JME Journal of Mining & Environment, Vol.2, No.2, 2011,

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1 JME Journal of Mining & Environmnt, Vol.2,.2, 2011, A nw algorithm for optimum opn pit dsign: Floating con mthod III E. Elahi zyni 1, R. Kakai 2*, A. Yousfi 3 1- Eastrn Alborz Coal Company, Shahrood 2- Faculty of Mining Enginring & Gophysics, ShahroodUnivrsity of Tchnology, Shahrood 1- Eastrn Alborz Coal Company, Shahrood Rcivd 4 Oct 2011; rcivd in rvisd form 9 Mar 2012; accptd 5 Apr 2012 *Corrsponding author: r_kakai@yahoo.com (R. Kakai) Abstract Ultimat limits of an opn pit, which dfin its siz and shap at th nd of th min s lif, is th pit with th highst profit. A numbr of algorithms such as floating or moving con mthod, floating con mthod II and th corrctd forms of this mthod, th Korobov algorithm and th corrctd form of this mthod, dynamic programming and th Lrchs and Grossmann algorithm basd on graph thory hav bn dvlopd to find out th optimum final pit limits. Each of ths mthods has spcial advantags and disadvantags. Among ths mthods, th floating con mthod is th simplst and fastst tchniqu to dtrmin optimum ultimat pit limits to which variabl slop angl can b asily applid. In contrast, this mthod fails to find out optimum final pit limits for all th cass. Thrfor, othr tchniqus such as floating con mthod II and th corrctd forms of this mthod hav bn dvlopd to ovrcom this shortcoming. Nvrthlss, ths mthods ar not always abl to yild th tru optimum pit. To ovrcom this problm, in this papr a nw algorithm calld floating con mthod III has bn introducd to dtrmin optimum ultimat pit limits. Th rsults show that this mthod is abl to produc good outcom. Kywords: Opn pit mining; Ultimat pit limit; Floating con mthod; Floating con mthod II; Floating con mthod III 1. Introduction Opn pit mining is an important gnral mining mthod that minral dposit will b mind via pits. Th shap of mining ara at th nd of mining opration or final limits of a min must b dsignd bfor starting th opration. According to th dsignd final pit limits, mining oprational paramtrs such as width, lngth and dpth of mind pit, opning track ways, location of wast dump, stripping ration, min lif, minabl or tonnag, wast tonnag and production schduling can also b dtrmind [1]. Optimum pit limits ar usually dsignd with th us of th block modls. Gological block modl, which prsnts th rsrv as a combination of numrous small blocks, is dtrmind by invrs distanc or gostatistical mthods. Thn th conomical block modl is calculatd by applying cost, pric and othr paramtrs to ach block. In this modl or blocks hav positiv s, wast blocks hav ngativ s and air blocks, and th blocks ovr th surfac topography hav zro s. Most of th optimum pit limits mthods us th conomical block modls to dtrmin th pit limits. Th mthodology is sarching for a combination of blocks with th maximum conomical at currnt conomical and tchnical condition [2]. Floating or moving con mthod [3], floating con II mthod [4], modifid floating con II mthods [5], dynamic programming [6], [7] and [8], th Lrchs and Grossmann algorithm basd on graph thory [9], Korobov algorithm [10], corrctd form of th Korobov algorithm [11] and gntic algorithm [12] ar som of th svral algorithms

