Helm ut ler lerch chss"'
Optimum Design
Ma nager of Scientific Services, Services, Mantreal Datacentre, Datacentre, lnternational Busi Busin n ess Machin es Ca.. Ltd Ca Ltd., .,
of
Open-Pit Mines
lngo F. F. Grossma nn Manager , Manage Manag ement Science !!lications, !!lications, lnternatianal Business Machines C o. Ltd .
oint C.#.$.S. ond #.$.S. .$.S.. . Conference, Montreal,, Mo% &(&) Montreal &(&),, *)+
-r onsoct onsoct ions, C. l . M ., /o l ume L0/ 1 1 1 , *)+2, pp pp.. * *(& (&
Bulletin for anuar%, *)+2. Montreal BS-$CBS -$C-
n o!en(! o!en(!iit minin mining g o!e o!er r ati ation ca can n 3e .vieed ª:' a !roce ss 3 !rocess 3% % 4 hi hic ch the 5*6 5*6en surfa urface ce of a mme is contmu ousl% deforme eformed d. -he !lanning of a minin g !rogram in rnlves the des design o f th the e final sha! ha!e e of this o ! !e en sur face. face. -he a!!roach de,7e de, 7elo!ed lo!ed in thi thiss ! !a a ! !er er is 38i 38ise sed d . on he follo4 in ing g ass ssum um !tio !tions8 l. th the e t% *6 of mate materi rial, al, it nune ,7aue and a nd it its s e9 e9tt rac ractt io ion n cost is g*ve g*ven n for eac each h !omt: &. r estr estr ictions ictions on the geometr geomet r % of the !it are s ! !e ecifi cifie ed ;sur face 3o 3ound und arie aries s an and d ma9imu m allo llo4a3le 4a3le 4a ll slo! lo!es< es< : =. the o3>ecti o3>ect ive i s to ma9i ma9 i m i?e total !r !r ofit ofit total mine min e value of mate material e9 e9tract tracte ed minus minu s total e9 e9tra tract ctio ion n cost.
(
-4o num numeric eric methods are !ro !ro!o !ose sed d 8 .A sim !le d%namic !rog !rogr r ammin amming g al go gor r ithm ithm for the t4o(dimensional t4o(dimens ional !it ;or a single ve vert rtical ical sec section tion of a m in ine<, e<, and. a moe ela3@rate ela3@rat e gra! ra!h h algo gorith rith m fo forr th the e ge gen neral thr thr ee( ee( d*me d*m ens*ona *onall !*t.
lntrod uction SA$FC mi ning !rog r am am i s a com!le9 co m!le9 o!era tion that ma% e9te e9tend nd ove overr man% %ea %ear r s, and in volve h uge ca!i ca!ita tall e9!e e9!end nd i tu re ress and ri ris. s. Before Befo re un dertaing such an o!eration, it m ust 3e n no4n o4n 4hat ore there i s to 3e mi ned ; t%!es, grades, uantities and s!atial d istri3ution < ancl h o4 much of the ore shou ld 3e m ined to mae the o!erati o!eration !rofita3 !rofita3lle.
-he rese reserves rves of ore and i ts s!atia s!atiall distri3ution are estimated 3% geological i nter!retation of the in informa forma tion o3tai ned from d rill cores. -he o3>ect of !it de sign then i s to determine the amou nt of o re to 3e mi ned. ssumi ng that the concentration of ores and im !urities is no4n at each !oint, the !r o3lem o3le m is to de cide 4hat the ultimate contour of the !it 4ill 4il l 3e and in 4hat stages this contou r is to 3e reached . Let us note that if, 4ith res!ect to the glo3al o3>ectives of a mining !rogram, an o!timum !it contour e9ists, and if the mi ning ning o!erat o!eratii on is to 3e o!timi? o!ti mi?ed, ed, then then this contour must 3e no4n, if on onll % to minimi?e the total cost of mining.
#!en(it Model Besides !it design, !lanning such as8
ma% 3ear on uestions
4hat ma r et to select : 4hat u !grad ing !lants to install : ( 4hat uanti ties to e9tract, as a function of time: ( 4hat mi ni ng m ethocls to use: 4hat trans!ortation facili ties to !rovide.
-here is an in ti mate relationsh i! 3et4een alI the a3ove !oints, and it is meani ngless to consider an% one com!onent of !la nni ng se!aratel%. mathema tical model tai ng i n to account alI !ossi3le alterna tives simu ltaneousl% 4ou ld, ho4ever, 3e of formida3le si?e ancl its solu t ion 4ou ld 3e 3e%ond the means of !resent no4ho4. -he model !ro!osed in this !a!er 4ill serve to e9!lore alternatives in !it design, given a real or a h%!othetical economical environment ;mar et si tuation, !lant configu ration, etc.<. -his en vironment is d escri 3ed 3% the mine value of ali ores !resent and the e9traction cost of ores and 4aste mater ials. -he o3 > ective then is to design the contour of a !it so as to ma9imi?e the d ifference 3et4een the
total mine val ue of ore e9trncted and the total e9trac tion cost of ore and 4aste. -he sote restrictions con cern the geom etr% of the !i t : the 4all slo!es of the !it m ust not e9ceed certai n given angles that ma% var% 4ith the de!th of the !it or 4ith the material. nal%ticall%, 4e can e9!ress the !ro3lem as follo4s8 Let v, e and m 3e three densit% functions def ined at each !oint of a three(dimensional s!ace. v ;9. %, ?< = mine J"alue of ore !er unit volume c;9. %, ?< ( e9traction cost !er unit volume m ;9. %, ?< v;9, %, ?< (c;9, %, ?< K !ro'it !er unit J olume.
