro! lem is to $e-ci$e #&at t&e ultimate contour o' t&e pit #ill !e an$

in #&at sta(es t&is contour is to !e reac&e$. Let usnote t&at i', #it& respect to t&e (lo!al o!/ecti"es o' a

minin( pro(ram, an optimum pit contour e*ists, an$i' t&e minin( operation is to !e optimi0e$, t&en t&iscontour must !e no#n, i' onl% to minimi0e t&e totalcost o' minin(.

:;o# enior esearc& Mat&ematician, <eneral Motorsesearc& La!orator%, =arren, Mic&.

8esi$es pit $esi(n, plannin( ma% !ear on uestionssuc& as)  #&a t ma r et to se l ec t )

− #&at up(ra$in( plants to install

  #&at ua nt i t ie s to e*t rac t , as a 'unc t ion o'time

  # & a t m i n i n ( m e t & o $ s t o u s e   #& at tr an sp or ta ti on 'a ci li ti es to pr o" i$ e.

T&ere is an intimate relations&ip !et#een all t&ea!o"e points, an$ it is meanin(less to consi$er an%one component o' plannin( separatel%. mat&ema-tical mo$el tain( into account all possi!le alterna-ti"es simultaneousl% #oul$, &o#e"er, !e o' 'ormi$a!les i0e a n$ i t s solution #oul$ !e !e%on$ t&e means o'  presen t no#&o#, T&e mo$ el propo se $ in t& is paper 

#ill ser"e to e*plore alternatives in  pi t $esi(n, (i"ena r e al o r a & %p ot & et i ca l e c on om i ca l en"i ronment

+maret situation, plant con'i(uration, etc.. T&is en-"ironment is $escri!e$ !% t&e mine "alue o' all ores pr es ent an $ t& e e* tr ac ti on co st o' or es an $ #a st ematerials. T&e o!/ecti"e t&en is to $esi(n t&e contour o' a pit so as to ma*imi0e t&e $i''erence !et#een t&etotal mine "alue o' ore e*tracte$ an$ t&e total e*trac-tion cost o' ore an$ #aste. T&e sole restrictions con-cern t&e (eometr% o' t&e pi t t&e #all s lopes o' t&e pi t mu st not e*c ee $ ce rta in (i "e n an (l es t& at ma %"ar% #it& t&e $ept& o' t&e pit or #it& t&e material.

nal%ticall%, #e can e*press t&e pro!lem as 'ollo#s)Let ", c an$ no !e t&ree $ensit% 'unctions $e'ine$ ateac& point o' a t&ree-$imensional space.

" +*. %. 0    mine "alue o' ore per unit "olumeci*, %, 0 > e*traction cost per unit "olumero+*. %, 0 " +*, %, 0 c+*, %, 0 > pro'it per unit "olume.

Let a +*, %, a $e'ine an an(le at eac& point an$ let !e t&e 'amil% o' sur'aces suc& t&at at no point $oest&eir slope, #it& respect to a. 'i*e$ &ori0ontal plane,e*cee$ a.

Let V be t&e 'amil% o' "olumes correspon$in( tot&e 'amil%, , o' sur'aces. T&e  pro!lem is to 'in$,amon( all "olumes, V, one t&at ma*imi0es t&e inte(ral

a $* $% cla

I

■

Helmut Lerchs4

#onoger of Scientific Services,#anteca$ %otacentre,&nternational Business #acines

Cu' td,'

IngoF. Grossmann

#anager,

#anagement Science Applications'&nternational Busness #acinesCo' td'

Optimum Design of 

Open-Pit Mines

?oint C.@... ari$ @..., Con'erence,

Montreal, Ma% 27-29, 1964

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