2 Elahizyni t al./ Journal of Mining & Environmnt, Vol.2,.2, 2011 dvlopd to dtrmin optimum pit limits. Each of ths mthods has spcial advantags and disadvantags. For xampl, dynamic programming is just a 2D modling mthod and although th Lrchs and Grossmann algorithm can always crat an optimizd and tru limit of th pit, th mthod is too complicat to b applid asily. Floating or moving con algorithm is on of th asist and fastst algorithms for dtrmining th final pit limits. In addition, mining oprational rstrictions on various slops can b applid to this mthod prfctly. Furthr, a fw algorithms such as floating con II mthod and modifid floating con II mthod ar dvlopd to ovrcom th shortcoming of this mthod. In dspit of ths corrctions, ths mthods ar not abl to obtain th tru optimum limit of th pit. Thrfor it nds mor corrctions. This papr prsnts a nw dition of th algorithm, calld floating con III mthod, in ordr to covr shortcomings of prvious mthods. 2- An ovrviw on floating con algorithms 2-1- Floating con mthod This mthod, which was first dscribd by Carlson, Erickson, O Brain and Pana (1966), works on an conomical block modl of th dposit [3]. For ach positiv (or) block, this mthod involvs constructing a con with sids orintd paralll to th pit slop angls, and thn dtrmining th of th con by summing th s of blocks nclosd within it. If th of th con is positiv, all blocks within th con ar mind. This procss starts from th upprmost lvl and movs downward sarching for positiv blocks. Th procss continus until no positiv cons rmain in th block modl. Although this algorithm is simpl and asy to undrstand, it is not abl to yild a tru optimum pit limit. For xampl, for th 2-D conomical block modl in Figur 1, whn final dip of pit is 1:1, floating con algorithm cannot produc tru optimum pit limit, as prsntd on Tabl 1. Nvrthlss, by applying dynamic programming mthod [6] to this modl an optimum pit limit with th of 2 would b obtaind (Figur 2) Figur 1. 2-D conomical block modl Figur 2. Optimum pit limit by dynamic programming mthod Tabl 1. Con for xampl shown in Figur 1 Con Stag Minabl? 1 (2,2) (2,4) (2,8) (3,4) Floating con II algorithm Floating con II algorithm was introducd and prsntd by Wright in 1999 [4]. Th mthodology is similar to th floating con approach xcpt that first s of th con of all or blocks ar calculatd in ach lvl and th con with maximum is rmovd from th block modl. Nxt cumulativ pit is calculatd and this procss is carrid out for rmaining or blocks. Thn all th xtraction cons of th block with highst cumulativ pit ar includd as a mmbr of th optimum solution st. For th block modl shown in Figur 1, whn final dip of pit is 1:1, th rsult of this algorithm is illustratd in Tabl 2 and Figur 3, with th of -3. A tru optimum pit limit of this modl is shown Figur 2 with th of +2. Tabl 2. Cumulativ pit for xampl shown in Figur 1 L v l Con Cumulativ Minabl? (2,8) (2,2) (2,4) (3,4) Figur 3. Optimum pit limit by floating con II algorithm 119

3 Elahizyni t al./ Journal of Mining & Environmnt, Vol.2,.2, Modifid floating con II, mthod 1 This algorithm is th sam as th algorithm of floating con II mthod xcpt that whn th highst cumulativ pit is positiv, th blocks ar addd as part of th optimum pit solution [5]. Thrfor using this mthod, for th block modl shown in Figur 1, as illustratd in Tabl 2 3, no blocks would b mind. Nvrthlss, th tru optimum limit for this modl is shown in Figur 2 with th of Modifid floating con II, mthod 2 This mthod is a dvlopmnt of first mthod of modifid floating con II algorithm [5]. In th scond mthod all lvls ar considrd togthr. In othr words, th of th cons of all or blocks ar valuatd conomically and th con with maximum is assumd to b as part of th pit limits and th cumulativ pit is calculatd. This procss is thn continud until no positiv block rmains in th block modl. Finally, th block with positiv and maximum cumulativ and all othr prvious blocks ar includd as a mmbr of th optimum solution st. As illustratd in Tabl 4 and Figur 4, this algorithm yilds a pit limit with th of +1 for th block modl shown in Figur Floating con III Although th floating con II mthod and its corrctions ovrcoms som waknsss of th floating con mthod, in som circumstancs ths mthods fail to yild a tru optimum pit limit and thrfor a nw dvlopmnt is ndd. In gnral, blocks in conomical block modls could b dividd into two groups of dpndnt and indpndnt blocks. Th or blocks which hav no common ovrlying block with othr or blocks in thir xtraction cons ar indpndnt othrwis thy ar classifid as dpndnt. Each of ths groups with rgard to th of thir xtraction cons ar also classifid as ffctiv or inffctiv. Effctiv blocks hav positiv and inffctiv blocks hav ngativ. Th optimum pit limit is dtrmind according to th cons on ffctiv blocks. A flowchart of th floating con III mthod is shown in Figur 5. Stags of th algorithm ar as follows: 3-1- Th algorithm is th sam as th floating con algorithm xcpt that aftr xtraction of ach mining cons, sarch for othr limits would b continud from th first lvl for rmaining blocks. Th aim of this stag is finding indpndnt ffctiv blocks in conomical modl Finding or blocks from th first lvl of conomical block modl to th othr lvls. If any or block is found in any lvl, othr or blocks ar considrd from this lvl to th first lvl. Th aim of this stag is to chck th ffct of lvls on ach othr Constructing xtraction cons for all or blocks, with rgard to th tchnical rstrictions, thn finding dpndnt and indpndnt blocks Finding inffctiv and indpndnt blocks which hav no positiv s. If positiv s ar assignd against ngativ d blocks, th ffct of such blocks on optimum pit limit will b rmovd. 3- Finding inffctiv and dpndnt blocks which hav no positiv s. If positiv s ar assignd against uncommon and ngativ ovrlying blocks, th ffct of such blocks on optimum pit limit will b rmovd or dcrasd Finding ffctiv and dpndnt blocks, ths ar all rmaining blocks aftr carrying out th forgoing stags. Finding optimum pit limit is as follows: Idntifying common blocks for ach of xtraction con and thn calculating thir wights, Th wight of ach block is qual to th numbr of cons nclosd within it. Tabl 3. Cumulativ pit by floating con II, mthod 1 L v l Con Cumulativ Minabl? (2,8) (2,2) (2,4) (3,4) Tabl 4: Cumulativ pit by floating con II, mthod 2 Stag Con Cumulativ Minab l? 1 (2,8) (2,2) (3,4) Figur 4. Optimum pit limit by floating con II, mthod 2 120