Let a ;9, %, ?< define an angle at each !oint and I et S 3e the famil% of surfaces such that at no !oint does their slo!e, 4ith r es!ect to a fi9ed hori?ontal !lane, e9ceed a. Let / 3e the fam il% of vol umes corres!onding to the famil%, S, of surfaces. -he !ro3lem is to find, among ali volum es, /, one that ma9imi?es the integral f.m;9, %, ?< d9 d% d? 47
Two-Dimensional Pit Let us select the units u, and ui of a rectangular grid s%stem such that K tan
a
U j
For each unit r ectangle ;i, j ) determine the uan tit% m *i = v, C::. Construct a ne4 ta3leau ;Fig ure &< 4ith the uantiti es
(
8/1:: =
m: . Ki
Figure
M,: re!resents the !rofit reali?ed in e9tracti ng a single column 4ith element ;i, > < at its 3ase. 1n a final ta3leau, add a ro4 ?ero and com!ute the follo4ing u antities8
l.
oi = # then, column 3% column starting 4ith column l8 :: K M:N m a9 ;:.N , ( *< 4ith K (*' #' *
1ndicate the ma9imum 3% an arro4 going from ;i, > < to ;i + , j-1). -he inter!retation of the ,: is as follo4s8
Generall%, there is no sim!le anal%tical r e !resenta tion for the functions v and e: conseuentl%, numeric methods must 3e used. -he traditional a!!roach is to ** is the ma9imum !ossi3le co ntr i 3u tion of columns divide the 4hole !it into !arallel vertical sections, * to j to an% feasi3le !i t that contains the element and to consider each section as a t4o(dimensional !it. ;i, j ) on its contour. It follo4s that if the element -he techniue used to determine the contour of a ; i, > < is !art of the o!timum contour, then this con section consists in moving three straight ines, re! tou(, to the lef t of element ; i, > < , can 3e traced 3% resenting the 3ottorn of the !it and t4o 4alls, at fol lo4rng the arro4s starting from element ; i, j ) . slo!es a ;see Figu re 1), and in evaluating the ore and o4, an% f easi3le !it contour must contain at least the e9traction cost of mater ials limited 3% the three one ele ment of the f irst ro4. If the ma9imum value lines. -he conf igu ration of lines %ielding the 3est re of in the f irst ro4 is !ositive, then the o!timum sults is then selected. ;Her e, a i s taen to 3e constant contou r is o3tained 3% follo4ing the arro4s from and over the entire !it < . to the Jef t of this element. If ali elements of the f irst -he follo4ing d%namic !r ogramming techniue is ro4 are negative, then there e9ists no contou r 4ith !ositive !rofit. sim!ler , faster and more accu rate.
Oi'?E
-4-
a = 2 ffi¡ j
=
M:: K
/:
(
Cil
1: m , K1
Po¡= O
F=F
= ;;
2
JC =c ]r= =3c==
= 2 O
8.
,
,_
....:._-.L .:
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!_:.!..!l:=.!JiE.L. '--J .!_=-..;_-=:.L.:::..L_J
Figur e &.(D%namic rogramming. 48
-he Canadian Mining and Metallurgical
(( ----:::: Three-Dimensional Pit
mass m, is associated to each verte9 9*, and if M r is the total mass of a set of vertices P, then the !ro3lem of o!timum !it design comes to find ing in a gra!h G a closure P 4ith ma9imum mass or, shortl%, a ma9i mum closure of G. ; See Figure =< .
hen the o!timum contou rs of ali the vertical sec tions are assem3led, it invaria3l% tu rns out that the% do not fit together 3ecause the 4all slo!es in a vertical section and at right angles 4ith the sections that 4ere o !timi?ed e9ceed the !ermissi3le angle ex. -he 4alls -his !ro3lem can 3e vie4ed as an e9treme case of and the 3ottom of the !it are then "smoothed out." the time(cost o!timi?ation !ro3lem in !ro>ect net -his taes a great amount of effort and the resu lting 4ors, to 4hich severa< solutions have 3een !ro!osed !it contour ma% 3e far from o!timum. Let us note ;&, =, , 2< . It can also 3e transformed into a that 3ecause the d%namic !rogramming a!!roach net4or flo4 !ro3lem. Ho4ever, there are o3vious %ields not onl% the o!timum contour 3ut also ?.11 alter com!uta tional advantages to 3e gained from a nate o!tima, if such e9ist, as 4ell as ne9t 3est solu direct a! !roach : these aclvantages 3ecome tions, i t can 3e of hel! i n "smoothing" the !it. im!ortant 4hen the gra!hs considerecl contain a -he d%namic !rogramming a!!roach 3ecomes im ver% large num3er of elements, as ma% 3e the case !ractical in three dimensions. 1nstead, a gra!h al for an o!en( !it model. gorithm can 3e a!!lied. -he model is derived as fol n eff ective algorithm to find the ma9imum clos o4s 8 Let the entire !i t 3e divided into a set of vol u re of a gra!h is develo!ed in the !!endi9 . -he u me elements /:. -his division can 3e u ite ar3itrar%, !rocedure starts 4ith the construction of a tree -Q 3u t ma% also 3e o3tained 3% taing for /: the unit in G. -Q is then transformed into successive trees volumes defined 3% a three(dimensional grid. ssoci -', -&, E E E -" follo4ing sim!le rnles until no further transfor mation is !ossi3le. -he ma9i mum closure of ate to each volume element /: a mass G is then given 3% the vertices of a set of 4ell(iden tif ied 3ranches of the final tree. ffi¡ K \"¡ - C; -3e decom!osition of the !it into elementar% vol umes 4here v: and C: are the mi ne value and the e9traction /: 4ill cle!end on the structu re of the !it itself and on cost of element /:. Let each element /: 3e re!resented the function a ;9, %, ?< . hen a is constant, as is the 3% a verte9 9, of a gra!h. Dra4 an are ;9*, 0< if case in most instances, one of the grid s%s tems / is ad>acen t to /:, that is, /: and /have at east sho4n in Figure can 3e taen, 4ith !ro!er selection one !oint in common, and if the mining of vol ume /: is not !ermissi3le unless volume /i is also mined. of un its on the a9is. e thus o3tain a directed three(dimensional gra!h G K ; 0, < 4ith a set of vertices 0 and a set of ares . n% feasi3le contour of the !it is re!resented 3% a closure of G, that is, a set of vertices P such that if a verte9 9* 3elongs to P and if the are ;9*, 9*< e9ists in then the verte9 9i must also 3elong to P. If a
-!
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1
_,
J
Figur e .
- I
-1
-"
1-
- t
I
1
!
- !
\
---
1
-
-1
-he three(d imensional !it model can 3e illustrated 3% a !h%sical analogue. In Figure 2, each 3loc has a grid !oint at its center through 4hich there is an u!4ard force ;the val ue of the ore in the 3loc < and a do4n4ard force ;the cost of removing the ore< . -he resulting force in a 3loc is indicated 4ith an arro4. If the s%stem is 1ef t to move freel% one unit along a vertical a9is, sorne of the 3locs 4ill 3e lif ted. -he total 4or done in this movement is F X 1= F", 4here F is the resu lting force of all 3locs that !ar tici !ate in the movement. Ho4ever, the movement of an% free mechanical s%stem is such as to ma9imi?e the 4or done. Hence, F is the ma9imum resulting force over an% set of 3locs that can freel% move u ! 4ard in this s%stem, and, retu rning to ou r model, the 3loc s 4ill se!arate along the o!timum !it contom.
1 •
Figure =.
Búlletin for January
1965 Montreal
Figure 5.
9
Three-Dimensional Pit
hen the o!timum contours of ali the vertical sec tions are assem3led, it invaria3l% turns out that the% do not fit together 3ecause the 4all slo!es in a vertical section and at right angles 4ith the sections that 4ere o!timi?ed e9ceed the !ermissi3 le angle a. -he 4alls and the 3ottom of the !it are then "smoothed out." -his taes a great amount of effort and the resulting !it contour ma% 3e far from o!timum . Let us note that 3ecause the d%namic !rogramming a!,roach %ields not onl% the o!timum contour 3ut also alI alter nate o!tima, if such e9ist, as 4ell as ne9t 3est solu tions, it can 3e of hel ! in "smoothing" the !it. -he d%namic !rogramming a !!roach 3ecomes im !ractical in three dimensions. 1nstead, a gra!h al gorithm can 3e a!!lied. -he model is derived as fol lo4s 8 Let the entire !i t 3e divided into a set of vol ume elements /,. -his division can 3e u ite ar3itrar%, 3ut ma% also 3e o3tained 3% taing for /, the unit J'#lumes defined 3% a three(dimensional grid. ssoci ate to each volume element /, a mass
4here v, and c, are the mine value and the e9traction cost of element /,. Let each element /, 3e re!resented 3% a verte9 9, of a gra!h. Dra4 an are ;9,, 9i< if /i is ad>acent to /,, that is, /i and /> have at least one !oint in common, and if the mining of volume /* is not !ermissi3le unless volume /i is also mined. e thus o3tain a directed three(dimensional gra!h G = ;0, < 4ith a set of vertices 0 and a set of ares . n% feasi 3le contour of the !it is re!resented 3% a closu1e of G, that is, a set of vertices P such that if a verte9 9, 3elongs to P and if the are ;9*, 0 < e9ists in then the verte9 0i must also 3elong to P. 1f a
-\
1._
- I
1
'
J J
1
-1
l
1-
!
l
-
- 1
! -1
mass m * is associated to each verte9 9,, and if M,. is the total mass of a set of vertices P, then the !ro3lem of o!timum !it design comes to finding in a gra!h G a closure P 4ith ma9imum mass or, shortl%, a ma9i mum closure of G. ; See Figure =< . -his !ro3lem can 3e vie4ed as an e9treme case of the time(cost o!timi ?ation !ro3lem in !ro>ect net 4ors, to 4hich severaI solutions have 3een !ro!osed ;&, =, , 2< . It can also 3e transformed i nto a net4or flo4 !ro3lem. Ho4ever, there are o3vious com!uta tional advantages to 3e gained from a direct a! !roach : these advantages 3ecome im !ortant 4hen the gra!hs considered contain a ver% large num3er of elements, as ma% 3e the case for an o!en(!it model. n eff ective algorithm to f ind the ma9imum clos ure of a gra!h is develo!ed in the !!endi9. -he !roced ure starts 4ith the construction of a tree -Q in G. -Q is then transformed into successive trees -', -&, E E E -n follo4ing sim!le rules until no further transformation is !ossi3le. -he ma9imum closure of G is then given 3% the vertices of a set of 4ell(iden tified 3ranches of the final tree.
-he decom!osition of the !it into elementar% vol umes /, 4ill de!end on the structure of the !it itself and on the function a ;9, %, ?< . hen a is constant, as is the case in most instances, one of the grid s%s tems sho4n in Figure can 3e taen, 4ith !ro!er selection of u nits on the a9is.
Figure .
-he three(dimensional !it model can 3e illustrated 3% a !h%sical analogue. In Figure 2, each 3loc has a grid !oint at its center through 4hich there is an u!4ard force ;the value of the ore in the 3loc < and a do4n4ard force ;the cost of removing the ore< . -he resulting force in a 3loc is indicated 4ith an arro4. If the s%stem is lef t to move freel% one unit along a vertical a9is, sorne of the 3locs 4ill 3e lif ted. -he total 4or done in this movement is F X 1= F', 4here F is the resulting force of ali 3locs that !ar tici!ate in the movement. Ho4ever, the movement of an% free mechanical s%stem is such as to ma9imi?e the 4or done. Hence, F is the ma9imum resulting force over an% set of 3locs that can freel% move u! 4ard in this s%stem, and, returning to ou r model, the 3locs 4ill se !arate along the o!timum !it contour.
LU
- - --- -
, ¡
1
1
t
i
Figure =.
Búlletin for January,
1965# Montreal
Figur e 2. 9
Porometri$ %nolysis -he esta3lished algorithm !rovides solutions to the final contour of a !it. -here are, ho4ever, virtualJ% unlimited nurn3er s of 4a%s of reaching a final con tour, each 4a% having a different cash flo4 !attern. Figure + illustrates sorne of the !ossi3le cash flo4s.
'((.88888.8 ,.c.(..(((((((((((: Figure +.(Cash Flo4 atterns
-
n o!tirnurn digging !attern rnight 3e one in 4hich the integral of the cash fJo4 curve is ma9imum. -he !ro3lem of designing intermediate !it contours can 3ecome e9tremel% com!le9. -he follo4ing anal%sis 4ill highlight sorne !ro!erties of the !it model, and the results rna% !rovide a 3asis for the selection of interrnediate contours. Let us add a restriction to our !it rnodel. Su!!osing that 4e 4ant to ma9irni?e the !rofit in the f i r st %ear of o!erations and that our rnining ca!acit% is limited to a total volume /. hat is the o!timum contour no4 @ -o ans4er this uestion 4e shall consider the function PK M- W 4here M is the mass of a closu re, / the volu me of the closu re and ,J a !ositi ve scalar. 1nstead of ma9i mi?ing M as 4e did in ou r 3asic rnodel, 4e no4 4ant to rna9imi?e . -his !ro3lern can 3e transforrned into the 3asic !ro3lem 3% su3stituti ng each elernenta r% rna ss 3% a ne4 mass For ,J
= # 4e
'( K , o3tain ou r ol d solution : 4hen ,J i n(
-
creases, decrea ses, 3u t for suff icientl% small incre sta% ments of ,J the o!timum contour and / 4ill constan t.
p
v., ¡.---
,
*
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-- -
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*
V.!- - (*( ( _ ,I _----...... 1
* *
" Mo
M, (
(((((
f'
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&
M
((
(
((
. #$%.
(:(,,...
.
$
*** (( ( (-f.-& *
Figur e .(Ste !s ;left and right, a3oPe< involvcd in determ ining the sha!e ;lo4 er ef t< of Curve M --= M ;/<.
*
*
*
* 1 1
"
/ /, -he Canadian Mining and Metallurgical
ith a suf ficientl% large ,J, the contour 4ill >um! to a smaller volume. -he function / = / ;,J< is a ste! function. = ;.< is !iece4ise linear and con ve9 : indeed, as long as / is constant, is linear 4ith ,J and the slo!e of the line is /. s / >um!s to a smaller value so does the slo!e. M K N / Hence, the value of M corres!onding to a volume / is given 3% the intersection of the segment of slo!e / and the a9is #. For each ine segment of ;,J<, 4e can o3tain a !oint of the curve M KM ;/< . -hese !oints corres!ond to o!timum contours for given vol umes /. -he total curve M M ;/< cannot 3e gen erated 3% this !rocess, 3ut its sha!e is sho4n in Fig u re . Bet4een an% t4o of its characteristic !oints ;M&, /&<, ;M,, /,<, the cu rve M ;/< is conve9. 1n deed, if 4e go 3ac to the cu rve / ;,J<, the intermedi ate volume /, def i nes the value ,J&. -he !oint B on the surface ,,J, re!resenti ng the o!timum contour for /i, must 3e situated 3elo4 the cu rve ;,J< . -o o3tain M*, 4e dra4, from B, a segment of slo!e /, and tae its intersection 4i th #. e can no4 4rite
K
M:
(
R.&/ M? R.&/& = M* <,&/*
(
(
From the euali t%
Ma&imum Closure of a 'ra(h
directed gra!h G K ;0, < is defined 3% a set of elements 0 called the vertices of G, together 4ith a set of ordered !airs of elements a, = ;9, %<, called the ares of G. -he gra!h G also defines a function r ma!!ing 0 into 0 and such that % -r
M
,J&
% M z +
;/
K tI
&
(
V2)
:
1)
But a !oint D on the segment ;M&, /&<, ;M,, /,<, has a value M'* = M z
+
;/:
(
;&<
/&<
Frorn 1) and 2), it results that !oint C must in deed 3e situated 3elo4 !oint D. -he !ro!osed gra!h algorithm can 3e easil% e9tended to !ermit such !ara metric studies.
1n summar%, 4e have esta3lished the sha!e of the curve M ;/< and sho4n ho4 its characteristic !oints can 3e o3tained. -o each !oint of this curve corres !onds a contour that is o!ti mum if the volume mined is e9actl% /. n interesting f eature of the curve M ;/< is that given t4o o!timum contours C. and C3 eorres!onding to t4o volumes /. and V* then, for /" / 3, the contour C. com!letel% encloses the con tour C., that is, an% volume element eontained in C. is also contained in cE. It follo4s that if no other restrictions are im!osed, the ore3od% can 3e de!leted along the cu rve M ;/<. -his rnining !attern 4ill ma9imi?e the integral of eash flo4 4ith res !eet to total volume mined TM ;/< indeed is a cash flo4U.
o3tained 3% su!!ressing an are a, in a rooted tree has t4o eom!onents. -he com!onent -, 0*, *< 4hich does not contain the root of the tree is ealled a b"nch of -. -he root of the 3ranch is the verte9 of the 3ranch that is ad>aeent to the are a*. 3ranch is a tree itself, and 3ranches of a 3ranch are ealled t$igs. =
Def init ions
+, %< A
Su3stituti ng for
0.
path is a seuenee of ares ;a,, a& , E E E a.< sueh that the termina l verte9 of each are eorres!onds to the initial verte9 of the suceeeding are. circuit i s a !ath in 4hieh the initial verte9 co incides 4ith the terminal verte9. n edge, e, = T9, %U of G, is a set of t4o elements sueh that ;9, %< V or ;%, 9< V . -his conce!t diff ers from that of an are, 4hich im!lies an orientation. chain is a seuence of edges Te,, e., . . ., e,,l in 4hich each edge has one verte9 in eommon 4ith the sueceeding edge. cycle is a chain i n 4hich the initial and final vertices co i ncide. subgraph G ;P< of G is a gra!h ;P, ,< defined 3% a set of vertiees of P e 0 and containing ali the ares that connect vertiees of P in G. partial graph G ; B< of G is a gra!h ;0, B< defined 3% a set of ares - and contain ing ali the vertices of G. closu!e of a directed gra!h G = ;0, < is a set of vertiees P e 0 such that 9;P r 9fP. If P is a closure of G, then G;P< is a closed subg"ph of G. B% definition, the null set, P K #,, is also a closure of G. tree is a connected and directed gra!h - = ;0, C< eontaining no c%cles. rooted tree is a tree 4ith one disti ngu ished verte9, the root. -he gra!h Bulletin for January, 1965 Montreal
The Pro)lem Given a directed gra!h G K ; 0, < and for each verte9 a numeric called themass. iass of 9,,9* f ind a closuvalue re P m, of G 4ith#, ma9imum 1n other 4ords, finds a set of elements P e 0 such that
6
0: P (((N r9,eP and Mv K
mis ma9imum X( /0
closure 4i th ma9imu m mass is also called a "a xi rnuni closure.
The %l*orithm -he gra!h G is first augmented 4ith a dumm% node 9.and dumm% ares ;9., 9*< . -he algorithm starts 4ith the eonstruction of a tree -Q in G. -Q is then transformed into sueeessive trees -', -&, E E E E E -Q follo4ing given rules, u ntil no further transforma tion is !ossi3le. -he ma9imum closu re is then given 3% the vertiees of a set of 4ell identified 3ranehes of the final tree. -he trees eonstructed during the iterative !rocess are characteri?ed 3% a given num3er of !ro!erties. -o highlight these !ro!erties and to avoid unneces sar% re!etitions 4e shall ne9t develo! sorne additional terminolog%.
R I 1 J
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J ! !
! !
Figu r e l.
Figur e &.
Defiition s
gl'a!h a du mm% verte9 9. 4ith negative mass and dumm% ares ;9., 9*< , >oining 9.to ever% verte9 9,. Be eause 9. eannot 3e !art of an% ma9imum elosure of G, the introduction of du mm% ares ;9., 9,< loes not affect the !ro3lem. -he verte9 9. 4ill 3e the root of ali trees considered. e shall ne9t esta3lish !ro!erties of normali?ed trees. -hese !ro!erties 4ill lead us to a 3asic theorem on ma9im um closu res of a direeted gra!h.
aeh edge e ;are .) of a tree - defines a 3raneh, noted as - ;0., < . It is also eonvenient to 4rite 9.for the root of the 3raneh -E. -he edge e. ;are a < is said to support the 3raneh -. -he mass M. of a 3ranch -. is the sum of the masses of ali vertiees of -.. -h is mass is assoeiated 4ith the edge e. ;are a.< and 4e sa% that the edge e. ;are .) supports a mass M2. 1n a tree - 4ith root 9., an edge e. ; 3raneh -< is &ropert y eharaeteri?ed 3% the orientation of the are . 4ith re s!eet to 9.: e. is ealled a p-edge ;!lus(edge < if the are If a verte9 9. 3elongs to the ma9imum closure W a !oints to4ard the 3raneh -., that is, if the ter minal verte9 of ª2 is !art of the 3raneh -E. -. then of a normali?ed tree -, then all the vertiees 0 .of the 3raneh -. also 3elong to is ealled a p-branch. If are a. !oints a4a% from 3ranch Proof : -., then e.is ealled an m(edge ;minus(edge< and - e shalI sho4 that if a verte9, sa% 9., of the an 1n-branch. Similarl%, ali t4igs of a 3ranch can 3e 3ranch -. does not 3elong to W, then W is not a ma9i divided into t4o elasses8 !(t4igs and m(t4igs. e shaIl also disti nguish 3et4een strong and $ea% edges mum closure. Let ; Figure =< -;W< and - ; 0(W < 3e the su3gra!hs ;3ranehes< . !(edge ;3raneh< is strong if it su! !orts a mass that is strietl% !ositive: an m(edge of - defined 3% the vertices of W and 0(W, res!ee ; 3raneh < is strong if it su!!orts a mass that is null tivel%, and assume or negative. dges ; 3ranehes < that are not strong 0*8 VW: 9, V0s: 9,t80 (& are said to 3e 4ea. verte9 0: is said to 3e strong if there e9ists at 1east one strong edge on the ehain of - >oining 0: to the root 9E. /ertices that are not "#$% strong are said to 3e 4ea. Finall%, a tree is norrnal ized if the root 9.is eommon to ali strong edges. n% tree - of a gra!h G can 3e normali?ed 3% re!laeing the are ;9. 9,< of a strong !(edge 4ith a dumm% are ;9., 9,<, the are ;9., 9.< of a strong m(edge 4ith a dumm% are ;9., 9.< and re!eating the !roeess until ali strong edges have 9.as one of their e9tremities.
K
?.
-he tree in Figure & has 3een o3tained 3% normal i?ing the tree in Figure l. ote that as ali dumm% edges are !(edges, ali strong edges of a normali?ed tree 4ill also 3e !(edges. -he gra!h G considered in the seueI 4ill 3e an augmented gra!h o3tained 3% adding to the original
T(X-e)
52 Figure
=.
li ares 87 of - that >o in vertices of 0(W 4ith vertices of W have thei r ter m inal verte9 in W, as W is a closure of -. t least one of the edges of X is an m( edge 3ecause the chain >oining 9. to 9. must go over 9. and thus contain at least one of the edges of A!. -he f irst such edge 3et4een X3 and 9, is an m edge. Let eD K T9,,, 9,,U 3e this m(edge 4ith 9,,cW and ; !ossi3l% 9., X3 andRor 9. K 9.<. -,, is 9.c0(W an m(3ranch of -. Let -.' 3e the com!onent of - ;0(W< containing the verte9 9E. -he edges of X connecting vertices of -.' to vertices of W, 4i th the e9ce!tion of ; 9., 9,,U, ar e all !(edges in -, other4ise th er e 4ou ld 3e a c%cle i n -. Hence -.,' is a 3r anch o3tained 3% re moving !(t4igs from the m(3ranch -. of -. Because - is normali?ed, the mass of -. is strictl% !ositive: the mass of an% !(t4ig of -., is negative or null. Hence, the mass of -'. is strictl% Figur e . !ositive and W is not a ma9imum closure ; th e closure W 0,,' has larg er mass< . -his com!letes the =.( orma li?e -". -his %ie ld s -' '. Go to ste! l. !roof . .(-erminate. P' is a ma9imum closu re of G. ;
&-''
K
+
&ropert y ' -he ma9imu m closu re of a norma l i?ed tr ee is the set W of its str ong vertices .
If 4e note that an% !(3ranch of - is a closed su3 gra!h of -, 3ut that an m(3r anch is not a closed su3gra!h of -, this !ro!ert% follo4s directl% from !ro!ert% l. If the tr ee has no str ong verte9, that is, no strong edge, then the ma9imum closu re is the em!t% set W c¡,.
See Figu re < .
*onstrnction of ( + -Q can 3e o3tained 3% constru cting an ar3itrar% tree in G and then nor mali?ing this tree as ou tlined earlier. mu ch sim!ler !rocedu re, ho4ever, is to constru ct the gra!h ;0, o< 4here o is the set of ali dumm% ares ;9., 9,< . -his gra!h is a tree and it is, of course, normali?ed.
(
(ransf or"ations (heore" )
1f, in a directed gra!h G, a norma l i?ed tree - can 3e constructed such that the set P of strong vertices of - is a closur e of G, then 0 is a ma9im um closur e (5f G. P1oof :
-he ste !s ou tlined a3ove do not indicate the amou n t of cilcu lation involved in each iterati on nor do the% esta3lish that the !rocess 4ill termYnate in a f in ite n um3er of ste!s. -o clarif% these !oints 4e sha ** anal%?e, in more detai *, the transformations taing !lace in ste!s & and = of the algor ithm.
;a< !ons"ruc"#on of (, e shall use the follo4ing ar gument 8 If S = ; 0, 2< is a !artial gr a!h of G and if W and P -he tree -" is o3tai ned from -' 3% re!lacing the are ma9imum closu res of S and G, res!ectivel%, are ;9., 9,.< 4 ith the are ;9., 9,< . then o3viousl%
M.Mv
If then 4e find a closure 0 of G and a !artial gra!h 4 of G for 4hich 0 is a ma9imum closu re then, 3ecause of the a3ove relation, 0 must also 3e a ma9imu m closure of G. Because - is a !artial gra!h of G and 3ecau se ;!ro !er t% & < 0 is a ma9imum closure of -, the theorem follo4s immediatel%. If, in !articu lar, the set of str ong vertices of a normal
i?ed tree of G is em!t%, then the ma9imum closure .of G is the em!t% set 0 = c¡,. teps of t he /lgorit hrn Constru ct a norma li?ed tree -Q in G and enter the iterative !rocess. lteration t N l transforms a nor mali?ed tree 5' into a ne4 normali? ed tree 5'+*. ach tree -' K ;0, '< is characteri?ed 3% its set of ares A' and its set of strong vertices 0'. -he !rocess ter minates 4hen 0 is a closu re of G. 1teration t N l contains the follo4ing ste!s8 1.-If there e9ists an are ;9., 9*< i n G such
-he are ;9., 0m < su!!orts in - a 3ranch 5.,,' 4ith mass M,,,'67. Let ;Figu r e 8 ) T9..., . . ., 9., n,, . . ., 9., 9.U 3e the chain of -E lini ng 0m to 9E. 9ce!t for this chain, the status of an edge of -E and the mass su !!or ted 3% the edge ar e u nchanged 3% this trans formation. #n the chain T9,., . . ., 9.U of -E 4e h ave the follo4i ng transformation of masses 8 For an edge e: on the ch ain T9m, . . ., 9. M:E = Mm' M:' For the edge T9., 9,U
lJlhE = MmE
7&.(Deter mine 9,,., the root of the strong 3ranch con taining 9.. Construct the tree 5' 3%
Y& <
For an edge e on the chain T9,, . . ., 9., 9.U Mm' + M,' ;=< 1n addi tion, ali the edges e* on the chain T9m, . . ., 9.U have changed their statu s 8 a !(edge i n -' 3ecomes an m(edge in 59 and vice versa . #n the chains T9,,,, . . ., 9.< and T9,, . . ., 9,,U in -', all !( edges su!!or t ?ero or negative masses and all m( edges su ! !ort str ictl% !ositive masses as -' is normal i?ed. Hence, 4e o3tain the follo4ing distri3ution of masses in -E8 MiE
th at P.e P', and 9* e 0(P', then go to ste ! &. #ther4 ise go to ste! .
Zl <
(
=
"-ed g e edge e on ' . . , 0EU [dge
& -edge l\I#¡
M E K Mm'
%
l/1...E
;<
;)
ste! =. 8Bulletin for January, 1965, Montreal
It results from these relation s.
2=
&roperty 0
1f, in -E, e" i s an m (edge on the cha in T9m, . ., 9.U then the mass M"E is strictl% !ositive and larger than an% mass su!orted 3% a !(edge that !recedes ea on the chain T9m, . . ., 9. . ; 3< $or%al#&a"#on of -E8
trees in the seuence -Q, -', . . ., -n 4ill differ either in their masses M% or in their sets of strong vertices. 1ndeed, et us see ho4 M% and P transform during an iteration. Because, in the normali?ation !rocess, 4e never generate a strong m(edge it is clear that the last !(edge removed from the chain T9m, . . ., 9.U is the !(edge that su!!orts the argest !ositive mass in -E. Let e4 3e this edge and 04 \ the vertices of the 3ranch -,.S. s a result of ste!s ;a< and ;3<, 4e no4 have
s -' 4as normali?ed, ali strong edges must 3e on the chain T9m , . . ., 9.U. e remove strong edges one 3% one starting from the fir st strong edge en cou ntered on the chain T9.,,. . . .,9.U. -his edge, sa% (7) e. = T9., 9 3U , must 3e a !(edge ; 3eca use of !ro!ert% =, all m(edges are 4ea < . e re!lace (8) e.4ith a strong dumm% edge ;9., 9.< . -hus, 4e l7emove a !(t4ig from the 3ranch -R and must 1n an% case, M4E Mm' 3ecause of ;< and ;+< su3tract its mass from all the edges of the chain <+.,, . . ., 9.U. Be If M4E Mrut then Mv'N* M,.t ca use of !ro!ert% =, !ro!ert% = 4ill remain valid on If M4E K Mm' then M%tN * K M:.t the chain T9 3, . . ., 9.U. e no4 search for the ne9t -he latter case can on l% occu r if the eu alit% a! strong !(edge on th e chain T9.,, . . ., 9.U and re!eat !lies i n ;+<, and thus e4 must 3e situated on the chain the !rocess u ntil the last strong !(edge has 3een re T9*, . . ., x 1 , 9.U of -'. -hen, ho4ever, the set 04' con moved from the chain. tains 0...' and the set =+ is larger than the set P'. 1n !ractice, transfor mations ;a< and ; 3< can 3e -h is com!letes the !roof. carried out simultaneousl% : 4e have anal%?ed them se!aratel% to 7esta3lish the follo4ing8 $ ef erences (heore" JI ;*< C. Berge, "-he -heor% of Gra!hs and its !!lica 1n follo4i ng the ste !s of the algorithm, a ma9i tions," ile%, *)+&. ;&< Ful erson, D. $ ., " et4or Flo4 Com!utation mum closure of G is o3tained in a finite num3er of for ro>ect 7Cost Curves," Manag e"ent cience, /ol. ste!s. , o. &, anuar%, *)+*. Proof : (3)Grossmann, l. F., and Lerchs, H., "n lgorithm s the num3er of trees in a finite gra!h is finite, 4e on l% have to sho4 that no tree can re!eat itself in the seuence -Q, -', . . ., -". ach normali?ed tree is characteri?ed 3% its set P of strong vertices and the mass M% of this set. e shall sho4 that either M r decreases du r ing an iteration or else M% sta%s constant 3ut the set P increases, so that an% t4o
for Directed Gra!3s 4it3 !!lication to t3e ro>ect Cost Curve and 1n(rocess 1nventor%, &roceedings of the -hird nnual Conference of the Canadian (#!eration a l $ esearch Societ%, #tta4a, Ma% (2, *)+*. ; ] ell%, ,. ., r., "CriticaI ath lanning and < Schedul ing8 Mathematical Basis," 2p et !ations 3esecirch ;A.S.<, /ol. ), ; *)+*< , o. =, ; Ma%( une<. ;) rager, illiam, " Structural Method of 7Com!uting ro>ect Cost ol%gons," M anag e"ent cience, !ril, *)+=.
Tenth Annual Minerals Symposium M a y 4 -1, 15
5
H l#th annual Minerals S%m!osium 4ill 3e held in Grand unction, Colorado, on Ma% , ^ and ), *)+2, under the s!on sorshi! of the Colorado lateau Section of the merican 1nstitute of Mining, Metallurgical and e troleum ngineers. -he Minerals S%m!osium, form erl% no4n as the "Aranium S%m !osium," originated ten %ears ago at Moa3, Atah. -he name 4as changed t4o %ears ago to Miner als S%m!osium and the sco!e 3roadened to inclu de other miner als such as oil, !otash and oil
shale. In addition to Moa3, the S%m!osi um has 3een held at Grants, e4 Me9ico, and at $iv erton, %oming, s!onsored 3% .1.M.. sections. SeveraU hu ndred !eo!le attend this annual event, 4hich no4 re! resents a 3road coverage of the mi ning and !rocessing industries of the estern states. t the *)+2 Grand unction meeting, s!ea ers 4ill cover technical and non techn ical su3>ects relating to m in ing, metall urg% and geolog%. Social events and field tri!s are also 3eing !lanned.
Committees are no4 at 4or on the *)+2 !rogram. C. H. $e%nold s, of Continental Materials Cor!., Grand u nction, is general chair man for the event. hili! Don nerstag, of merican Metal Cli ma9, Grand unction, is chai rman of the host grou!, the Colorado lateau Section of the .1.M.. dvance !rograms 4ill 3e mailed out. n%one desiring more information ma% 4rite to $o3ert G. Beverl%, S%m!osiu m secretar% treasu rer, .#. Bo9 &^, Grand unction, Colorado.
;8
-he Canadian Mining and Metallurgical