4 Elahizyni t al./ Journal of Mining & Environmnt, Vol.2,.2, 2011 Start Continu for rmaining blocks with original Tak first lvl Includ this con as part of th pit Is thr any indpndnt and ffctiv or block? Tak nxt lvl All lvls? Tak first lvl with original Is thr any indpndnt and inffctiv or block? Construct con and allocat positiv against ngativ Is thr any dpndnt and inffctiv or block? Construct con and allocat positiv against ngativ for uncommon block Is thr any positiv block? Construct con and calculat importanc of common block, of con and final importanc Sort or blocks in ascnding ordr on thir importanc and con Rmov cons in ascnding ordr and calculat cumulativ s Is maximum cumulativ positiv? Tak nxt lvl Includ all th cons as part of th pit for th maximum cumulativ All lvls? End Figur 5. Flowchart of Floating Con III algorithm 121

5 Elahizyni t al./ Journal of Mining & Environmnt, Vol.2,.2, Calculating th wight of mining cons. It is th sum of wight of common blocks of th con Calculating th con s for ach of mining con Calculating th final importanc for ach of mining con.final importanc is th ratio of th wight of con to th absolut of its con Sorting minral blocks on final importanc, thn on of mining cons in dscnding ordr and supposing xtraction of first or block Supposing xtraction of othr blocks would b continud according to fist block xtraction. At this stag th of othr cons will b valuatd and th mining con with maximum will b xtractd, a supposd xtraction offcours. Th cumulativ of vry supposd xtraction must b calculatd now Finding th maximum and positiv cumulativ and dtrmining optimum pit limit by including blocks from fist mining con to this con. Thn w sarch for anothr limit from nxt lvls by using original block modl. If th maximum cumulativ is not positiv, this mans that thr is no optimum limit to this lvl and th sarch will b continud from nxt lvl. Th floating con III algorithm can bst b xplaind by a simpl xampl applid to an conomical modl shown in Figur 1. As shown blow, this algorithm producs a final pit with th of +2 (Figur 13) which is a tru optimum pit limit. a) Th first lvl containing or blocks is th scond lvl and th algorithm applid for this lvl is as follows: First stag: finding ffctiv or blocks. As can b sn in Tabl 1, sinc th xtraction con of all th or blocks ar ngativ, thr is no indpndnt ffctiv block in th modl. Scond stag: Rmoving indpndnt and inffctiv or blocks from modl, as shown in Figur Figur 6. Rmoving indpndnt and inffctiv or blocks Third stag: dcrasing or rmoving th ffct of dpndnt and inffctiv blocks, as illustratd in Figur 7. Forth stag: Dtrmining th importanc of common blocks. According to Figur 7, only on or block rmains. Thrfor thr is no common block hr and th initial and final importanc of this con is 0, as is shown in Figur Figur 7Dcrasing or rmoving th ffct of dpndnt and inffctiv blocks Figur 8. Dtrmining importanc of common blocks Fifth stag: Supposing th xtraction of only rmaining or block which its of xtraction con and its cumulativ of both ar -2. Sinc th is lss than zro, this con is not includd as part of pit limit. In othr words, thr is no optimum limit to th scond lvl. Hnc th algorithm will b continud for th conomical block with its original as follows: b) Th scond lvl with or blocks is th third lvl and th stags of th algorithm for this lvl ar as follows: Sixth stag: finding ffctiv or blocks. As it can b sn from Tabl 1, thr is no indpndnt ffctiv block in th modl. Svnth stag: Rmoving indpndnt and inffctiv or blocks, as illustratd in Figur Figur 9. Rmoving indpndnt and inffctiv or blocks Eighth stag: dcrasing or rmoving th ffct of dpndnt and inffctiv blocks, as shown in Figur Figur 10. Dcrasing or rmoving th ffct of dpndnt and inffctiv blocks Ninth stag: Dtrmining th importanc of common blocks, as shown in Figur Figur 11. Dtrmining importanc of common blocks 122

6 Elahizyni t al./ Journal of Mining & Environmnt, Vol.2,.2, 2011 Tnth stag: Dscnding sort of or blocks on thir final importanc of mining cons, and thn on th of cons, as shown in Tabl 5. Tabl 5. Dscnding sort of th or blocks Rf Final importanc Con 1 (3,4) (2,4) (2,2) Elvnth stag: Supposd xtraction of or blocks as illustratd in Figur Figur 12. First supposd limits Twlfth stag: Finding maximum of rmaining blocks from sortd list and calculating thir cumulativ (Tabl 6). Rf Tabl 6. Calculation of cumulativ s Con Cumulativ Minabl? 1 (3,4) (2,2) Thirtnth stag: With rgard to Tabl 6, maximum cumulativ is positiv. Thrfor rlatd mining cons is includd as part of th pit and optimum limit is obtaind with th of +2 (Figur 13). Th rsult of this algorithm is th sam as th Dynamic programming mthod Figur 13.Final Optimum pit limit by floating Con III Exampl 2 In ordr to show th ability of th floating con III algorithm, anothr simpl xampl as shown in Figur 14, Whn final dip of pit is 1:1, is mployd. Floating con II algorithm and its modification produc a pit with of +1 as illustratd in Figur 15, whras floating con III mthod as shown in Tabl 8 and Figur 16-c crats a tru optimum pit with of +2 as th sam as th dynamic programming tchniqu Figur 14. An conomical block modl L v l 3 Tabl 7. Stags of floating con II and its modification algorithm Con Cumulativ Minabl? (3,7) +17 (3,4) (3,3) Figur 15. Optimum pit limit by of floating con II and its modification algorithm Figur 16-a. Dpndnt inffctiv blocks rmovalfloating con III algorithm Figur 16-b. Dtrmining th importanc of dpndnt and inffctiv common blocks - floating con III algorithm Figur 16-c. Optimum pit limit by floating con III algorithm 3-7- Cas study In this sction, floating con algorithms ar applid for a ral data of a gold min, locatd at 35 kilomtrs north-ast of Swdn. First of all, an conomical block modl of this min has bn cratd using Pitwin32 softwar. This softwar with using grad block modl and tchnical and conomical paramtrs such as cut-off grad, dimnsion of blocks, or and wast dnsity, pric, 123

7 Elahizyni t al./ Journal of Mining & Environmnt, Vol.2,.2, 2011 cost and tc crats an conomical block modl of dposit. Th dposit is dividd into 15m (ast-wst) x 10m (north-south) x 5m (vrtical) blocks and th lock numbrs bin th ast wst, north south and vrtical dirctions ar 101, 82 and 36 rspctivly. Each block is assignd th stimatd (krigd) rcovrabl tonnag of or abov a cut-off grad and th stimatd (krigd) avrag grad of this tonnag. Tabl 9 shows th ovrall rsults of this cas study with th us of floating con, floating con II and its modifications mthods. In addition, for implmntation of th floating con III mthod a C++ cod was dvlopd by using Visual C++ programming languag. Th rsult of this algorithm for this cas is also shown in Tabl 9. It can b concludd from this tabl that compard with othr floating con mthods, floating con III producs a final pit limit with th highst. 4- Summary Although floating con II algorithm and its modifications ovrcom som of th shortcomings of th floating con mthod, for xampls shown in this papr thy produc a pit with lss and fail to dtrmin tru optimum pit limits. Sinc ths mthods do not tak into account th ffct of indpndnt and dpndnt block to ach othr. On th othr hand, th floating con III mthod tak into considration this shortcoming and always crat a pit with positiv and high. Th algorithm is straightforward and using diffrnt pit slops in diffrnt parts of th orbody is vry simpl. Tabl 8. Dscnding sorting of dpndnt ffctiv blocks - floating con III algorithm Rf Final Con Cumulativ Minabl? importanc of con 1 (3,4) (3,3) Tabl 9. Ovrall rsults by diffrnt mthods Mthod Numbr of blocks Valu (*10000) Pit Or Wast Or Wast Nt Floating con Floating con II Floating con II- First modification Floating con II- Scond modification Floating con III Lrchs& Grossmann Rfrncs [1] Khalokakai, R., Dowd, P. A. and Fowll, R. J., 2000, Lrchs-Grossmann algorithm with variabl slop angls, Trans. Instn Min. Mtall. (Sct. A: Min. industry),. 109, p.p. A77-A85. [2] Khalokakai, R.; Dowd, P. A. and Fowll, R. J.; 2000; A windows program for optimal opn pit dsign with variabl slop angls, Intrnational Journal of Surfac Mining, Rclamation and Environmnt, 14, p.p [3] Carlson, T. R.; Erickson, J. D.; O Brain D. T. and Pana, M. T.; 1966; Computr tchniqus in min planning, Mining Enginring, Vol. 18,. 5, p.p [4] Wright, E. A.; 1999; MOVING CONE II - A Simpl Algorithm for Optimum Pit Limits Dsign, Procdings of th 28th Symposium on th application of computrs and oprations rsarch in th minral industris (APCOM), (Colorado USA), p.p [5] Khalokakai, R. 2006; Optimum opn pit dsign with modifid moving con II mthods, Journal of nginring in Thran univrsity, Vol. 4,. 3 p.p (in Prsian). [6]Konigsbrg E.; 1982; Th optimum contours of an opn pit min: an application of dynamic programming, Procdings of th 17th Symposium on th application of computrs and oprations rsarch in th minral industris (APCOM), (Nw York: AIME), p.p [7] Wilk, F. L. and Wright, E. A.; 1984; Dtrmining th optimal ultimat pit dsign for hard rock opn pit mins using dynamic programming, Erzmtall,. 37, p.p [8] Yamatomi, J.; Mogi, G.; Akaik, A. and Yamaguchi, U.; 1995; Slctiv xtraction dynamic con algorithm for thr-dimnsional opn pit dsigns, Procdings of th 25th Symposium on th application of computrs and oprations rsarch in th minral industris (APCOM), Brisban, p.p

8 Elahizyni t al./ Journal of Mining & Environmnt, Vol.2,.2, 2011 [9] Lrchs, H. and Grossmann, I. F.; 1965; Optimumm dsign of opn pit Min ; CIM Bulltin,. 58, pp [10] David, M.; Dowd, P. A. and Korobov, S.; 1974; Forcasting dpartur from planning in opn pit dsign and grad control, Procdings of th 12th Symposium on th application of computrs and oprations rsarch in th minral industris (APCOM), (Goldn, Colo: Colorado School of Mins), Vol. 2, p.p. F131-F142. [11] Dowd, P. A. and Onur, A. H.; 1993; Opn pit optimization - part 1: optimal opn pit dsign, Trans. Instn Min. Mtall. (Sct. A: Min. industry),. 102, p.p. A95-A104. [12] Dnby, B. and Schofild, D.; 1994; Opn-pit dsign and schduling by us of gntic algorithms, Trans. Instn Min. Mtall. (Sct. A: Min. industry),. 103, p. p. A21-A

4.2 Design of Sections for Flexure

4.2 Design of Sections for Flexure 4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt