37
Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales. Robust models for simultaneous open pit and underground mines. RR-5044, INRIA. 2003. <inria-00071539> HAL Id: inria-00071539 https://hal.inria.fr/inria-00071539 Submitted on 23 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.

Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

  • Upload
    others

  • View
    5

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

Robust models for simultaneous open pit and

underground mines

Nelson Morales

To cite this version:

Nelson Morales. Robust models for simultaneous open pit and underground mines. RR-5044,INRIA. 2003. <inria-00071539>

HAL Id: inria-00071539

https://hal.inria.fr/inria-00071539

Submitted on 23 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Page 2: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--50

44--

FR

+E

NG

ap por t de r ech er ch e

THÈME 1

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Robust models for simultaneous open pit andunderground mines

Nelson Morales

N° 5044

December 2003

Page 3: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales
Page 4: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

"!#$%&!'% "!!#(!)*!#

+,.-0/13254613798:-0,./∗

;=<?>@BABCEDGF#HIJAKMLONAQPRIJSOITPU>@BAIVXWUY[ZTAQPR\^]E_?`#aE;#;#b

FKMcdc:Y.WePfOAgWUAh<?AWh<dAjilkXmMn[oMopDGqrAhA@"s:AWrt[n.nMuvDwuMuxc?K[y.AI

z|~dMM \idi?ycdK[i?idi?yIvKhAiPJWK[cdWUYMsdA@iPJ<?A"@Bidi?ysdL?IJidAIUIR#PJ<PJ<dAxyMYK[YMIJAQPJPJidyB<dyM<WJAMAiL?AfANOcdY.PKPUY.iKMyMAi:fdKMIXYAMAWOhLdWJWUAiP#@BYOfOAI=K.hWUYMsdL?IePJi?AIUIPJ<?ASfdYidY[PvhQY.i?IefOAWLdi?hAWJPUK[iPTSYMPJ<dALOPJLdWUAMIJYPJ<?AcdK[i?IrK[WUAM:i?fdAAflWUAhK[hQLdKPUAfA.AWUSS.AK[WK.hhYMWfOi?yPUY"PU<[email protected];=<dI#YMWUcdWUAIJAiPUI#IJYM@BAK[c?cdWJYKMh<dAIPUYIeY..ARPU<dAKMs:Y.AQ@xAiPJYMidAfcdWUYMsdA@XKP#:WUIeP#PJ<dA

hLdWUWJAi.PB@BYOfOAIK[WUAAQNOc:YIeAf=ITPUYh<:KMIePJhcdWUYMy.WUKM@B@xidypIBL:IeAfPU<dAi PUY¡IJAQPL?c i?A@BYOfOAIhYMi?IJfOAWUi?y¢Ldi?hAWJPUK[idPTSM$]RIKIJAhQY.i?f£hYMiPJWUs?LOPJYMiPJ<dA¤idA¥@BYOfOAIK[IeYyMAi?AWK[¦A^PJ<dAhLdWUWJAi.PY.idAIs:AhK[L:IeA#PU<dASBK[Y hYM@s?idAfxAQNOcdYMPUKPUY.i.IJAQPJPJidyLdcc?KMi?I#<dAWJA#L?i?fOAWUyMWUYML?i?fKMi?fYMclAicdP#PJAh<?idYMYMy.AIhK[is:AgL:IeAfIJ@"LdPUK[i?AYML:IeSM\Y.WJAYMAW§K[¨PJAWK©fOAAc:AWWJAIJY.iªYMrPU<dAhQLdWUWUAiP@xYOfOAIK[i?fPJ<dAIePJYOh<?K.ITPUhcdWUYMy.WUKM@I

@BAi.PUY.idAf3KIeAhQY.i?fK[c?cdWJYKMh<IRANclY.IJAfE;=<dIr@xYOfOA03iPU<dA"<?YMclAY[«@xY.WJAWJY.sdL?IePr@Bi?idyIUh<dAfOLdAIIJAKMWUh<dAIXY.W=IUh<dAfOLdAIIJK[PJITSidy"fOA@K[i?fdI=KMi?fhKMc?KMhPUAI§@K[N@B¦i?yPU<dArWUAIJfOL?K[Y.WJA"YMWvLdPJLdWUAMi?IePJAKMf¤Y[«PJ<dA¬rVX­®K[LdABL?IeAf&ipPJ<dABcdWUAYML?IEY.idAI¯i?KMSMK<dALdWUIePJhxIc?WJAIeAi.PUAfPJYIJYMMArPJ<dI#@BYOfOA°±^²³3´Uµ¶ ~·|M @BididyIJh<?AfOLdidy?OWUYMsdL:ITPUidAIUIOIePJYOh<?K.ITPUhEcdWUYMy.WUKM@x@Bidy?

∗ ¸¹º¼»¾½¿ÁÀ0®½Ã»£»0Ä9Å9Å[¿ÁÀÇÆ0ÈTÉËÊQÌËÍÎÏÐÇÍ3ѸÓÒ͹º¼Ô¼ÈUÕÖÐÏ×ШØÓÃÙÉÚvØÛ9Íθ¸ÜÝÒ Þ0¿Áº¼ÙQÆ£Å9ÀÇ¿Þ0ÈTßàÆÍÏ×ÛáUÐÏ×ШØXáâÏÛØ«ã Õ

Page 5: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

Q#j# %¤$*!#j# !# ^#

| KRcdK[i?:hKPUY.ifOAIANOcdY.PKPUY.i?I@Bid>WUAIAIePL?idA«[email protected]«f?K[i?IAI«AiPUWJAcdWJIJAI§fdLIJAhÁPUALdW.L?lIeAgPUWJY.LdMAW«fOAI«c?WJY.yMWK[@B@BAIf AQNOcdYMPUK[PJYMiKMLONcdWUY[dPI§cdL?IHAMHIX`«AclAi?fdKMiPAIX@BYfd>AIK.hÁPJL?AIidAIJYMiPc?K.I:K[s?AIàIidAcdWUAidi?AiPc?KMIAihQY.@BcOPJA#AIi?hAWJPJPJL?fOAIgfOAB K9MAidW AIvcdK[i?:hKPUY.i?IvfOYMMAi.PgfOY.i?hQPJWUAxWJAhK[hQL?HAIvK[[email protected]:ifdAvhYMi?IJfOHWUAW=AI#[email protected]?I

`«AvPUWUK9K[cdWUHIJAiPJAgfOAgi?YMLd.AAI=c:AWUIJc:AhÁPU.AI=clYMLdWWUAIJYML:fOWJAEAgcdWJY.sd>@BAMbXi^L?ic?WJA@xAWPUA@Bc?I.Y.icdWUHIJAiPJA#AI§@xYOfO>AI§K.hÁPJL?AIclYMLdW«Ai?IJLdPJAMvPUWUK9.AWIKvcdWUYMy.WUKM@KPJYMiBIePJYOh<?KMIePJLdAMc?WJY.c:YIeAW«fOAvidY.LdMAK[LONB@BYfd>AILdhQY.i?IefO>WJAi.PArÇKMhQPJALdW#fOAr i?hAWJPJPJL?fdAEAPLdy.HidAWUKMIJAiPAI"K[iPJAWJAL?WUIgc?LdIL I"IJYMiP"hK[c?KMsdAIfOAPJWK[PJAWAIANOcdY.PKPUY.i?IhYM@s?idHAIbXi¢A"[email protected]?KMi?IvY$#pAIv@BHQPJ<?YfdAIgfOAIvAQNOcdYMPUKPUY.i?IvIeY.LOPJAWJWK[idAIRAP%hAXYMLd.AWJPclAL?MAiP&QPUWJAgLdPJIeHAI#IJ@"LdPUK[i?A@BAiP¯i?K[@BAiPKMcdWU>IELdidAxWJAIJY.i&cdL?IEcdWUY[Y.i?fOABfOAIv@BYOfO>AIvhYMi?IJfdHWUHIlL?idAfOALONO>@BAK[cO

c?WJYOh<dAgAITPRc?WJHIeAi.PUHA.`«A@xYOfO>A.:[email protected]?IePJAIJIJAM?h<?AWh<dAfOAIRcdK[id:hK[PJYMi?I@Bid>WUAI'LdIUKPJIeÇKMIUIeAi.PrAIRfOA@BKMi?fOAIRAPEhKMc?KMhPUHIrhQYMi:IeAWJK[iPclYMLdWEAgLOPJL?WEKcdL?IryMWK[i?fdAL?KMiPJPJHrfdAr@BidHWK[:idYMiOAQNOcdYMPJHEK[LALfdAR@KNO@xIJAWK­rV¬hYM@B@BAÇKMIUK[AiP«AI«@BYOfO>AIc?WJHhQAf?K[iPUI(RidAg<dALdWJIePJ.L?AgAIeP=c?WJHIeAi.PUHAgK[YMWIc:Y.LdW#WUAIJYML?fOWUAvhQAv@BYfd>AM) ¶ | ´ $* | VXK[id:hKPJYMifOAI=@xidAIdWUYMs?L?ITPUAIUIeA.cdWJY.yMWK[@[email protected]<?KMIePJLdAM

Page 6: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ u

2 "!!

3546 ~ ¶ ·[7 ¶ 6 8

9;: ~~ ² 6 ¶ · ² *| <tdC arclAicdP@Bi?AI ggvgvgvggEgvgvggvgvgEgvggvgvgvggEg =

tdC.C ;=<dAvLdPU@K[PJAgcdPRcdWUYMsdA@ gvgvggvgvgEgvggvgvgvggEg >tdC. t arc:AicdPrIJA LdAi?hQidyx@xYOfOAvgvggvgvgEgvggvgvgvggEg ?

td t (Ri?fOAWJy.WJY.Ldi?f@xidAIRvgvgvggEgvgvggvgvgEgvggvgvgvggEg Cntd tdC ] @xYOfOARggvgvgvggEgvgvggvgvgEgvggvgvgvggEg C9t

@BA ¶ "Cd|D7 ² * ² ² 6 | 3@u?C FYMsdL?IePJi?AIUI gvggvgvgvggEgvgvggvgvgEgvggvgvgvggEg Co

E A ¶ ²^² * ² ² 6 | ¶+F | ¶ "Cd|7HG~ ¶+I ~9 J7 6 I 3KE

8 ) 7 6 7 6 I |"C ² · *L7 6 I ¶ · ² *| µ 7¨C¾| ¶ "Cd|D7HG~ ¶)I ~ J7 6 I 38mdC _OYM@BAgyMAidAWK[3IJIJLdAIEvgvgvggEgvgvggvgvgEgvggvgvgvggEg C=

mdC.C ;WUAK[PJ@BAiP#Y[PJ<dAgLdi:hQAWJPUKMiPTSIJYML?WUhAIBvgvgEgvggvgvgvggEg C=mdC. t F#AQsdYhidy vgvgvggEgvgvggvgvgEgvggvgvgvggEg C"MmdC. u _YM@BAgKMIUIeL?@xcdPJYMi?I vggEgvgvggvgvgEgvggvgvgvggEg C"M

md t ]5:WUIeP#@xYOfOAvggvgvgvggEgvgvggvgvgEgvggvgvgvggEg C"Mmd tdC ­K[WUKMsdAI*gvgvgvggEgvgvggvgvgEgvggvgvgvggEg C"Mmd td t _PJYOh<?K.ITPUhEK[WUKMsdAI)ggEgvgvggvgvgEgvggvgvgvggEg C?md td u qRAPJAWU@Bi?IePJhgfdKPKBK[i?fc?K[WK[@BAPJAWI¡gvgvgEgvggvgvgvggEg¥tMnmd td o arsOZTAhQPJMAELdi?hQPJYMiÝvggEgvgvggvgvgEgvggvgvgvggEg¥tMnmd td m `«YMi?IePJWK[iPUIjvgvgvggEgvgvggvgvgEgvggvgvgvggEg¥tMnmd td = ars?IeAWJK[PJYMi?IY[PJ<dAv:WUIeP#@xYOfOA vggvgvgEgvggvgvgvggEg¥t.t

md u ]jIJAhYMi?fONÇIJ@xc?AW$P@BYOfOAvggEgvgvggvgvgEgvggvgvgvggEg¥t.tmd o _OLd@B@BKMWJS EgvggvgvgvggEgvgvggvgvgEgvggvgvgvggEg¥tMu

<RQ ² ¡ 6 ·TS : UGd7¨ ³ ¶ · ² *| 9'E=?C \KPJ<?A@KPUhK[3YMWU@L?K[PJYMiËvggEgvgvggvgvgEgvggvgvgvggEg¥t.m

=?C.C ­K[WUKMsdAI*gvgvgvggEgvgvggvgvgEgvggvgvgvggEg¥t.m=?C. t qrK[PUKBK[i?fc?KMWUKM@BAQPJAWUIggEgvgvggvgvgEgvggvgvgvggEg¥t.m=?C. u arsOZTAhQPJMAELdi?hQPJYMiÝvggEgvgvggvgvgEgvggvgvgvggEg¥t==?C. o `«YMi?IePJWK[iPUIjvgvgvggEgvgvggvgvgEgvggvgvgvggEg¥t=

=? t \^YfdAIhQY.@xc:K[WUIJYMijvgvgvggEgvgvggvgvgEgvggvgvgvggEg¥t>

V A ¶ *LWX7 6 I C ² · ² ¡ 6 ·TSd.'G?7¨ ³ ¶ · ² * 9UYMOC ¯PUAWUi?y gEgvggvgvgvggEgvgvggvgvgEgvggvgvgvggEg¥t?MO t VAWJYOfdI=c?K[WJPJPJYMiÝgvgvgvggEgvgvggvgvgEgvggvgvgvggEg¥t?

YB: ¶ 6 * |7 ¶ 6 | @)Z

××Ù"[U\]$^^

Page 7: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

o " "

z G G ² 6 · 7X0G G ¶ ~9 6 7¨ ³ ¶ | @)@

ÐÏ×ШØ

Page 8: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ m

!§Q"!

;=<?AxhYMcdclAWv@Bi?idyi?fOL?IePJWUS<?K.IRslAAi¤@Bc:Y.WePK[iPEi¢`«<dAxAMAi&slAQY.WJA"PJ<dAhYMLdiPJWUSAQNOITPUAf;=<?A@BidA$"0" N°`«<?APx=KMIBANc?Y.PUAfªsS¢PJ<dA ài?hKMIB#<dAiªi?Y¡bLdWJY.c:AK[i AAP<?KMIPUYML?h<?AfPJ<dA"hQYMiPUidAiPRSMAP:KMi?fA.Ai^idY=KMfdK9SOIPRAQNOITPIK[i?fI=PU<dAsdyMy.AIePYMclAicdPr@BidAiPU<dA§YMWUfYA.AWKMi?ffOAIecdPJA§PJ<dA«@BclYMWJPUKMi?hQAYMOPJ<dA«i:fOL?IePJWUS NÇ#<dh<WJAcdWUAIJAiPUIKMs:Y.LOPonYM?PU<dA EWJYIJI¬RK[PJYMi?KMOVXWJYOfOL:hÁP$PQM@KPU<dA@K[PJhK[cdWUYMs?A@BIhYMi?hAWUidi?yRPU<dA=@xididygcdWUYOhQAIUIJAIK[WUA.AWUSvWJAhQAiP[sdLOP[YMi"PJ<dA=YMPJ<dAW<:K[i?f .L?PUA=fd.AWIeA.;=<dI«Y.WJvIYhL?IJAf"YMi(1!U!-$" !K-#<?h<yMYK[3I#PJYxdNYMi?y"PUAWU@ AQNOcdYMPUK[PJYMiK[y.Ai?fdK.I«YMWKx@BidA.;=<dAWJABK[WUAgPTYi?fdIrY[«hYMcdclAWE@Bi?AIRYMclAi¤cdPE@BidAI N#<?AWUA"ANc?Y.PKPJYMi&IEclAWJYMWU@xAf

Y.i PJ<dAIJLdWJÇKMhQA PÁ=K[i?fªPU<dAY[PU<dAWI§PU<dA^Ldi?fOAWJy.WJY.Ldi?fª@BidAIjarc:Ai¾cdP@xididy©@BYOfOAI<?K9.AslAAi@L:h<^@xY.WJA,NÇ<dITPUYMWUhK[S'P#fOAMAY.c:AfPU<?K[i&Ldi?fOAWJy.WJY.Ldi?f@BididyKMi?fPU<dA " . "$"PJYIJAQPvLdcp<dyM<&c?WJYMdPJidyY.c:Ai¤c?PgAQNOcdYMPUK[PJYMic?KMi?IE<?KMIEslAAi&PJ<dA 0" &)&)/K x#<dh<hYMWUWJAIeclYMi:fdIgPUY¡fdAQPJAWJ@BidAPJ<dAcdP#PU<ª<?y.<dAWBAhYMidY.@BhhQYMiPUAiPLdi?fOAW"PU<dA^KMIUIeL?@xcdPJYMiYMiO?i?PUARhKMc?KMhPTS.aRhQYML?WUIJA=ixPJ<?AWJAK[OYMWUfPJ<?I§<SclY[PU<dAIJIXfdYAIXidYMP<dYMfMs?LOPPJ<?A#AQNOIePJAi?hQAYM«K[i&AhQAiPgK[yMYMWUPJ<[email protected]^sS AWUh<?IEK[i:fEWUY.IUIe@K[i^iPU<dA =Mn I o°3PUYIJYMMAPg<[email protected]<dIcdWUYMs?A@ PJ<dAEMAWJidA?YMK[@BY.IeP«A.AWUSB<dALdWUITPUhPJWUSidyPJYBK[cdcdWUY.K.h<BPJ<dArPJWULdAEcdWUYMsdA@iWJAK[3YMWUfi:ITPK[i?hAI

(Ri?fOAWUyMWUYML?i?f@Bididy©YMcOPU@B¦K[PJYMi¾@BYfdAIK[WUA^IeY©WUAhAiPBPJ<?K[PPJ<dAWJA^IidYMPhQYM@B@BAWhQK[IJY[¨PT=K[WUAv@[email protected]?yPJ<?A@ËSMAP3K[i?fY.idSJNÇIJA.AWK[P<dAL?WJITPUhI<?K.IslAAiL?IJAfPJYIeAPr@Bi?idyIUh<dAfOLdAIi&WUAKM@BidAI m°v¬AMAWePU<dAAIUI:PJ<?AIJA@BYOfOAIR<:K9MAgPU<dAxK.fOK[iPUKMyMAvPU<?KPEPJ<dAS^IeY..AK@BYMWUAWUAK[ITPUhcdWJY.sdA@3i?hL:fOi?yhK[c?K.hQPTShYMi?IePJWK[iPUIgari&PJ<dAxY[PU<dAWE<?K[i:flPJ<dAS^idAAf^PJYfdYIeY.@xABcdWUAc?WJYOhQAIJIJidyPUY^@K[MAxPU<dAiOYMWU@KPJYMip@BYMWUAK[yMy.WJAy.K[PJAf¤K[i?f¡KMWJABid.Afp#PJ<¡YMidAIJclAhQ:hELdi?fdAWUyMWUYMLdi?fAQNOcdYMPUK[PJYMi@xididyBPJAh<didY.Y.yMSM;=<dA@Bididy&IUh<dAfOL?AIvYMsOPK[idAfpsS^PJ<?AhQLdWUWUAiPg@xYOfOAIK[WUA LdPJAi?YMiOWJY.sdL?IePari?hABPJ<dA

hYMi?fdPUY.i?I NY.i cdWUhAM#fOA@KMi?fKMI«AEKMIBPU<dAAQNOclAhQPJAf¾Y.WJA^y.WUK.fOAI$PBh<?KMidyMA.§PU<dAS s:AhQY.@xAIJLdslYMcOPU@KMKMi?fPJ<dAWJAYMWUAMl@L?IePrslAWUAhK[hQLdKPUAf]@KZTY.WyMYK[YMPJ<dIE«Y.WJIRPJYiPJWUYOfOL?hQAWUYMs?L?ITPUidAIUI"iPJ<dAcdK[idi?idypcdWUYOhQAIUI¡;=<?IxIxfdYMidAiªPTY¤fO !lAWJAiPx=K9SOIL:Ieidy¡ITPUYh<:KMIePJhc?WJY.yMWK[@B@Bidy¢KMi?f£[email protected] idYMPPJY¾¬RV­ @KNO@B¦KPJYMisdLOPPJY hK[c:KMhÁPTSfOA@K[i?fIUKPUIeÇKMhQPJYMiE]RIvK[i¤KMfdfOPJYMi:K[hQYMiPUWJsdLOPUY.ilPJ<dIE@xYOfOAIEK[WUA"fdAMAY.c:Af^i&IeL:h<Kx=K9SBPJ<dASKMY£slY[PJ<Ldi:fOAWUyMWUYMLdi:fKMi?fY.c:AicdP#PUAh<di?YMYMyMAI«PUYBhYAQNOIeP;=<dI"WJAc:Y.WePI"YMWUy.K[i?¦AfpKMIYMYI]P?WIePXhQLdWUWUAiP NÇIJAc:K[WKPJAfUPvY.c:Ai¢cdPxK[i:f¡Ldi?fOAWe

y.WJY.Ldi?f"@BYfdAIXK[WUAWUAIeAf]#¨PJAWXPJ<:KPITPUYOh<?KMIePJhAA@BAiPUI§K[WUAANOc:YIeAfiBYMWfOAWPUYEiPJWUYOfOL?hQAIJYM@[email protected][WUShQY.i?hQAcOPUIvYM§PJ<?AK[WUAKK[i?f¤<dY WUYMs?L?ITPUidAIUIvK[cdclAKMWUIEi©ITPUYh<:KMIePJhcdWUYMy.WUKM@x@Bidy?¬RAQNP:PJ<?AAQNOclY.IJAfAA@xAiPUIRKMWJAKMcdcdAfPJYPJ<dAhKMIJAY[X@Bidi?yIUh<dAfOL?idy?§PJ<dA"@KZTYMWIJYML?WUhAIYMdLdi?hQAWePK[iPTSriPU<dA§cdWUYMs?A@ KMi?fEPTYrIePJYOh<?KMIePJhXcdWUYMy.WUKM@B@xidy#@BYOfOAIKMWJA«fOA.AYMclAf;=<dAgKMs:Y.AR@BAiPJYMi?Af@BYOfOAI#KMWJAEITPUYh<:KMIePJhRANPUAi?IJYMi?I«Y[¬RVX­ @BYOfOAIPU<[email protected]#fOAS

KMcdcdWUY.K.h<PJYPJ<dAx@BidAIUh<[email protected]?PU<dAWUAxKMWJA"Y[PU<dAWIEK9SOIiPU<dA LdAIePEYM $@xYOfOAI"Y.Wx@BidAIJh<dAfOLdi?y?¢;=<?K[PxI"PJ<dAWUAKMIJYMi¢YMWBIeAPePJidypLdc¾K&fO !lAWJAiPBi?f¢[email protected][c?K.hQPTSK[i?ffOA@K[i:fIUKPUIeÇKMhQPJYMii?IePJAK.fYM¬RV­5@K[N@B¦KPUY.iX¯i?KMSMKx<dALdWUIePJhrPJYIJYMMArPJ<dI#cdWUYMsdA@I=cdWUYMclY.IJAf

××Ù"[U\]$^^

Page 9: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

= " "

w

vi

i

¯yML?WJABCMYOh@xYOfOA

2 !Ë

¯?YMW=PU<dAIJKMMAEY[Ie@BcdhQPTSM:K[3PJ<dAhYMi?hAcOPI#<dAWUAgK[WUArc?WJAIeAi.PUAfi^Kt[qR«Y.WJf

`«L?WJWUAiPKMcdcdWUY.K.h<dAIPJYEPJ<dA=Y.c:AicdPX@xidA=cdWUYMsdA@ KMWJA«s?KMIJAfYMiPJ<dA=sdYOhg@xYOfOA0KR@K[y.i?KMWJSc:K[WJPJPJYMiY[PJ<dA@xidAgiPJYWUAhÁPK[idy.AI«¯dYMW=PU<dA

i°PJ<sdYh3OPTY L?K[iPUPUAIK[WUAghQYMi:IefOAWUAfONÇIJAA

¯yCPE

CM=PI0 $!+"1"" Uvi ∈ IR

#<?h<phK[i¤s:AxidAy.KPU.APJ<?A"K[LdAxY[§PUIvhYMiPUKMidAf^YMWUAIidY[PAidYML?yM<PJYBc?K9SPU<dAgAQNPJWKMhQPJYMiY[PJ<dAgsdYOh3

tOwi ∈ IR+

dPJ<dA -K=Y[PJ<dAgsdYOh3

]rIJIJYOhQKPJAf#PJ<PU<dAvsdYOh@BYOfOA3PJ<dAWJAvIK[IeYBK"yMWK[cd<G = (E, V )

Nǯy?§tPQdIeL:h<PU<?KPEIPJ<?AIeAPY[s?YOhOI#K[i?f

(i, j) ∈ V ⇔ j@L?IePslAgAQNOcdYMPJAfslAQYMWUA

i.¯yMLdWUAIuEK[i?foEIJ<dY¡AK.IesdAKMi?fiOAKMIJsdA=cdPUI NANc?Y.PKPJYMixITPKPUAI$PÁ[WUAIJc:AhÁPU.ASM;=<?WJAA

YMPJ<dAWhQY.i?hQAcOPUI#KMWJAvIe<?Y#iiPJ<dAIeAgcdhÁPULdWUAI#K[IJY?

;=<dAgidPUKMIJLdWeÇK.hQAvfOYAI#i?Y[P#idAAfPJYBs:AlKP

AhKML?IJABPJ<dA«Y.WJf¤fdYAIidYMPAi:f¡#<dAWUAPJ<?AsdYhp@BYOfOAfOYAIPU<dAPJY.c¢hQY.WJidAWsdYOhINy.WJASPhK[ididYMP#s:AgWUA@BYMAf N0IeLdWJÇKMhAgIe@BYY[PU<didAIUI=YML?yM<P=PJYBs:Ag<?YMfUPÁ

¯dY.W"Y.c:AicdPcdKMidididy&c?LdWJclY.IJAIxK[i:f¢fOLdAPJYpcdWUAhQAfOAi:hQAhYMi?IePJWK[iPUIPJ<dAWJAK[WUA@K[iSsdYhOIPU<?KP#idA.AW=s:AgANPJWKMhÁPUAf

ài¯y.LdWJAroPU<dAEAQNOcdYMPUKPUY.iYMhQWUhA°@KMWJ.AfBsdYOhI«Y.K[PJAI§cdWUAhAfOAi?hQArY[PU<dAvYMidAI§#PJ<PU<dAhQWUY.IUIJAfc:KPePUAWUi

ÐÏ×ШØ

Page 10: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ M

¯y.LdWUAgtdXVXWJAhQAfOAi?hAvyMWK[cd<

¯y.LdWJAgu?¯?AK.IesdAEc?PI

××Ù"[U\]$^^

Page 11: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

> " "

¯yMLdWUAvo? àiOAKMIJsdAgcdPUI

9 3 3 C ² *¨7 ¡O ² G 7¨,G~ ¶ * ² PxhQYMWUWUAIJc:Y.i?fdIvPUY^?i?f¡PU<dAIJLds?IJAQP

SY[

E@KNO@B¦idy^PU<dAIJLd@ Y[=PJ<dAK[LdAIY[=PJ<dAsdYhOI

hYMiPUKMi?AfiK[i?f¤<dYMfOidycdWUAhQAfOAi:hQABWJAITPUWJhÁPUY.i?IEY[G\^YMWUAxc?WJAhQIeAS.

xi = IIS(i)N¨PU<dA

i?fOANLdi?hQPJYMi^YMSPQOPJ<dAi ∑

i∈S vi =∑

i∈E vixi,K[i?fPJ<?AgcdWJY.sdA@hK[is:Av#WUPJPJAiK.I

NUP ) max

x

i∈E

vixi

I Pxi ≤ xj (∀(i, j) ∈ E)

x ∈ 0, 1|V |

;=<dIcdWUYMsdA@®I=hK[AfPU<dA1.0" &)X&+/ u Os:AhK[L:IeAEPI=IeY.LOPUY.iI«PJ<dAE@KNO@"Ld@c?P NÇi^i?hQL?IJY.iIJAi:IeA P«PJYslAgAQNOcdYMPJAfd#<dAi^hK[c?K.hQPTShQY.i?ITPUWUKMiPI#K[WUAEi?Y[PcdWUAIJAiP Os?LOPRPIKMIJYxi?Y#iK.IPJ<dA 0 /-/&cdWJY.sdA@O#<dh<hK[is:AgWUAfdL?hQAfPUY\]r¯ a¾\ T¬E` (R; =qRAIecdPJAY[PU<?KP:hYM@B@BAWhQK[IeYM¨PTKMWJAE@xc?A@xAiPUIIJc:AhQ:hgK[yMY.WJPJ<?@BI«PUYIeY..AP;=<?Ag@xYITP##fOASL:IeAfI=PJ<?A AWh<?I [email protected]<d@ o°¬RY[PJhQArPJ<:KPPJ<dA

wi

hQYAhQAiPI«fOYxidYMP=KMcdc:AK[WiPJ<dI@BYfdA°Os:AhK[L:IeARPJ<dI@xYOfOAfOYAIidY[Pi?hL?fdA"hKMc?KMhPTS.#YAMAW?i^PU<dAWUAKMYMWUfPU<dAWUAAQNOITPvhKMc?KMhPTShQYMi:ITPUWUKMiPUI NY[ANPJWKMhÁPUY.iKMi?fc?WJYOhQAIJIJidyKPPJ<?K[PK9.YMfvPJ<dA#<dY.A=IeY.LdPJYMiYM N (rVPs:[email protected]ÁPJAfKPYMi:hQAM;=<dI@BAK[i?IPU<?KPgY.idAB<?KMIrPJY^fOAhQfOAB#<dh<¤clYMWJPJYMi¤Y[§PU<dAIJYMLOPJYMi

S<?K.IrPJYs:ABANc?Y.PUAf&?WUIePK[i?f&PJ<L?I

0BK[cdclAKMWUIRi&PJ<dAxcdWUYMsdA@§sdYOhiIvIeY.fi&PJ<dAxidAQNPvc:AWJYOflPJ<dAxcdWUY[dP0N@BAKMIJLdWJAf^i

PUYOfdK9S [email protected]'PvY.sOPUKMi?Af©sS¤PJ<dAhQYM@Bc?KMiS¤#slAYMi?Sαvi

#<dAWJAα ∈ (0, 1)

hQY.WJWUAIJc:Y.i?fdIPUY^PJ<?AO$&&.!) "0®aRRhQY.LdWUIJAMfO !lAWJAiP, U! $ (&+ " " "! SAf©fO !lAWJAi.PxidAQPc?WJAIeAi.PK[LdAI NǬrVX­ PQdIeYPJ<dArcdWJY.sdA@ s:AhQY.@xAI§AMAi<?K[WfOAWP=I§idYMPAidYML?yM<BPJYBfOAhQfOA "PUYBAQNOcdYMPdsdLdPRK[IeY !PUYBfdYxP

ÛÈTÈÆ0¹9ȧÃÅ9Å[ÈTÙ9É9º ¿ÁÀÃÊ[ÈàÆÇÆ°ÈTÀÄÙ9É9ÈTÀ0»ÇÆÃÙÉ9º¼Ù¿OÆ°¹º¼»ßT¿ÁÙ9ßTÈTÅÆeã

ÐÏ×ШØ

Page 12: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ ?

9 3 9 0G ² 6 G 7| ² ² 6 7 6 I ¶ · ² *AhK[L:IeAEY[PJ<dAgc?WJAYML:I«AA@BAiPUIdPI=idAhQAIJIUK[WUSBPJYhQY.i?IJfOAW

]jfOIJhYMLdiP#WUK[PJAvY[ α ∈ (0, 1)

T ∈ INc:AWJYOfdI=Y[PJ@BAM

qRAhQIeYMiK[WUK[sdAI

xit =

1

YOhiI=AQNPJWKMhQPJAfK[Pc:AWJYOf

t

0aRPJ<dAWJ#IJAM

Ct

PU<dAghK[c:KMhQPTSK[P=clAWUYft

PJ<PU<dAIJAgfOAQ?i?PUY.i?IdPJ<dA(&!,&+ U! !K-%&) ËhQY.LdfslAIeAPRKMI

N°aEV«_'Pmax

~x

¬rVX­=

i,t

αt vi xit

s≤t

(xi,s − xj,s) ≤ 1 (∀(i, j) ∈ E)(∀t)

i

wi xit ≤ Ct (∀t)

t

xi,t ≤ 1 (∀i)

xi,t ∈ 0, 1

AhK[L?IJAvY[PJ<dAg<Ldy.AEiLd@"s:AW#Y[K[WUKMsdAIOhK.IJIJhK[K[c?cdWJYKMh<dAIPUYIeY..AaEV«_hYMLdfAKMIJSslAhYM@BAK[i iOAKMIJsdA=K9S3

|V |IY[¨PUAi Y.AW^[email protected]#IeY:ZTL?ITPYMWK¾Cn¡c:AWJYOfdIYMrPU@BA

i?IePUKMi?hQA.+NaEV§_'PXslAhYM@BAIKCnd nMnMn? n.nMnrsdi?K[WUS"idAKMW§YMcdPJ@x¦K[PJYMicdWUYMs?A@XqrLdAPUYPU<dI§ÇKMhQP@KMiS<dALdWUIePJhIr<?K9MA"s:AAipfOA.AYMclAf^i&Y.WUfOAWRPUY?i:f&y.YYOfAKMIJs?AxIeA .L?Ai?hAIrY[NaEV§_'PÁ]hYMLdc?AvY[AQNdKM@xc?AIKMWJAgK9K[K[sdAri > K[i?f ? ]Rs:Y.LOPPJ<?ABPJ<dAYMWUAQPJhKMXhQYM@BcdAQNOPTSMPIvc:YIJIJsdAxPUY^cdWJY.APU<?KPgPU<dAAK.IeAWgKMi?f¡Ie@Bcd?Af

.AWIeYMiON#<dh<^K[cdclAKMWUIiPJ<dAg@BPUIhKMIJAIY[α = 0, 1

NUPC) max

x

i∈E

vixi

I Pxi ≤ xj (∀(i, j) ∈ E)∑

i

wixi ≤ C

x ∈ 0, 1|V |

I#¬RVXrK[Wf >

××Ù"[U\]$^^

Page 13: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

Cn " "

¯y.LdWJAmdài:fOfOL?KMsdYhAQNOcdYMPUKPUY.i

VWJY.yMWK[@B@Bidy"@BYfdAIYMW=Ldi?fOAWJy.WJY.Ldi?f@xididyHN (E\ WUYM@ i?Y P«KMWJAEMAWJSBWUAhAiPK[i?fKMfdKMcOPJAfPUYrY.idA«IJclAhQ:hi:fY[:AQNOcdYMPUKPUY.igPJAh<didY.Y.yMSM9i?KM@BAf !K- m°YA.AWPU<dI@BAK[i?IPU<?KP#PUYIeAPRK[ii?IePUKMi?hQAvYMPU<dAIJAv@xYOfOAIRIeY.@xAv@Bc:Y.WePK[iP#cdWJAcdWUYhAIUIeidy"I#WUA LdWUAf

ài [email protected]§s?YOhOIK[WUA^hK[Af $&KMi?f¾K[WUAyMWUYMLdclAfªiPUY©sdy©s?YOhOI#<?h<&KMWJAAQNOcdYMPJAfTi?fOAc:Ai?fOAiPUSWUYM@ËAK.h<^Y[PJ<?AWIE;=<dAANc?Y.PKPJYMi^YM«KIJc:AhQ:hs?YOhIePUKMWePI#KP=PJ<?AgsdYOh I=s?KMIJAvYMW*DÁd#<dAWJAEWKMhÁPULdWUAI#K[WUAEc:AWeY.WJ@BAf?IeYxPU<dAg@KPJAWJK[hYMKMc?IeAI:YIfdY#ifOL?AvPJYByMWK9PTSBY.WUhAEKMi?fIhYMAhÁPJAfX;=<dIhQY.AhQPJYMic?WJYOfOL?hAI#K.fdfOPJYMi?KMA@BcOPTSIJc?K.hQAMdIJYsdyMy.AW=A !3YMWJPUIKMcdclAK[W#iPJ<dAWJYOh#<dh<^hQYMK[c?IJAI#KMy.K[iK[i?fIJYxY.i§¯yMLdWUAmBIe<dYIPMAWJSIUh<dA@BK[PJhKMS Ny.WJASc:Y.WePUY.i?I#K[[email protected]<IePUKMyMA PÁYhOIrK[WUAyMWUYMLdclAfiPJYIJA.AWK[IJAhQPJY.WUI?#<dh<&hK[i^slAcdKMhAfK[PEfO !lAWJAiPRAMAIlIJY?3fOLdA

PUYgPJ<?ARsdWUAKM9KMyMA=Y[@K[PJAWUK[°.K[iS"iPJAWUK.hÁPUY.is:APT«AAiAMAI§@L?IeP§slArK9.YMfOAf\YMWUAcdWUAhIJASMYMidAvsdYOhI#IJLdidKP#PJ@BA

tOidYY[PU<dAWIYOhK[¦AfJ "#hKMislAgAQNOcdYMPJAfK[¨PJAW=PJ<dI#clAWUYOf

YOhjI#YMAW«sdYOh

iPI=s?KMIJAviPJAWUIJAhQPUIPU<dAhQY.idAgITPK[WJPJidyKP#sdYh

i I=s?KMIJAM

ÐÏ×ШØ

Page 14: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ C.C

SECTOR 1

SECTOR 3

SECTOR 2

SECTOR 1

SECTOR 3

SECTOR 2

FIR

ST P

ER

IOD

SEC

ON

D P

ER

IOD

¯yMLdWUA=d AMAI=ANOhL:IeYMi

××Ù"[U\]$^^

Page 15: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

C9t " "

ài¯y.LdWJA =dPJ<?ArAi.PUWUAg_Ob«`;RaEF C Ny.WJAS'PXslAhQY.@BAI "D !¤s:AhK[L:IeArY[PU<dAvAQNOcdYMPUKPUY.iYMPJ<dAEPTYsdYOhOI§i_Ob`;RaEF6tOsdLOP#sdYOhOI«K[P=_Ob«`;RaEF6u"KMWJARAQNOcdYMPJAfiPJ<dAEidANP=clAWUYOf¬RY[PUhArPU<?KP#PU<dAgcdWUAIJAi?hAvY[PU<dAhQY.i?IePJWK[i.PfOAc:Ai?fdI#Y.idSYMiPJ<dAgs:KMIJA N ?YYMWP«YOhK[PJYMi]RidY[PU<dAWhQYMi:ITPUWUKMiPcdWUAIJAiPi (E\ÓIBPU<?KPIeY.@xAhYMidi?AhÁPUPTSIK[IJY¢fOAIJWK[s?A.«i¾PJ<dA

IJAi:IeA#PU<?KP«PJ<dAEAQNOcdYMPUK[PJYMiYMsdYOhOI#<?h<KMWJARidY[PidAyM<slYMWIXY[KMidY[PU<dAWK[WJAKMfOS"AQNOcdYMPJAfIANOc:Ai?IeMA0NÇAPJ<dAW#@xY.idASYMW=WUIJ'PÁ;YBhYMiPJWUYM3POPU<dAvYMY#idyxWULdAI=<?K9MAEPUYxslAvYMY«Af

bXMAWJSsdYOhiPJ<dAIJAhQPJY.W=hK[islAK( " !K-&!U°

;=<dAPUY[PK[§iLd@"s:AWYM#IePUKMWePUi?yc:Y.iPIi¢K^yMMAi¢IeAhÁPJY.WhKMididYMPs:AIeL?c:AWJYMWgPUY&KdNOAfIe@KMiLd@"s:AW%NTCvYMWRtPÁ]#hQPJL?KMSMPU<dAITPK[WJPJidyBc:Y.iPI=@BK9Ss:AvdNOAfiKMfOK[i:hQAM

KsdYOhIri?Y[PEKIePUK[WJPJidyc:Y.iP3PhK[i&s:AxAQNOcdYMPJAfYMi?S§YMidAxY[PUIEidAy.<s:Y.WUIr<?KMIs:AAiAQNOcdYMPJAfs:AYMWUAM

aRhQY.LdWIeA.?hKMc?KMhPTShYMi?IePJWK[iPUIY.WUhAvPJ<dAi?hL?IJYMi^Y[XY.cdc:Y.WePULdidPTShY.IePRhQYMi:IefOAWKPUY.i?I=i(E\ cdKMidididyK[IeY:

9 9 3 z ¶ · ² *¯?YMW#AKMh<s?YOh

iPU<dAvYMY#idyxidYMWU@BK[PJYMi<?KMIPUYs:AgK9K[K[sdA m

]iAhYMidY.@BhEK[LdA vi ∈ IR:K[i?fKx«Ay.<.P

wi ∈ IR+

;=<dAIJAQP Ui

YMPU<dAgsdYhOI#K !lAhÁPJAfsSPI#IeidK[y.AM

Ni

PJ<dAIJAQPYMPUI#i?AyM<s:Y.WUI

;=<dAWJAKMWJAKMIJYS = Sgg

PU<dAIJAQPYM=IJAhQPJYMWIK[i:f&Y.WKyMMAi©IeAhÁPJY.WSg

PUI@KNO@"Ld@iLd@"s:AW#Y[XITPK[WJPJidyxc:Y.iPUI

sg ∈ IN

;=<dAyMYMs?KM§c?KMWUKM@BAQPJAWUIY[#PJ<?I"@BYfdA=K[WUAMK[i¢<?YMWU¦YMi¡Y[TclAWUYOfdIKfOIUhQYML?i.PxWUK[PJAYM

α ∈ (0, 1)dK[i:fKBhKMc?KMhPTS

Ct

Y.W#AKMh<clAWUYOfi, j

fOAidYMPJAgsdYOhOIgIJAhÁPUYMWI=K[i?f

t, sc:AWJYOfdIOWUAIJclAhÁPU.ASM

_Y:OYMW=PU<dAgK[WUKMsdAI

xit =

1

s?YOhiI#AQNOcdYMPJAfK[P

t,

0i?Y[P

KMi?f

fi =

1

sdYOhiI#KMiIePUKMWePUidyBclYMiP

,

0idY[P

,PU<dAgcdWUYMsdA@Y[IJA LdAi?hQidyK[iLdi?fdAWUyMWUYMLdi?f@Bi?AEWUAK.fdI

ÐÏ×ШØ

Page 16: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ Cu

maxxit, fi

¬RV­=

i,t

αt vi xit

xit ≤ fi +∑

j∈Ni

s<t

xjs (∀i)(∀t)

i∈Sg

fi ≤ sg (∀g)

i

wi xit ≤ Ct (∀t)

s≤t

xis ≤∑

s≥t

xjs (∀i)(∀t)(∀j ∈ Ui)

t

xit ≤ 1 (∀i)

xit, fi ∈ 0, 1 (∀i)(∀t).

_AMAWUKMOAQNPJAi?IeYMi:IXhK[ixslARfOYMi?A«PUYvPJ<dIX@BYOfOA°X]iB@xclYMWJPUKMiPYMidA=IPJYKMY¢PU<?KPclYMWJPJYMi?IYMPU<dAgsdYhOI=slAhYM@BAvAQNOcdYMPJAfK[P=AKMh<c:AWJYOf]RiK[i?KMY.yMYML:I§cdWUYMAEPUYBPJ<dAgY.idAgL?IJAfYMWIJ<dY#idyxPJ<?K[PPJ<dAgY.c:AicdPr@xididyIJA LdAi:hQidyBI

¬rVàRK[Wf > hK[islAK[cdc?AfPJYcdWUYMARPJ<?K[P#PJ<dI#cdWUYMs?A@ <?KMIPU<dAIUK[@BAghQYM@BcdAQNOPTSM

Q !

;=<?AWUAK[WUARIeAMAWUKMdIeY.LdWUhAIXYMLdi?hQAWePK[iPTSiPJ<dAR@xidArcdKMidididysdL:IeidAIUI;=<dARYMidAI§hQY.i?IJfOAWJAf<?AWUAvKMWJA.

* ¶ ¶ · ² * ² ~~ ¶ ~|M \AK.IeL?WJidyy.WUK.fOAIKMi?fBPU<dArWUAIeP«YMPJ<dAEiOYMWU@KPJYMiPJYs?LdfPU<dARs?YOh@BYfdA3IANclAi:IeMA.IJYY.idSK@BPJAfiLd@slAW=YMfOWUidyINÇAQNOcdYMWKPJYMi:I P§K[WUARclAWJYMWU@xAfPJY@xAKMIJLdWUAPJ<?AIJAK[WUKMsdAIR;=<dAxWJAITPRYMXPJ<dABfdK[PUKIrAIePJ@BK[PJAf^sSIePUK[PJITPUhK[@BAQPJ<?Yf?IYA.AWPIi?Y#igPJ<?K[PPU<dAIJA§@BAPJ<dYOfdI 9AIePJ@BK[PJYMi?IiPUWJYOfOL?hA«IJYM@BA§@Bc:Y.WePK[iPAWJWUYMWI

~"7 ² |M ;=<dAghQY.cdclAW#cdWUhAghK[i^KMIJY"KMWJS.dfOAclAi?fdidyY.iPU<dAgYMWUfhQYMc?c:AW#idAAfdI ² "C 6 ¶ * ¶)I:³ ;Ah<didYMYMy.SK !3AhQPUI#hQYITPINÇfOAhQWUAK.Ieidy"PJ<dA@YMAWPJ@xA PÁzB.7· ² 6 |. _YM@BAQPU@BAIdK"sdYhWJA@BKMi:I "!-$dPJ<dAvWKMhQPJLdWUAfWJYOhhQY.LdfWUAK.h<K[iA LdsO

WJLd@$ITPKPJAKMi?fx@KPJAWJK[:fOYAIXidYMP ?Yg]rIA0.P§hYMLdfx<?K[c?c:AiBPJ<?K[PcdP«KMIXhQY.K[c?IJAMK9MY.fOidyPJ<dIgIeAhÁPUYMWrPJYs:ABAQNOcdYMPJAf YMPJ<¤IJPJL?K[PJYMi?IE@BAKMiPJ<:KPvPJ<dAB@BidAIJh<dAfOLdi?y<?KMIPUYs:Ag@BYOfO?Af?K !3AhQPJidyIJh<dAfOLdi?yBKMi?fhQYM@Bc?KMiS I=WUAMAiLdAI

××Ù"[U\]$^^

Page 17: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

Co " "

;=<?A@BK[ZTYMWvyMYK[XYM«c?K9Sidy^KPePUAiPJYMipPUYIJL?h<©KcdWUYMs?KMsdITPUhxA.AiPUIvIvPUYs?Ldf¡@BYMWUABWJY.sdL?IePc?KMi?IEYMWvPU<[email protected][email protected]«PU<dAIJABAA@BAiPIK[WUAPUWJAKPUAf¤YMidS¤fOY.i?yIJYM@BAKMi?K[SOIeI#s?K.IeAfYMi^cdK[idi?AW IRiPJL?PUY.i:idY#AfdyMAM?cdWJYMWUPJAI?AQNOclAWUAi?hQA.?APUh[]RiS=K9SMK[P=PJ<dAgAi?fY[PU<dAfdK9SPU<dAgWUAIJLdP#IK0-c?KMiY.W=PJ<dAgc?KMididAW I§[email protected]?ILOPM#<?K[PIXWUYMs?L?ITPUidAIUI ài?IJcdWUK[PJYMi"PJYgK[i:IeAWPU<dI .L?AIePJYMiBhK[ixslA=YMLdi?fBi t ° ./

! $! $!#$ T K !K- &'$ &L"0"! O"! "! - ! &+/ " /-!+. O "" "$ "0 $ J&+/ $$" $" $! "! !+ ! /!) !# O ! /!U0!) $"!#" "!U %/$" 0!+$ T . " !+#&$ K ! ,!+ &)0" (0 -!+ $" $ !+$]rIA§KMIgi¡i?KPULdWUAMWJY.sdL?IePJidAIJIvi¡@xididycdK[i?idi?y^IgWUAKPUAfp#PJ<¢IJLdWJK[°`«<?K[i?yMAI

iPJ<dAcdWJhQA.dK[WUK[PJYMi?I=iPJ<dA"fOA@K[i:f:KMi?fJNc?KMWePUhLdKMWJS'P«[email protected]#hK[i@K[.AvKc?WJYMZTAhÁPgslAhYM@BABLdidcdWUY[dPK[sdAM#<dh<pIv.APIgfOAK[PJ< NY.WgAQNPJi?hÁPUY.iPJ<dAc:K[WK[AIe@ #PJ<AMY.LOPUY.iBI§MAcdP$PÁX;=<dI@BAK[i?IPJ<?K[P§WJY.sdL?IeP§cdK[i?I<?K9MA=PUYs:A * . QOKMi?fBcdWUAIJAWUMA#IeY.@BA $&'!D$giPJ<?AhKMIJAEPJ<?K[P#PJ<didyI#fOYBidY[Py.YxK.I=AQNOc:AhÁPUAf

" ! ª " %( *!(

;=<?ArPTYBWJAKMIJYMi?I«PUYxiPJWUYOfOL?hQAgIePJYOh<?K.ITPUhEcdWUYMy.WUKM@x@Bidy"K.I#K"PJYYMY.W#@BididyIUh<dAfOL?idyK[WUA

PIeAA@IPJYEs:As:APePUAWYMWK[WUyMA«IUhKMA«cdWUYMsdA@Is:AhK[L:[email protected]@BAPJ<dYOfdIhKMislAK[[email protected]@xYOfOAIeY.idyIeclAAf C

]RIvPU<dIK[i:fPU<dAidANPIJAhQPJYMi¡#L?IePJWKPUAMPJ<dAIJYMLOPJYMi:IvY[#ITPUYh<:KMIePJh"c?WJY.yMWK[@IvK[WUA@BcdhQPJSWUYMsdL:ITP

_PUYOh<?KMIePJhcdWUYMy.WUKM@B@xidy CMM OfOAKMI#PJ<gPU<dAIJAKMWUh<gYMdYMcOPU@KMfOAhQIeYMiLdi?fOAWLdi:hQAWJPUKMiPTSc?WJAIeAi.PUAfiPU<dAvidAQNPRIUh<dA@KdCM_YM@BAIJAQP#Y[XfOAhQIeYMi?I

xKMWJAE@KMfOAvPUYOfdK9SM

tO#qRL?WJidyBPJ<?Agidy.<PdIJYM@BAvWUKMi?fOYM@AMAiPUIω<:K[cdclAi

ud#]P#PJ<dAgi?AQNP@BYMWUidi?y?dhQY.WJWUAhQPJMAEKMhÁPUY.i?IyhK[is:AgclAWJYMWU@BAf

­KMWJK[sdAIWJAcdWJAIeAiPJAfpsSxK[WUAidY#i©KMI$$ -KK[WUKMsdAI#<dA

yIgPJ<?AMAhÁPUYMWYM

$"!# D- "" YMWZTL?IeP%/" $;=<dAR@xYITP«A@s?A@BK[PJh=AQNdK[@BcdA#Y[3PU<dI§i?fBYMlc?WJY.sdA@IXIPJ<dArIeYMhK[Af "0 &)/K

C §AWJAgKxÇKMWJ@BAWIJclAhQK[¦AIi^WUKMIJidy"#<?AKP?hQY.WJiK[i?fIJLdy.KMW=s:AAQPUIRK[i?f<?K.IPJYfOAhQfOAg<dY@"L?h<pKMi?fPUYfdAMYMPJAxPJYAKMh<¤hWJY.csdLOPfdAclAi?fOidyYMi¤PJ<?AxAK[PJ<dAW3PU<dAcdWUYOfOL?hÁPUPTSY[«PJ<dAK[i:f^h<?K[idy.AI#]IeY::KPrPJ<dA"Ai?fYMXPJ<dASMAKMW:<dAhK[iIJAYMWrsdLdSK[iSY[PJ<dAxhQWUYMc?I?s?LOPr<dA"<?KMIPUYBs:AIJLdWUArPU<?KP#hQY.AW=IeY.@xAv@Bid@"Ld@WJA .L?WUA@BAiPI«YM#<dAKPRKMi?fhYMWUiY.WhK[PePJAEAAf

ài PJ<?IcdWJY.sdA@ PU<dA?WITPIePUKMyMA^fOAhIJYMi?IxKMWJAPJ<?AK[@BYMLdiPIY[EKMi?f¾fOA.Y[PJAf¢Y.WAK.h<

hLdPJLdWUA PJ<?AWK[i:fOYM@ AA@xAiPUIBK[WUAPU<dASAfdI"Y.WAKMh<cdWJYOfOL:hÁP KMi?f©:i?K[S.PU<dA^hQYMWUWUAhÁPU.A

ÐÏ×ШØ

Page 18: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ C9m

fdAhQIJY.i?Iy

N#<dh<K[WUArhKMAf /" $ PhQYMWUWUAIJc:Y.i?fBPJYxPU<dAgIeAi?y 9s?LdSidyBK[@BY.LdiPUIidAAfOAfPJYIUKPUIeS@Bid@"Ld@c?WJYOfOL?hQPJYMiWJA [email protected]#K[i?f@KNO@x¦idyxÇK[WU@BAW Ii?hQY.@BAM

àix@BYMWUA«@KPU<dA@K[PJhK[PUAWU@BIPU<dA#IeYMhK[AfgPTY[àITPK[yMA§idAKMWXITPUYh<:KMIePJh§cdWUYMy.WUKM@ #PU<dNOAfWUAIJYML?WUhARWUAKMf?I

N0;_ _dVPmin cT x + Q(x)

NeCP

s.t. Ax = b

x ≥ 0,

#<?AWUA

Q(x) = IEω

[min qT (ω)y(ω) | Wy(ω) = h(w) − T (ω)x, y(ω) ≥ 0

],

N0tPIBhKMAf¡PU<dAO/ $ $! . !?§KMi?f

ω ∈ ΩK[WUAPU<dAc:YIJIJs?AIJhAi?KMWJY.I(NÇKMhhQY.WUfOidy&PJY¡IeY.@xA

c?WJY.s?K[sdPTSIJc?KMhA(Ω, IP )

!!( #!( * %( *!(

]P«PJ<dIclYMiP§PU<[email protected]?K[WKPUAfUPL?i?fOAWUyMWUYML?i?fBK[i?fBY.c:AicdP«@Bi?AI<:K9MAslAAic?WJAIeAi.PUAf AIJfdAIMPU<dArhYMi?hAcOP§YMWJY.sdL?IePJidAIJI§K[i?fIJYM@BAAA@BAiPUKMWJSxITPUYh<:KMIePJh#cdWUYMyMWK[@B@BidyAA@BAiPUIgKMWJABK[WJAKMfOS^ANOcdhQPK[IeY:B_OY?PgIvPU@BAxPJYcdLdPK[PJ<didy.IEPUYMy.AQPJ<?AWxRY«AMAW3YMidAK.fdfOPJYMi?KM[PU<di?yR#slA«fOY.idAMPU<dA§@BYOfOAIK[WUAyMY.i?y#PJYrs:A=fOA.AYMclAfEY.WPU<dA«hKMIJAY[?K 0!+$1!+[PU<?KPI.Kv@Bi?A#L?IeidygAPU<dAWL?i?fOAWUyMWUYML?i?fY.WXYMclAiBcdPPJAh<didYMYMy.AI;=<dA#fOAK?[Y[hQY.LdWUIJAMP=I«idY[PPJYBfOAMAYMcPT«YIJAc:K[WKPJAr@BYfdAIWULdiPJ<?A@®sSBPJ<dAWY#iKMi?fK[¨PJAW«PJ<:KP#hQYM@Bc?KMWJARPJ<dAWUAIJLdPUI§PJYx@K[MArPJ<?AR?i:K[3PJAh<didY.Y.yMhK[lfOAhIJYMi ài?IePJAKMfYMPJ<:KPPU<dAE@xYOfOAIe<?YMLdfKMY¾PJ<dAhYAQNOIePJAi?hQArY[slY[PU<PJAh<didY.Y.yMAIKMi?fBPJY"?i?fK[i &) $ $" ;Y"IJAQP=LdcIJL?h<KhYM@"sdi?AfONYMW#@BNOAfUP@BYOfOAYMslASOIPJ<dAvÇK.hÁPIPJ<?K[P

AhKML?IJAYMRPJ<?Ai?hWJAKMIJAiPU<dAK[@BYML?i.PBY[rK.ITPUAPU<?KP@L:[email protected]¢i YMclAic?P@BidAIPJY&c?KMIUIEWJY.@¥PU<dI@BAQPJ<?Yf¡YMANOcdY.PKPUY.i¤PUYPJ<dALdi?fOAWJy.WJY.Ldi?f¤YMidAhYMLdf¡s:AKyMYYOf^YMcdPJYMi^PUY.AAc&@xidAIEKMMA.E;=<?AWUAQYMWUAMl@BYOfOAIrKMY#idyPJ<?AxhYANOIePJAi?hQA"Y[§slY[PU<PJAh<didYMYMy.AIvK[WUAxKc:YAWJLdPJYY.PJY^fOAhfdA"PU<dABWJyM<Pv@BYM@BAiPvPJY^fOYPJ<?IEPUAh<didY.Y.yMhKMh<?K[i?yMAMOY.WPJYB?i?fPU<dAgK9SYMs:K[K[i?hARPU<dA@cdWUYMclAWUS.

PIgAK.IeS&PJYKMf?K[cOPgPU<dAfOAK.IvAQNOc:YIeAfp<dAWUABPJYPU<dAhQKMIUIJhK[hK.IeAIvY[#KYMidA0PUAh<di?YMYMyMS@BidA.

ài&PJ<dIvIeAhÁPJYMi&PT«YK[cdc?WJYKMh<dAI=PJYIeAPEL?c&@Bi?idyIUh<dAfOL?idy@BYOfOAIEs?K.IeAf^YMipITPUYh<:KMIePJhc?WJY.yMWK[@B@BidyvK[WUA#cdWUAIJAiPJAf;=<dA#?WITPYM3PJ<dA@$hYMWUWJAIeclYMi?f?IPJYKg@"LdPUIePUKMyMAIePJYOh<?K.ITPUh=sdi?KMWJS ׿ÁÊ9Ä9»ÇÆ°Ù9ÈT»0»º¼»º#Å9Ô¼º¼ßTº ƹ9ÈTÀÇÈÁã

××Ù"[U\]$^^

Page 19: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

C= " "

idAKMWrcdWUYMy.WUKM@ #PJ<pK[WUKMsdAIrY[PU<dAIJKM@xAxi:fY[§PJ<dABcdWUAYML?I1NÇhLdWJWUAiPP@BYOfOAI;=<?KPvIfdAhQfOidy«IJYM@BAIJc:AhQ:hclYMWJPJYMi^YMXPJ<dA"@xidA"<?KMIPJYslAANOcdY.PUAfi&IJYM@BA"IJc:AhQ:hgclAWUYOf^[email protected][P;=<dAIJAhYMi?fp@BYOfOAI [email protected]@BcdAPJ<:K[ipPJ<dA?WUIePgYMidA.PK.IJIJLd@BAIvPU<?KPPU<dAcdK[iIXL?SfOAhfOAfBiKMfOK[i:hQA%N#PJ<?WUIeP«IePUKMyMA#K[WUKMsdAI$PÁ[PU<dAiPJ<dAWK[i:fOYM@AMAiPUIXPK[.Ac?K.hQAMdKMi?f?i?K[SPJ<?AhQYMWUWUAhÁPU.ArK.hÁPJYMi:I=iPJ<?AIJh<dAfOLdAgK[WUAEfOY.idAM;=<dAIeAg@BYOfOAI#fOWUAhÁPUSh<?K[AidyMAEPU<dAhQLdWUWUAiPY.idAI=slAhK[L?IJAM

;=<dAEYMsOZTAhÁPJMALdi?hÁPUY.iI .X;=<dAvfO !lAWJAi?hQArI§PJ<?K[P«iPU<dAEhKMIJAY[PU<dAvITPUYh<:KMIePJh@BYfdAIdPI=idY[PclY.IUIesdAEPJYB@KNO@B¦AvPU<dAgidAQPc?WJAIeAi.P=K[LdAM?sdLOPPUI#ANclAhQPUK[PJYMi

]RIxK[i©iPJWUi?IJhcdWUYMclAWJPTS&Y[=PU<dA@BYOfOAidy?IeY.LdPJYMi?IY[=PJ<dAITPUYOh<?KMIePJhB@BYfdAIK[WUAAQNc:AhÁPUAfxPJYgslA@BYMWUA=WJY.sdL?IeP;=<dASxK[WUA=idY[P?NAfxcdKMi?IMsdLOP& rfOA?ididyg<dYªPJYK.hÁPiPJ<dABfO !lAWJAiPrclY.IUIesdAIJhAi?KMWJY.IM °l#<dA"PJ<[email protected]&IeY.LOPUY.i?I#PU<?KPK[WUA / " $S.AKMWJSYMiPJ<dAgs:KMIJAEYMPU<dA0!+ £LOPULdWUAgAQNOc:AhÁPKPJYMi:I

]#hQPJL?KMSMPJ<dAPT«YIePJYOh<?K.ITPUhxidAK[W@BYOfOAIgc?WJAIeAi.PUAf¡K[WUAByMAidAWK[¦KPUY.i?IvY[PJ<dAhQLdWUWJAiP@BYOfOAIM#PJ<BPU<dARKMfd9KMiPUK[y.A=Y[YY.idyEYMW§@xY.WJA#WUYMs?L?ITPc?KMi?I.s:AhKML?IeA=PU<dARIJYMLOPJYMi:IK[WUA#@BYMWUA:AQNOs?A.vPJ<dAS<?K9MAIJAhYMi?fpITPK[y.A"K[WUK[sdAIrPJYhYMWUWJAhÁPEPJ<dAc?KMipfOAc:Ai?fOidy^YMipPU<dAA.AiPUIKMIAKMIr?WITPvITPK[[email protected]<dAIeA"LOPJL?WJAxc:YIJIJsdPJAI"ari^PJ<dABYMPJ<dAWr<?KMi?fPvIEAQNOclAhQPJAfPJ<?K[PrPU<dAxWJAIeLdPUIrYMsdPUK[idAf#PJ<PU<dAIJAx@xYOfOAI3@BAKMIJLdWUAf^Y.idS^i&PJ<dAxYMsOZTAhÁPU.AL?i?hÁPUY.iªPUAWU@I#slA^c:YY.WJAW«sdLOPPJ<:KPIs:AhKML?IeAPU<dA^K[LdAIBY.sOPUKMidAf i PJ<?IB=K9SK[[email protected][ITPUhrPJ<:K[iPJ<dAgY.idAI#hYM@BidyWUYM@PJ<dAgY.WJyMi?KMl@BYOfOAI;=<dAWJAK[WUA§K[IJYIeY.@xA«@BclYMWJPUKMi.PfOIUKMfd9KMiPUK[y.AIKMi?fgcdWUYMs?A@BI#PU<IJAQPePUi?yRLdcPJ<dAIeA§IePJYOh<?K.IT

PUhcdWUYMyMWK[@B@Bidy@BYfdAIv;=<dAS^KMWJAfdIUhQL?IUIJAf^fOLdWUidyPU<dAY.WJ@"LdKPJYMiKMi?fKMIJYIJLd@B@BKMWJ¦AfK[P=PJ<dAgAi?fY[PU<dAIJAhÁPUY.i

;=<?A«:WUIePXcdWJY.sdA@jPUYEslAIJYMMAf"IidY[PKPJYMi àixPJ<dA=PTY[°PJAh<[email protected]<dAW YMW $KMWJA"L?IJAf^Y.Wri:K[@Bi?ysdYOhOIRYMPJ<dAxsdYOh@BYOfOA0KMi?f^PU<dA"s?ysdYhOIrY[§L?i?fOAWUyMWUYML?i?f^@Bi?idy#s:AhK[Af 0/K DÁ§;=<[email protected]=K[WUAEyMWUYMLdclAfiPUY $;=<dA¡IJAhQY.i?f¾clYMi.PslAQYMWUA¤IJAQPJPJidyL?c6K©@BYOfOAvIPUYªIUK9Sª<?YËPJAh<?idYMYMy.AI#Ldi:hQAWJPUKMiPTS

IJYML?WUhAI=K[i?fIePJYOh<?KMIePJhEcdWUYMyMWK[@B@Bidy#PUK[.Avc?K[WJP#iPJ<?Ag@xYOfOAidy:

8 3 3 ~ ² O ² 6 ¶+F C ² 6 ² ~9'7 6 ³ | ¶ ~9 ² |VWJhQAIrKMWJA"idY[PvK[iANc?hQPEc:K[WJPrY[§PJ<dAx@BYfdA°ls?LOPrPU<dASK !lAhÁPEPJ<dA"AhYMidY.@xhKMLdAIRYM§[email protected]<dAifd !3AWUAiP#c:YIJIJsdAgIUhQAi?K[WUYI«KMWJAvhQY.i?IJfOAWJAfiPU<[email protected][PJ<dYIeAKMLdAIx;=<dABIJKM@xABK[c?cdAIEPUYPU<dABLdi?hAWJPUK[iPTShQY.@xidyWJY.@ËPJ<dABsdYOh@BYOfOA°EfdAIJcdPUABi¤PJ<dAWUAKM?YMWUfBP=I«c:YIJIJsdAPJY 1!D"0" "!fdYMidy@BYMWUARANc?Y.WUK[[email protected]#AIePJ@KPJYMi:IPU<dI#YMcOPUY.iI=idYMPRhQYMi:IefOAWUAfiPU<dAg@BYfdAidy:

]Rs:Y.LOPgPJ<dABPUAh<di?YMYMyMhKMKMWJKPUY.i?I3PU<dAS¤#XidYMPvslAi?hQL?fOAfslAhK[L?IJAxPJ<dAWJABAQNOITPIEPJ<dAclYMhQSPJ<?K[PrPU<dAxWJAIeLdPUI1NǬrVX­ K[LdAI$P#@"L?ITPvslA"WUAK.h<?K[sdA#PJ<¤cdWJY.Ai NÇAQNOIePJidyP=PJAh<didY.Y.yMSMiYMWfOAWPUYxslAhQY.@Bc?K[WUAf#PU<AKMh<Y[PU<dAWI

ÐÏ×ШØ

Page 20: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ C"M

¯i?K[S.A.AigPU<dYMLdy.<KMhhQfOAiPIKMWJA«c:YIJIJs?A.ANOc:AWePI<dY.fPJ<?K[PPU<dAWA !3AhÁPIidYMPIJyMid:hKMiPiPJ<dAgYMi?yPUAWU@;=<L?IOPU<dASKMWJAvidYMP#yMYMidyxPJYBslAgi:hQL?fOAfiPJ<dAg@BYOfOA°

8 3 9 ².´ * ¶ X7 6 I

;YWUAfOL:hQA#PU<dAEKM@BYMLdiP§YM9KMWJK[s?AIhQY.WJWUAIJc:Y.i?fOidyvPJYYMclAicdPOKvPUAh<di? LdAEhK[Af, / !K-#lslAvL?IJAf =KMIJhK[S..WUAQsdYOhidyxhQYMWUWUAIJc:Y.i?fdIXPUY"y.WJY.LdcsdYOhOI«iPJYxsdyMyMAW«Y.idAIOK[Y#[email protected][y.yMWUAyKPJAfBcdWUYMsdA@®PUYs:AhYMi?IJfOAWUAf

8 3 @ A ¶ ² d|| JG7 ¶ 6 | ² "C 6 ¶ * ¶)I 7M * ² * |D7 ¶ 6 ¯dY.WIJ@BcdhPTS.[K#@KMhWJY.sdYOhA.A.Y[dAQNdhL?IJYMig#.s:AIeLdc?c:YIeAf

YMW«PJ<dAEA@BcdYS@BAiP«YMs:YMPJ<PUAh<di?YMYMyMAIP@BAKMi?I§PJ<?K[P§PJ<?AShKMididYMP«slAEKMcdcdAfKPPJ<dAIJKM@[email protected];=<dI#I=@BYMWUAEWUAIePJWUhÁPJAfPU<?K[iPU<dAgWJAK[YMWUfO#<dAWUAEPJ<dI#AQNdhL?IJYMiIc:YIeAfKP=PJ<dAgs?YOhA.A°

~ ² G~ ¶ ² ||7 6 I [email protected][i?fIJAhQPJY.W§K[WUAyMMAiMi?Y[P«Y.idS"iPJ<?AW§clY.IJPUY.iK[i?ffO@BAi?IJYMiOsdLdPRK[IeYB#PJ<PJ<dAW#[email protected]¦K[PJYMi

\^YfdAidy&Is?KMIJAf¡Y.i¢[email protected]@BYfdA«K.IIJ<dY#i¡iª¯y? MJNÇsdLOPxidY[PxidAhAIUIJKMWJS?PePJidyANOK.hÁPUSK.I#K[cdclAKMWJidy"PJ<dAWJA PÁ

P=@BAK[i:IXPJ<:KPPT«Y esdYOh"@BYOfOAI R<?K9MA#PUYANIePKIJAQP«YMsdYhOINi?fOAQNOAfsSiKMi?f

jPÁOK[i:f

K"IeAP«[email protected] N#PJ<i?fdhAImKMi?f

nPÁ;=<?AWUAIK[IJY"K[i<?YMWU¦YMi

TIeY

1 ≤ s, t ≤ TKMWJA

i?fOhQAI=WJAIeAWJ.AfYMW=PJ@BAgc:AWJYOfdI

8 9 3 d~ 7Çd * ² |¯?YMW#AKMh<s?YOh

iK[i?fc:AWJYOf

tOAPs:AvPU<dAgidAQNPK[WUK[sdAIWUAKPJAfPUYBYMclAicdP

rit =

1

XhQAiIKMsdArPUYBs:AgANc?Y.PUAfIei?hA

tsSYMclAicdP#PJAh<didY.Y.yMSM

0idYMP

KMi?f

pit =VAWhQAiPUK[y.ArANPJWKMhÁPUAfWJY.@

i0PU<s?YOhK[P=clAWUYf

t.ariPJ<?AgY[PJ<?AW#<?KMi?fYMW#L?i?fOAWUyMWUYML?i?f@Bidi?y

umt =

1@KMhQWUYMs?YOh

mI=AQNOcdYMPJAfKP#PU@BA

tsSLdi?fOAWJy.WJY.Ldi?f@xAPJ<dYOfdI0

iY[PU<dAWhKMIJAMKMi?f

××Ù"[U\]$^^

Page 21: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

C> " "

¯yMLdWUA M YOh[\^K.hQWUYMsdYOh"@BYOfOAYMW#@BNAf@xididy?

ÐÏ×ШØ

Page 22: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ C?

fm =

1

sdYhmIKMii?fOAc:Ai?fOAiPRIePUKMWePUidyBclYMiP

0Y[PJ<?AWU#IJAM

KMWJAvfOA?idAf P#IKMIJYxL?IeALdPUYhQYMi:IefOAW

Pit =∑

s≤t

pis, Umt =∑

s≤t

ums,

#<?h<WUAIJc:AhÁPJMASWJAcdWUAIJAiPUIPU<dAKMhhQLd@"LdKPJAfc:AWUhAiPUKMyMAgAQNOcdYMPJAfWJY.@ sdYOhiKPPU@BA

t

KMi?f@KMhQWUYMs?YOhm<?K.I=s:AAiAQNOcdYMPJAfiPJ<dAgiPJAWJK[

1, . . . , t

¬RY[PJhQAxPU<?KPvPJ<?AxY.idS^?WITPJIePUKMyMAKMWJK[sdAIEK[WUA"PJ<dAIePUK[WJPJidyYOhQL?IJAIEYMi?AIfm

x;=<dABWUAIePEYMPU<dA@K[WUAEIePJYOh<?K.ITPUhvKMi?ffOAclAi?fY.iPU<dAgcdWUAY.L?I=ITPK[y.AI¯?YMW#i?IePUK[i:hQA

pit = pit

(pjs, vis, wis, ums, vus, wms

)t

s=1.

8 9 9 A ¶ "Cd|D7ÇHWdO~"7d* ² |ài^Y.WUfdAW#PUYIJAQPPU<dA"IePJYOh<?K.ITPUhvcdWUYMy.WUKM@B@xidy@BYOfOA N#<dh<Iy.YMidyBPJYslA"K@LdPJITPK[y.AYMidA PÁP«Ii?Y[P§clY.IUIesdAPUYL?IJAWUKMi?fOY.@K[WUK[sdAIIJL?h<K.I

vi

KMi?fvm

YMW§AhQY.idYM@Bh#K[LdAI«K[i?fBAyM<PUIwi

K[i?fwm

WUAIJc:AhÁPJMAS.;=<?ArcdWUYMs?A@#PJ<PJ<dII§PJ<?K[PY.idS"PU<dARANOcdY.PUAfs?YOhOI§#lWUA.AKMPU<dAWWUAKMdAhYMidY.@xh=K[LdARKMi?fx«Ay.<.P;=<?I§@xAK[i?IXPJ<?K[P3PJ<?AWUKMi?fOY.@hQYAhAiPUIKMWJAh<dYIeAiK.IZTL:ITPgAQNOcdK[idAf3PJ<dAxK[WUKMsdAIri¤PJ<dAxcdWUYMsdA@¥KMWJAxIeAAhÁPUidy#<dh<&WK[i?fdYM@ hQYAhAi.PIvK[WUAWUAKM¦Af#<dh<I=idYMPRK[YAfsSPJ<dAIePJYOh<?K.ITPUhEcdWUYMy.WUKM@x@BidyWK[@BA«Y.WJ3aridAvK9SPUYIeY..APIPJYIJAAhÁP=PU<[email protected]#KMIY.YI

vit

slAidyBPJ<dAgAhQY.idYM@BhRK[LdAgYMsdYOhiKP#PU@BA

t

vmt

PJ<dAgAhQYMi?YM@BhrK[LdAgY[@KMhQWUYMs?YOhmKP#PU@BA

t

wit

d#<?h<IPJ<dAgAyM<P#Y[sdYOhiKP=PJ@BA

t

wmt

dPJ<dAKMi?K[YMy.YML?I«K[WUKMsdArYMW#@K.hQWUYMsdYOhm

RY«AMAWPYMWhQAIfOITPUWJsdLOPUY.i?IITPUWJY.idyMS fOAclAi:fOi?yªY.i£PJ<?A¡fdAhQIJY.i£K[WUKMsdAI\^YMWUAc?WJAhQIeAS.O

Vi

IPJ<dAfOIePJWUsdLdPJYMiYMW=PU<dAgAhYMidY.@xhr9KML?AgY[s?YOhiOPU<dAiYMW#AMAWJS

t > 1

vit ∼ Pi t−1vit−1 + (1 − Pi t−1)Vi,NÇuP

PU<?KP#Ivit = vis

PJ<dAgsdYOh<?K.I=s:AAiAQNOcdYMPJAfK[P#KMiSBPJ@xAs < t

OY.W#P#3PK[.AvKxWK[i?fdYM@KMLdAYMY#idyKgfOITPUWJsdLOPUY.i

Vi

idYMP§aRhQYML?WUIJAMMIJ@xK[W§fOA?idPJYMi?I«@L:ITP§slARhK[WUWJAf"YMW§PJ<dAWUAIeP#Y[PJ<dAgWK[i?fdYM@hYAhQAiPUI

××Ù"[U\]$^^

Page 23: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

tMn " "

8 9 @ Q ² ² ~ J7 6 7|D7·O¤ 6 · GO~ ² ² ~|_OAMAWUKMfOA?idPJYMi?IEK[WUAWUA LdWJAfiY.WUfOAWPUY#WJPJAPJ<dAxhQY.i?ITPUWUKMiPIK[i?f^IJAQPELdc^PU<dA@BYOfOA°r]P:WUIePOAQP

Im = i |sdYOh

ii.PUAWIeAhÁ[email protected]

m]rIA0OPU<dAIJAQPI

Ni

KMi?fNm

YMPJ<dAi?AyM<s:Y.WUI=YMs?YOhiK[[email protected]

mWJAIeclAhQPJMAS

KMWJAvfOA?idAf]Rs:Y.LOPA.AAQNdhQL?IJY.i9PXhQY.i?hQAWJi:IslY[PJ<PJAh<didYMYMy.AI;=<[email protected]

mK !3AhQPUI

PU<dA«IJAQPUU

m

[email protected][i?fgPJ<dAIeAPUP

m

YMdsdYOhOIYOhK[PJAfi?IJfOA§PUIhQY.idA§YMdi :LdAi?hAM]rIL:IeL?KM0KrfOIUhQY.LdiPWKPUAY[

αK[i:fK[i<?YMWU¦YMi

TK[WUA§hYMi?IJfdAWUAfPJ<L?I

s, tWUAcdWUAIJAiPclAWUYOfdI

]RIJY?yMMAiBPT«YidAyM<slYMWUidy 1!'Y[s?YOhOI.PJ<?AEfO !3AWUAi?hARslAQPTAAiPJ<dAW<dAyM<PI<?K9.APUYBs:AslYML?i?fOAfsSKxclAWhQAiPK[yMA

δY[K[iLdidPUKMWJSs?YOh I=<?AyM<P%NsdYOhIRK[WUAEhYMi?IJfdAWUAf<dYM@BY[

y.AidAYML?IiPJ<dAWRfd@BAi:IeYMi?I$PÁ]Rs:Y.LOPPJ<?AvhK[c?K.hQPJAIPJ<dAWJAgKMWJAEPJ<?WJAAEiMY..AfK[P#PJ@BA

t

;=<dAhKMc?KMhPTSYMY.c:Aic?PRANOcdY.PKPUY.i CPt

CUt

X@BPJidyLdi?fOAWJy.WJY.Ldi?f@xididyK[i:f

;=<dAgyMYMs:K[3cdK[iPRhKMc?KMhPTS Ct

¯i?K[S.OiI=KxsdYOh"PU<dAi

i−fdAidYMPJAI«PU<dArs?YOh@x@BAfOK[PJASY.AW

iK ∈ IN

I«PJ<dAv@KN@L?@iLd@slAWRY[K.hÁ[email protected]=K[PAKMh<PJ@BAM?K.I#AKMI

Sg

KMi?fsg

fOA?idAM?WJAIeclAhQPJMASMIJAhQPJY.WUI=K[i:fPU<dAW@KNO@Ld@ iLd@"s:AW#Y[XITPK[WJPJidyBc:Y.iPI

8 9 E ² MD7W ² F 6 MD7 ¶ 6ari?SY.W=PJ<dAIUK[.ArYMhAK[W#K[i:fIJ<dY.WeP=#WJPJidy:dAP

Ft =∑

i

vitpit +∑

m

vmtumt,

PU<dAhK.Ie< ?Y5KP#c:AWJYOft;=<?AvLdi?hÁPUY.iPJYslAg@BK[NO@B¦AfIPJ<dAgANclAhQPJAf¬rVX­

V = IE

[

F1 + αIE[

F2 + αIE[F3 + αIE[· · · ] | I2

]| I1

]]

.NoKP

RAWUAIt

WUAcdWUAIJAiPUIPJ<dA !D . ! $" $ & .0tlPJ<?K[PEIlPU<dA"KMWJK[sdAI

pis

K[i:fums

dKMI=AKMIPU<[email protected]

wis

K[i:fvms

wms

YMW#AK.h<s ≤ t

8 9 8 : ¶ 6 |~9U7 6 |;=<?AIJAvhK[is:AhK.IJIJ?AfiPUYBIJA.AWK[3yMWUYML?c?I * ² ² 6 O~ ³ ¶ 6 |~'7 6 |YhOI#[email protected]:AgANc?Y.PUAfYMidSYMi:hQAM

ÐÏ×ШØ

Page 24: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ tdC

(∀i)∑

t

pit ≤ 1N°mP

(∀m)∑

t

umt ≤ 1.N.=P

,G ² 6 G 7¨ ² XG* ¶ 7O7 ¶ 6 ¶ 6 |~'7 6 |RAWUARcdWUAhAfOAi?hQAEhQYMi:ITPUWUKMiPUI«K[WUArIJAQP=KMI§YMYI¯WUIeP=Y[K[°KsdYhB@"L?ITP=slArANc?Y.PK[sdAM

pit ≤ rit (∀i)(∀t),N MP

s?LOPOPUYBs:AgANc?Y.PK[sdAMPU<dAhQAWUyM<P#YMAWPUIJA@"L?IeP=slAgAQNPUWUK.hÁPJAfhYM@BcdAQPUASs:AYMWUAM

rit ≤∑

s≤t

pi−s (∀i)(∀t).N.>P

]#f9ZeK.hQAi.PRhAAQNOcdYMPUK[PJYMi?I=<?K9.ARPUYs:AIJ@xK[W

(∀i)(∀t)(∀j ∈ Ni)∑

s≤t

pis − pjs ≤ δ,N.?P

PU<dAWUAgK[WUAghK[c:KMhQPTSslYMLdi?f?I=PJYBslAIJK[PJIT?Af

(∀t)∑

i

wit pit ≤ CPt ,

NeCnP

KMi?f?i?K[SMOPU<dAgidAQNPhQY.i?IePJWK[i.P#@Bc:YIeAI=AMAANdhQL?IeYMiYMW=PU<dAgYMclAicdP@BAQPJ<?Yf∑

s≥t

pis +∑

s≤t

ums ≤ 1 (∀m)(∀s ∈ UPm)(∀t).

NTC.CP

6 · ² ~ I ~ ¶ 6 · ² XG * ¶ 7¨OD7 ¶ 6 ¶ 6 |~'7 6 |\^K.hQWUYMsdYOhOIhKMis:AEAQNOcdYMPJAfPJ<dASK[WUARITPK[WJPJidyclYMiPUI«YMWYMidAYMPUI«idAyM<slYMWI<?K.I§slAAi

ANOcdY.PUAfs:AYMWUAM

umt ≤ fm +∑

n∈Nm

s<t

uns (∀m)(∀t).NTC9tP

;=<dAgiLd@"s:AW#Y[XITPK[WJPJidyxc:Y.iPUII=slYMLdi:fOAfiAKMh<IeAhÁPUYMW#KMIY.YI

m∈Sg

fm ≤ sg (∀g).NTCuP

AMAAQNdhQL?IJY.iANOcdY.PKPUY.i@[email protected]

s≥t

ums +∑

s≤t

uns ≤ 1 (∀m)(∀n ∈ UUm)(∀t).

NTCoKP

××Ù"[U\]$^^

Page 25: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

t.t " "

]#fdfdPUY.i?K[S.OPJ<?AgiL?@slAW#Y[@KMhQWUYMs?YOhOIs:AidyBANc?Y.PUAfIe@L?PK[idAYML?IJSI#slYMLdi:fOAf

(∀t)∑

m

umt ≤ K.NTC9mP

¯i?K[SPJ<dAhK[c?K.hQPTShQY.i?ITPUWUKMiP=WJAKMfdI

(∀t)∑

m

wmt umt ≤ CUt .

NeCD=P

: ¶ ¶ 6 S?|CO~ ² ·ª ¶ 6 |~'7 6 | ;=<dAgcdK[iP#@BPKPUY.iI=#WUPJPJAi^K.I

(∀t)∑

i

witpit +∑

m

wmt umt ≤ Ct,NeC MP

KMi?fPJ<dAvPUAh<didY.Y.yMhKMlANdhQL?IeYMiWUAKMf?I

(∀m)(∀i ∈ Im) UmT + PiT ≤ 1.NTC>P

8 9 < | ² ~ WdOD7 ¶ 6 | ¶)F C ² ~| ¶ · ² *]rIK[i¡ANPUAi?IJYMi¡Y[PJ<dAhQLdWUWJAiPJSL?IJAfpcdWUYMyMWK[@I@BYOfOA NÇoKPE1NTCD>KPEWJAcdWUAIJAiPUIEcdWUYMclAWUS&PJ<dAc?WJY.sdA@ PK.IIJAQPLdcYMWPJYEAIePUKMsdIJ<gAK.IesdA«ANc?Y.PKPJYMi"IeA .L?Ai?hAIPJYRY.sOPUKMi"<dyM<WUA.AiLdAIY.W=PJ<dAhYM@Bc?KMiS.

¯?WJY.@ PU<dAIePJYOh<?KMIePJhBcdWUYMy.WUKM@x@BidyPJ<dAYMWUAQPJhKMclYMi.P"Y[=AgPJ<?idyIKMWJAidYMPPJ<:KPy.YYOf]P?WITP#PJ<dAp@BYOfOAEIidYMPK / $(&)/K xrslAhK[L?IJA^PJ<?A¤@KPUWJN¾@LdPJcdSidyªPJ<dAIJAhYMi?f^IePUKMyMAfOAhQIeYMi^KMWJK[sdAIRIrITPUYOh<?KMIePJh[AIJfOAI:fOA?ididyWK[i:fOYM@ËhQYAhAi.PI#Y.WRAK.h<IePJWUL?hQPJLdWUA"KMi?fclAWUYOf¤fOYAIEidY[PgIJAA@ i:KPJL?WUKMY.WvKIePJYOh<?KMIePJhc?WJY.yMWK[@gari&PJ<dAxY[PU<dAWv<?KMi?fIJL?h<pfOIePJWUsdLOPJYMi:IEKMWJA"MAWJS^fO hQLdPvPJYsdLdf&cdWUYMclAWUS.s:AhK[L:IeAxY[N0uPrK[i?fPJ<dABWUAKPUY.i?Ie<?c?IPU<?KPv<?K9.AgPJYAQNOIePRipKyMAYM@BAQPUWJhKM=K9SON0hQY.IJA"IePJWUL?hÁPULdWUAIr<?K9MAWUAKPUAf^AhQY.idYM@BhK[K[LdAI$PÁKMi?f^i¤KPJA@xclYMWK[YMi?A N§PU<dABhQYMc?c:AWrcdWUhQABh<?K[idy.AI:Pgh<?K[idy.AIRYMWrPJ<dAxidYMPvIJYMf^sdYOhIEK[i:[email protected]$PÁ

]s:APePUAWK[cdc?WJYKMh< N¨WUYM@ PJ<?APU<dAY.WJAPJhK[IefOAKP"AK.ITPPhK[i©slAfdYMidAPU<dAi?AQNPBIUh<dA@BKIhYMi?IJfOAWUAf

CM#;=<dAvLdIJA LdAi:hQAvY[AQNOcdYMPUK[PJYMiIdNOAftO#]PJ<?AgWUKMi?fOYM@9KMWJK[s?AIKMWJAvWUAKM¦Afud_YM@BAhYMWUWJAhÁPJMAEKMhQPJYMi?I#KMWJAvclAWJYMWU@xAf

ÐÏ×ШØ

Page 26: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ tMu

]Ri"iPJAWUAIePJidyrYMs?IJAWUKPJYMigIPJ<?K[PPU<dA=YMidSvc:YIJIJs?A«iO°AK.IesdPTSK[WUIeidy#<dAiPU<dA«WK[i:fOYM@KMWJK[sdAIRK[WUAg?i?KMSidY#i^I#PUYANOhAAfPJ<dAxhK[c:KMhQPTSM#RY«AMAWdPU<dI@KPU<dA@KPUhK[cdWUYMsdA@I-^Y.WrPU<dAxsdL?IJidAIJIslAhKML?IJA"Pg@xAK[i?IRPJ<:KPvPJ<dAB@BidA"IEWJh<dAWRPU<?K[ipPv=KMIEAQNOclAhÁPUAf NPU<dAsdYh«Ay.<.PIBK[WUAsdyMy.AWxPJ<:K[i YMidAIBKMPUAfYMW§PBIBslAhKML?IJAPU<[email protected]¡s:Ac?WJYOhQAIJIJAf)PÁ;=<dAIJ@BcdhQPTS^Y[PU<dAB@xYOfOAyMMAIEIJYM@BA<dY.c:ABYM§cdWUYidyc?WJY.c:AWePUAIEK.I $ &+ !+ "YM

PU<dAWUAIJYMLdWhQA.K[i?f¤PU<dAiYMipPU<dAPJWUAK[PUKMsdPTSY[=PJ<?AcdWJY.sdA@;=<dIgPUWJAKPUKMsdPTSIgWJAKWU@xAfsSBPU<dAEÇKMhÁP=PU<?KP=PJ<?AEWUAIJLdPJidyx@xYOfOAI#KPTY[àITPK[yMAI§IePJYOh<?K.ITPUhrcdWUYMyMWK[@ N0K[i?fidY[PK

TIePUKMyMAI

Y.idAPQariPJ<dAY[PU<dAW<?K[i:f9PJ<dAWJA=IJAA@5PUYrslA§PU<dWUAAc:YIJIJs?PUAIKMs:Y.LOPPJ<dA=IeAhQY.i?fIePUK[y.A$ IfdAhQIJY.i?I

;YB<?K9.AgKxhYM@BcdAQPUAvIJAQPYMidA6KMWJK[sdAIAPL?I#IUK9S p2it

r2it

KMi?fu2

mt

OPJYhQY.iO?WU@OWJAeZTAhÁPYMW=KMfdfidA£clY.IUIJsdARANPJWKMhÁPUY.i?I àiPJ<dI=hKMIJA#PU<dAWUAEIJ<dY.Ldfs:[email protected]<dAriLd@"s:AW«YMh<?K[i?yMAI NPJYK9MY.fKBhQYM@BcdAQPUAvLdidYMWU@BK[PJYMiWJAsdLdfdidyBYMPU<dAgcdK[i)PQ

]Y#idyBYMi?SWJAyMWUAQPePUi?y?WIePeàITPK[yMAEAQNOcdYMPUK[PJYMifOAhIJYMi?IOiY.WUfOAWPJYBdNPU<dAclY.IUIesdPTS¤Y[=Ldi?IUKPUIe?AfphKMc?KMhPTS¤hQYMi:ITPUWUKMiPUIB;=<dI<?KMIEPU<dAKMfOKMi.PK[y.A"YMclAWU@BPJPJidy^c?K[WJPJK[fOAhWJAKMIJi?yY[PU<dAEAQNOcdYMPUKPUY.iY[@BK.hQWUYMsdYOhOI[#<?h<@xAK[i?IKhYMiPJiLdYML:I«IJAhYMi?fàIePUK[y.AcdWUYMyMWK[@OK[i:fPU<L:I?KBIJ@BcdAW#c?WJY.yMWK[@

] 0!+. !Y[PJ<dAE?WITPPTYYMcdPJYMi?IPJYBK[Y£IJYM@BARi?fY[Y.cOPJYMi?I«Y.W«PU<dAEYMclAic?Pc?K[WJP[KMi?fgYMPJ<dAWUIYMWPJ<[email protected]?IePUK[i:hQA§igPU<dA«c?<SOIJhK[.i?KPULdWJAY[PJ<dAc?WJY.sdA@EPIJAA@IgfO hQLdPgPJYi?hQWUAK.IeAxPU<dAIe¦AYM§PJ<?AcdP#PJ<?YMLOPiPJAWIJAhÁPUi?yPJ<dAgL?i?fOAWUyMWUYML?i?f@BidAM

]Ri¤@Bc:Y.WePK[iPEc?WJY.sdA@ #PU<&PU<dIv@xYOfOAXI<dYA.AW?PU<?KPgPvIEidY[Pg.AWUSK.hhLdWUK[PJAPJYPJ<dA=K9S&@Bididy¤IfdYMidA.Xs:AhK[L:IeAPJ<dAWJAI"idY[PBK=K9SPJY¤idY®K[§PU<dAWK[i?fdYM@ AA@BAiPUI"i¢YMidAIJ<dYMP©;=<dAiPJ<?AIJYMLOPJYMi¢YMLdi?fªIe<?YMLdf¢slAL?i?fOAWIePJYYOf¢idYMPx.AK&?i?KM«cdK[iªsdLOPKMI"yMi?yIJYM@BA¤yMLdfdK[i:hQA¡K[i?f [email protected]:Y.LOPWUYMs?L?ITPhYMi?IJAWUKPJMA^clYMhQAIPJY s:A©K[c?cdAf£PJYªPJ<dAANOcdY.PKPUY.iYMPJ<?Ag@xidA.

`«L?WJWUAiPR@BYOfOAI#Y.Wr@BididyIUh<dAfOL?idyKMWJAgy.AidAWUKM¦AfL?IJidyITPUYh<:KMIePJhgcdWUYMyMWK[@B@BidyxPJYIJAQPL?c&idAj@xYOfOAIr#PJ<PU<dABKMfOK[iPK[yMAY[§hQY.i?IJfOAWJidyPU<dAxc:YIJIJs?A"[email protected][slYMWKPUAWUYMs?L?ITPIUh<dAfdLdAIPU<dI=Id@BYMWUA :AQNOs?AIJYMLOPJYMi:I=@xi?fOidyfO !lAWJAiPc:YIJIJs?AgIUhQAi:K[WUYI§Y.W=PJ<dALdPJLdWUAMX;=<dArcdWJAAQNOITPUidyg@BYOfOAIK[WUARK[IJYgyMAi?AWK[¦AfxiPJ<dAEIeAi?IeAPJ<?K[P§PJ<?ARidA @xYOfOAI«hKMis:AKMcdcdAfPUYKx@xidAhYM@s?ididyBLdi?fdAWUyMWUYMLdi?fK[i?fYMclAicdP#PJAh<didY.Y.yMAIYMcOPU@KMSM

ari^PU<dAxY[PJ<?AWR<:K[i?fPJ<dAWJAxK[WUAIeY.@BAfO hLdPJAIEL?IJidyITPUYOh<?KMIePJhgcdWUYMy.WUKM@x@BidyBYMWE@Bi?idyIUh<dAfOLdidy: àiBPU<dA=?WITP@BYOfOA°MP«fOYAIidY[P§IJAA@ PJYgslAAKMIJSKP§KMPUYIeAP§KMhhLdWKPJA#fOIePJWUsdLdPJYMi?IY.WPJ<dAEWUKMi?fOY.@®AA@BAiPUIiPU<dAvcdWJY.sdA@X;=<dAEIJAhYMi?f@BYfdA3ILdPJAgIJ@BcdAEKMi?fIJL?hhQAIUIeLdSK9.YMfdIRPJ<dABcdWUYMsdA@ ZTL:ITPg@BAiPJYMi?Afs?LOPgPJ<dAIJYMLOPJYMi¤YMsOPK[idAf&idAAfdIEPJYslAITPUL?fOAfpfOAAc:AW

Ø»ÇÆ°¿ß¹Ã»ÇÆ°º¼ßXÅÀ0¿ ÁÀ0à ª¹9ûße¿ #ÅÔ¼ÈTÆ0ÈÀÇÈT»°¿ÁÄ9ÀÇßTÈXº ¿ÁÀ3ÈQÈTÀÇÌRÅú¼Àx¿9À0»ÇÆ»0Æ°ÃÁÈÉ9ÈTßTº¼»°º¼¿ÁÙ9»ÃÙ9ÉÈQÈTÀÇÌRÀÇÈeÃÔ¼ºeÈTÉ

ωÖÆ°¹ÈeÀÇȧº¼»Ã QÈTßTÆ0¿ÁÀ

y(ω) ¿dßT¿ÁÀÇÀ0ÈTßàÆ°º QÈÃßàÆ°º¼¿ÁÙ9»Æ0¹ÃUÆ ÃÂÁÈT»Æ0¹9È»°¿ÁÔ¼ÄÆ0º¼¿ÁÙ (x, y(ω)) Èeû°º¼Ê9Ô¼ÈÁã

××Ù"[U\]$^^

Page 27: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

t[o " "

slAhK[L?IJAxPJ<dAWK[i:fOYM@ AA@xAiPUIvi¤PU<dA@BYOfOAKMWJABWUAK[¦Af&i¡Y.idAIe<dYMP#<?h<¡IvidY[PvPJ<?AK9SPU<didy.I<?KMcdc:AiiPU<dArWUAKM:YMWUfXaRhQYML?WUIJAMK[iSBY[PU<dAEPT«Y"@BYfdAI§cdWUAIJAiPUAf#li?AAf@[email protected]?K[A"!lY.WePIPJ<?KMiPU<dAhQLdWUWUAiP#YMidAIPJYBs:AIJYMMAf;Y?i?IJ<©PJ<?IxIeAhÁPUY.i¢iªK@BYMWUAYMcdPJ@xIePJhÇKMIJ<dYMiPxI"@BclYMWJPUK[iPPUY&@BAiPUY.iIeAMAWUKM

) &)/".rPU<?KPhKMis:AgANPUWUK.hÁPUAfWUYM@PJ<dAgfOAKMI=cdWUAIJAiPJAf<dAWJA.

KM3PU<dAIePJYOh<?KMIePJhrAA@BAiPUI#KMWJAEYM@BPePUAfOPU<dAvWJAIeL?P#I#K"@BYOfOAK[Y#idy"APJ<dAW#Ldi?fOAWeyMWUYMLdi:fK[i?fYMclAicdPEPJAh<?idYMYMy.AI=PJYhQYAQNOITPEYMcOPU@K[SJNWJY.@PJ<dAB¬RVX­$clYMi.PvY[AgKPAK.ITPPÁ

?PU<dA#YMclAi"cdPXc?KMWePXI* $MPJ<dA=WUAIJLdPXYMLdfslA#ITPUYOh<?KMIePJh§c?WJY.yMWK[@IY.WL?i?fOAWUyMWUYML?i?f@Bididy:]RIvA0K[ipYMclAi¡cdPITPUYh<:KMIePJhcdWUYMy.WUKM@ËIgY.sOPUKMi?Af¤#<dAi¤PJ<dAL?i?fOAWUyMWUYML?i?fc?K[WJP#[email protected]

`«YMiPJiLdYML:I9KMWJK[s?AIfdAIUhQWUsdidyEsdYh IAQNPJWKMhQPJYMixK¨PJAWXWJA°s?YOhi?yrIidAg AQY.WJAPJ<?I«Y.WJ3lWJA°s?YOhi?y<?KMIrslAAiZTL?IePrPUYy.WJY.Ldc^sdYhOIriPUYsdyMy.AWrYMidAIrKMi?fPJ<?Ai .-KrPJ<dAYMWUy.i:K[OsdYOhI[iPJWUYfdL?hQidygK[[email protected][3cdWUAhIJY.i"@KMidyEPJ<?A=PJWUAKPU@BAiPY[lPJ<dAhQY.K[c?IJidyK[idy.AIfd hQL?Pd#<dh<I#K9MY.fdAf#PJ<PU<dAhQY.i.PUiLdY.L?I#K[c?cdWJYKMh<

F#YMs?L?ITPUidAIUIri@BididyIUh<dAfdLdidyIEK[IeYKidA hYMi?hAcOP:#<dh<&KMcdc:AK[WIKMIEK[i^@BcdhPAK[PJLdWUAgY[PJ<dAIePJYOh<?K.ITPUhEcdWUYMy.WUKM@x@Bidy"IJYMLOPUY.i?I

! 2 % ÁG*

àixPU<dI§IeAhÁPJYMi.KvidAªi?fxY[3@xYOfOA?IXAQNOclY.IJAfM@xYMPJKPJAfxiKvfOAAclAWWUAIeYMixY[lPJ<dArhQLdWUWJAiP@BYOfOAI=iYMWfOAWPJYWUAIUhQL?ARPU<[email protected]@xAiPUI=WUYM@®PU<dAgWUYMsdL?IePJi?AIUI«clYMiPY[Ag]?WITPg@Bc:Y.WePK[iP LdAITPUY.i¤KMWJIeidyI ,/ "!UU*1 $"&! "U ,/

. $H" ,!' "!U 1 U( !K- K 1 0 ]#hÁPJL:K[S.PJ<dAWJAxAQNOITPIEPJ<dAclY.IUIJsdPTS¡PUY¡IUK9MAONÇKMi?fi?Y[PIeAP"AQNPJWKMhQPJAf¢@Bi?AWK[=i YMWfOAWPUYpi ?L?Ai?hAPU<dAcdWUhAMª;=<?I@BAK[i?IPJ<?K[P«IeY.@BARIeAidyclYMhAIAQNOITPKMi?fxPJ<?AiPJ<?A@BYM@BAiP«#<?AiBPJ<?ARYMWUA#I§ANPUWUK.hÁPUAfBfOYAIi?Y[PhQY.i?hfdAg#PU<PU<dAgi?ITPK[iP#Y[PIIJKMA.;=<dAIJAhYMi?fª[email protected][WIBiPU<dA^YMsOZTAhÁPU.AL?i?hÁPUY.i;=<dA^fOIJhYMLdiPWUK[PJA.¾;=<?I

AA@BAiPriPJWUYfdL?hQAI -/$$^iPJ<?A"ANc?Y.PKPJYMi&cdKMi?IlWUAfdL?hQidy@BidA$ IrAMKMi?f^PU<?KPgfOYAIridY[PIJYML?i?fpMABWUYMsdL?IePJi?AIUIgKPK[°]#hÁPUL?K[S.K[PgPJ<dIgclYMiPYMidAWUAKM¦AIEPJ<?K[PgPJ<dAhAiPJWK[c?KMididAWc?WJYMWUPTSpIidYMPxi?AhQAIJIUK[WUS&PJ<dAAhYMidY.@xhKM§K[LdA.©_OLdWJAS¡PxIxK[i¢@Bc:Y.WePK[iP"IJIJLdA.sdLOPxPJ<dAIJYOhQK[?KMi?fxc:Y.PJhKM:hY.IePUIXYMhYIeidyKv@BidARK[WUA#<LdyMA..IJY?K[y.KMiMyMWUAAfOSIUh<dAfdLdAIXWUAfOL?hi?ygPJ<dAAgY[PU<dAg@Bi?AIeAA@L?i?fOAIJWJAf;=<dAidAK[c?cdWJYKMh<¢hYMi?IJfdAWIPU<dA?WUIePBc:[email protected]?ypIJYM@BA@Bid@Ld@ WUA LdWUA@BAiPUI

N0fOA@KMi?fdI$PYMYMWUAEK[PAKMh<clAWUYfslAhK[L?IJArPJ<?K[P#K9.YMfdIXPU<dA ""! " 0-K" ÇK[IeAc?WJi?hQcdAMdKMY#i?yPJ<?AEc?KMididAW=PUY"iPJWUYOfOL?hQAgIePJYOhK[i:fIJAidyxc:Y.hQAI=@BcdhPUS#<?AidNOidyIJL?h<@Bid@L?@AQNOyMAi?hQAI;=<dAIJAhYMi?fclYMiPIhK[WUWJAfsSPJ<?AYMsOZTAhÁPU.ArL?i?hÁPUY.i?#<dh<iPU<dIRK[cdcdWUY.K.h<I#idYMP#PJ<dA

¬rVX­ Y[XPU<dAxIJA LdAi?hQA.3sdLOPEPJ<dABK[@BYML?i.PvY[XPU<dAxWJAIefOL?KMY.WJA"i^PU<dAB@BidAx#<dAi^PJ<dAx<dY.WJ¦Y.iT

ÐÏ×ШØ

Page 28: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ t.m

I#KMh<?AMAfPU<dAgYMi?AE#<?h<<?KMI=i?Y[PslAAiANc?Y.PUAf;=<L?IPJ<dAv:i?K[cdWUYMs?A@ I#PJYIJK[PJITSPJ<dAfdA@K[i?f?I#PJ< e@Bid@L?@ hY.IeP i¡PU<dAIJAi:IeAPU<?KPBKMIg@"L?h<K.Igc:YIJIJsdAYMWUAI"IJK9.Af¤YMWPJ<dALdPJLdWUAM àiÇKMhQPPU<dAgYMWUAvI#IUK9MAf !H OOPJ<L?IPU<dAWUAgI=idYITPUYhidyhQYITP=PJYBslAgc?K[f

Ry.<xWUA.AiLdAIKMWJA#IJLdslYMWfOi:KPJAfPUYvPJ<dAIUKPUIeÇKMhQPJYMixY[:PJ<dARfdA@K[i?f?IXKPAK.h<"clAWUYf"i?ITPUAK.fYMO@BK[NO@B¦idy#PJ<?A@ fOWUAhQPJS1NÇYMWhQA§MAWIUKdlPJY<:K9MAPJ<?Ac:[email protected]<dA§c?KMididAWY.idSidAAfdIPJYBi?hQWUAK.IeArPJ<?AfOA@K[i:fdI PQ àiPU<dIIeAi?IJAMcdWUYMsdA@ IAKMIJsdPTS#fOAc:Ai?fYMiPJ<dAc?KMididAW IRy.WJAAf#PU<dAx<dy.<dAWvfOA@[email protected]<dAx@BYMWUA"fO hQLdPvPJYIJK[PJITSPJ<?A@IeclAhKMSPJ<dAg@BidAfOYAI=idYMP#WJAIeLdP#PJYBslAKMI=WUh<KMI=ANOc:AhÁPJAf

AhK[L:IeAYM#PJ<dAi?Y[P"ANOcdhQP"c?WJAIeAi?hQAYMR¬RVX­ i©PU<dA@BYOfOA0idYMPAhYMidY.@xhKM§K[LdAIgYMWPJ<dAIePJWUL?hQPJLdWUAIEK[WUAidAAfOAf ài?ITPUAK.fY[XPU<?KPAPEL:IEhYMi?IJfOAWEhQYAhQAiPI

vi

WUAcdWUAIJAiPJidyPU<dA&)//hYMiPUKMi?AfsS

i

]rIigPU<dAhKMIJAY[?¬rVX­@BYOfOAIMPIc:YIJIJsdA§PJYr#WJPJAK#PTY[àITPK[yMAI@BYOfOAYMWKTàIePUK[y.AI3Y.idAM

¬RA.AWJPJ<dAAIJIMfOA@BKMi?fBhQY.i?ITPUWUKMiPI@BKMMAPJ<?A#cdWJY.sdA@ YM3IJAhYMi?fBIePUKMyMAIL?i?hQAWePK[i.PTS .L?PUA#@xYITPfd hQL?PEPJYs:ABIJYMMAf:A@Bi?K[PJidyPU<dAxyMYYOfPJ<dAYMWUAQPUhK[cdWJY.c:AWePUAIRANOc:AhÁPJAf^Y.WEKITPUYh<:KMIePJhKMcdcdWUY.K.h<¯?YMW=PU<?KP#WUAKMIJYMiY.idSKBfOAQPUAWU@BidIePJhr@BYOfOAIcdWJAIeAiPJAf

< 3 3 d~ 7Çd * ² |;=<?Ag9KMWJK[s?AIK[WUAEPJ<dAL:IeL?KM0

rit ∈ 0, 1WJAcdWJAIeAiPJidyBPJ<?AsdYh

iI#WUAK.fOSPUYs:AANOcdY.PUAf

Y.WidY[P§K[PPU@BAtpit ∈ [0, 1]

slAidyvPJ<?AclAWhQAiPUK[y.A«ANOcdY.PUAf"WUYM@jPU<dIs?YOhK[PXPJ<dAt°PJ<Bc:AWJYOf

KMi?fumt ∈ 0, 1

IEPU<dAfOAhIJYMi&YM«ANc?Y.PUidy NÇYMWvidY[PPrPJ<[email protected][PEPU<dAi?ITPK[iP

t

]rI=«A°OPU<dAITPK[WJPJidyxc:Y.iPUI9KMWJK[s?AIfm

IePUK9SL?i?h<?K[i?yMAf

< 3 9 QO¡ 6 · GO~ ² ² ~|¯?YMWPJ<?AvhA

iOAQP#slAQP

vi

K[i:fwi

PJ<dAghYAhQAiPUIWUAcdWUAIJAiPJidyPJ<dAvcdLdWUAvYMWUArKMi?f=KMIePJAEK[@BYML?i.Pc?WJAIeAi.PEKPPU<dIrsdYOhl3K[i:f

vm

PU<dA"hYMiPJAi.PrY[XYMWUAi@KMhQWUYMs?YOhmR]PJ<dAIeA.L:K[iPJPJAIrK[WUA

@BAKMIJLdWJAfiPJYMi:I]rI#i@BYOfOANÇoKPà$NeCD>PQdAQP#s:A

Im

PU<dAIJAQP#Y[sdYhOI=iPJAWIJAhÁPUi?yPU<dAm°PJ<@KMhWJY.sdYh3

;=<dAxidAy.<slYMWU<dYYOfdI ?fOAQ:idPUY.i?IvK[WUAK[IJYcdWUAIJAWUMAfNi

KMi?fNm

KMIRPJ<?A"IJAQPUIrY[i?AyM<s:Y.WUIYMsdYOh

[email protected]

mWJAIeclAhQPJMAS.

AMAAQNdhQL?IJY.i&IrAITPK[sdIe<dAfKPU<dAIeAPUU

m

Y[@BK.hQWUYMsdYOhOIYMAWrPK[i?fPJ<dAIJAQPUP

m

YMs?YOhOIYOhKPUAfBi?IJfOArPI§hYMidArY[i ?LdAi:hQAMOK.IA:KMIPJ<dAWJArI«KvPJY.AWUKMi?hQARY[

δYMWKMf9ZeK.hQAiP§c?P

hYMLd@Bi?I]rIK[slYMAE@BAiPJYMidAfdiPJ<dI=@BYfdAPU<dAWUAgK[WUAvfdA@K[i?f?I#K[i?fhKMc?KMhPUAI«YMW=PJ@BA

t

Dt

;=<?AfOA@K[i:fiYMWUArPUYMi?I#K[P#PJ@BAt

CPt

;=<dAgANPUWUK.hÁPUY.ihKMc?KMhPTSBY.W=PJ<dAgY.c:AicdP@BidAKP=PU@BAt

CUt

«`«YMWUWUAIJc:Y.i?fOidyxhYAhQAiP=YMW#Ldi:fOAWUyMWUYMLdi:f@Bi?AM

××Ù"[U\]$^^

Page 29: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

t= " "

;=<dAgyMYMs:K[3cdK[iPRhKMc?KMhPTS Ct

¯i?K[S.KMIslAQY.WJA.

i−fOAidY[PUAIRPJ<dAxhQA@B@BAfOKPUASYMAW

iKMi?f

KslYML?i?fdI#PU<dAxiL?@slAWRYM

K.hÁ[email protected]

< 3 @ ² MD7W ² F 6 MD7 ¶ 6;=<?AEY.sOZTAhQPJMAELdi?hQPJYMiI=PJ<dAgWUA@KMididyxYMWUA0N¨PJ<?AgidY[PANc?Y.PUAfYMidA PÁO#<dh<^hK[iWUAKMfKMI

S =∑

i

vi +∑

m

vm −

[∑

i,t

vi pit +∑

m,t

vm umt

]NeCD?P

#<?AWUABPJ<dA:WUIePPJAWJ@ IvPJ<dAPJYMPUK[§YMWUAhQYMiPK[idAfpi¡PJ<dA@BidA N0K^hQY.i?ITPK[iPiLd@slAWPÁXKMi?fpPJ<dAIJAhYMi?fIPJ<dAK.fdfOPJYMiY[PJ<dAgANPUWUK.hÁPUAfKxYMclAicdPrK[i?fLdi?fdAWUyMWUYMLdi?f@Bididy:dWJAIeclAhQPJMAS.;=<dI#I=idY[P#PU<dAgYMi?Sh<?YMhQAM§aRPU<dAWI=YMcOPUY.i?IYMW=PU<dAgYMsdZTAhÁPU.ARL?i?hÁPUY.iKMWJA. ;Yvy..A«IJYM@BA«c?WJAAWUAi?hA§PUYrPU<dA /$ &+ " $ " QPU<?KPXI[PJ<dA=YMi?A§Y.WX#<dh<PJ<dAWJAIKMWJWK[iPJAfKMhhQAIJIvKP

TsdLOPP<?KMIvi?Y[Pgs:AAi¡AQNOcdYMPJAf&S.AQP " "rPJ<dI L?K[iPUPTS

hKMislAg#WJPePUAiidAK[WUSiPJ<?AhQLdWUWJAiPK[WUK[sdAI

;=<dA !#"0 &+ " $ !+ "" K "/=PJYYMWhQAPU<dA&@BYOfOARPJY¢@Bi?f¾PU<?KPK[i idYMPMAWJSYMWJPJLdi:KPJASAQNOcdYMPUK[PJYMiY[XK"@K.hQWUYMsdYOhhK[iAK9.ArsdyclYMWJPJYMi?I=YMPU<dAg@BidAgL?i?K[sdAvPJYs:A#ANOcdY.PUAf;YgfOYrPJ<:KPMPJ<dAPUWJhgIPUYghQYMi:IefOAWY.idAK.fdfOPJYMi?KMc:AWJYOfYM?PJ@BAM.#PU<dYMLOPhKMc?KMhPTShYMi?IePJWK[iPUIOK[i?f@KNO@B¦AEPJ<?AgYMWUAEANOcdY.PUAfPJ<dAi

< 3 E : ¶ 6 |~9U7 6 |]rI#fOYMi?Ags:AYMWUAMPJ<?ASKMWJAvIJAc?KMWUK[PJAfi * ² ² 6 O~ ³ ¶ 6 |~'7 6 |YhOI#[email protected]:AgANc?Y.PUAfYMidSYMi:hQAM

(∀i)∑

t

pit ≤ 1N0tMnP

(∀m)∑

t

umt ≤ 1.N0tdCP

,G ² 6 G 7¨ ² XG* ¶ 7O7 ¶ 6 ¶ 6 |~'7 6 |VXWUAhAfOAi?hQAhYMi?IePJWK[iPUI=K[WUAvIeAPRKMIYMYI¯WITPYMKM°?KxsdYh@L?IePs:AgANOcdY.PK[sdAM

pit ≤ rit (∀i)(∀t),N0t.tP

ÐÏ×ШØ

Page 30: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ tKM

s?LOPOPUYBs:AgANc?Y.PK[sdAMPU<dAhQAWUyM<P#YMAWPUIJA@"L?IeP=slAgAQNPUWUK.hÁPJAfhYM@BcdAQPUASs:AYMWUAM

rit ≤∑

s≤t

pi−s (∀i)(∀t).N0tMuP

]#f9ZeK.hQAi.PRhAI=ANOcdY.PKPUY.i<?K9.ARPUYs:AIJ@xK[W

(∀i)(∀t)(∀j ∈ Ni)∑

s≤t

pis − pjs ≤ δ,N0t[oKP

PU<dAhKMc?KMhPTSslYML?i?f<?K9MArPJYslAIJK[PJIT:Af

(∀t)∑

i

(vi + wi)pit ≤ CBt ,

N°tMmP

KMi?f?i?K[SMOPU<dAgidAQNPhQY.i?IePJWK[i.P#@Bc:YIeAI=AMAANdhQL?IeYMiYMW#Y.c:Aic?P@BAQPU<dYOf∑

s≥t

pis +∑

s≤t

ums ≤ 1 (∀m)(∀s ∈ UPm)(∀t).

N0t=P

6 · ² ~ I ~ ¶ 6 · ² XG * ¶ 7¨OD7 ¶ 6 ¶ 6 |~'7 6 |\^K.hQWUYMsdYOhOIhK[islAAQNOcdYMPJAfPJ<?ASKMWJAIePUKMWePUi?yxclYMiPUI#Y.WRKPAK.ITPY.idAgY[XPUI#i?AyM<s:Y.WUI

<:KMI=slAAiANc?Y.PUAfs:AYMWUAM

umt ≤ fm +∑

n∈Nm

s<t

uns (∀m)(∀t).N0tKMP

;=<dAgiLd@"s:AW#Y[XITPK[WJPJidyxc:Y.iPUII=slYMLdi:fOAfiAKMh<IeAhÁPUYMW#KMIY.YI

m∈Sg

fm ≤ sg (∀g).N0t>P

AMAAQNdhQL?IJY.iANOcdY.PKPUY.i@[email protected]

s≥t

ums +∑

s≤t

uns ≤ 1 (∀m)(∀n ∈ UUm)(∀t).

N0t?P

;=<dAgiLd@"s:AW#Y[@KMhQWUYMs?YOhOI§[email protected]?IJSI=slYMLdi?fdAf

(∀t)∑

m

umt ≤ K.NÇu.nP

¯i?K[SPJ<dAhK[c?K.hQPTShQY.i?ITPUWUKMiP=WJAKMfdI

(∀t)∑

m

wmtumt ≤ CUt .

N0udCP

××Ù"[U\]$^^

Page 31: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

t> " "

: ¶ ¶ 6 S?|CO~ ² ·ª ¶ 6 |~'7 6 |;=<dAvPUAh<di?YMYMyMhKM3AQNdhQL?IJY.iI

(∀m)(∀i ∈ Im)∑

t

umt + pit ≤ 1,NÇutP

IJ@xK[WUSBPUYBPJ<dAgY.WJyMi?K[lY.WJ@"LdK[PJYMiPU<dAgcdK[iPRhKMc?KMhPTSWUAIePJWUhQPJYMiWJAKMfdI

(∀t)∑

i

vi pit +∑

m

vm umt ≤ Ct,N0uMuP

KMi?fOYMhYMLdWIeA.PJ<dAfOA@K[i?fhQY.i?IePJWK[i.P

(∀t)∑

i

vi pit +∑

m

vm umt ≥ Dt,N0u[oKP

;=<dAhK[c?K.hQPTSK[cdc?WJYKMh<I#@BY.WJAgWUAKMITPUhviPJ<?AIeAi?IJAvPJ<?K[PPvfOYAI#idYMPRKMIUIeL?@xAvPU<?KPrYMWUAIgIJYMfpKPvPJ<dAIUK[@BAxPJ@BAPgIvAQNOcdYMPJAfx;=<dIKMYIPU<dAhQAiPJWK[cdK[ididAWEPJYAIePUKMsdIJ<IeY.@BAvIJAidyBclYMhAIK[i:fPUYBK.fdK[cdP#PJ<dAgANc?Y.PKPJYMiPJYBPU<[email protected]#hQAv.AWIJK?

;=<dAWUAIrKhQYM@BcdWUYM@BIeA#PU<^PJ<dA"AYMXPJ<dA"@BidA.:slAhK[L?IJALdPJLdWUA NÇ#<?KPE<?K[cdclAi:IRK¨PUAWPJ<dA&<dYMWU¦YMi

TPxIhYMi?IJfOAWUAfªi PU<dA&fOAhQIeYMi cdWUYhAIUI ¯?YMWPU<dIWJAKMIJYMi§PJ<dIi:fªYM

@[email protected]?IePPU<?K[iPJ<?A¬RVX­ Y.WJAiPUAfYMidAI

;=<dAK[@¥Y[«PJ<dAB@BYOfOAXIrPJYcdWUYfOABIJYM@BA"@Bid@Ld@¥=K[WUWK[iPJAI1N¨PUYIUKPUIeSfOA@BKMi?fdI$PÁ#<dh<JN0K[yK[i)P«I#KxclYMi.PiÇK9MY.WYMPU<dAgWJY.sdL?IePJidAIJI=YMPU<dAIJAg@BYfdAI

VXK[iP I§IJ¦ARK[i:f"PUI§WJAK[PJYMi?IJ<dcB#PU<BPJ<dAr@xidAPUIJAK[cdclAKMWUIXhAK[WUAWifOA@BKMi?fdIPUAWU@I

PgIEclY.IUIesdA NYMWEi?IePUKMi?hQA PRPJY $ U PJ<?Ax@BidAMiPU<dAIeAi?IJAY[ 1 $"! - .PJ<dA<dy.<dAW=PU<dAfOA@K[i?fdIOPU<dA@BYMWUAgfO hLdPRPJYIUKPJIeSPJ<?A@:IJYidAKMIJsdPTShKMi^KMhhL?IJAPJ<dAg@K[N@L?@AQNOcdYMPUK[PJYMiWUK[PJAvY[PJ<dAg@BidAM

<dAgiPU<dA¬RV­j@xYOfOAIPU<dAghQYITPIKMi?fs:AidAQdPIK[WUAEAQNOcdhPOiPJ<dAfdA@K[i?f hKMc?KMhPTS@BYfdAI"KM§PJ<dAAhQY.idYM@[email protected]@BcdhPxi©PU<dAfdA@K[i?f?IK[i:f©hK[c:KMhQPJAIY[#AQNcdYMPUKPUY.i

" !( ! « % = G

ài PU<dIIeAhÁPUY.i¾K©IJ@xc?A^<dALdWUIePJhIfOA.AYMclAfªi Y.WUfdAWBYMWIeY.idypPJ<dA&fdA@K[i?f hKMc?KMhPTS@BYOfOA0;=<dI<dALdWUITPUh§PJWUAIPJYvIeY..A§PU<dA=i:ITPK[i?hAI LdhSgsSgWJAfOL?hidyEPU<dA#Ie¦AY[:PJ<dA=cdWUYMsdA@P<:KMIPTY"@KZTY.WITPUAc?I

ÐÏ×ШØ

Page 32: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ t?

;=<?AE@BYMPJ9K[PJYMiY[PJ<dI#c?KMWePhYM@BAIWUYM@®PU<dAgidANPCM_YMMA"PJ<?AxY.WJyMi?KMc?WJY.sdA@¥K[i?fphKM

z∗PIgIeY.LOPUY.i NPJ<dIgWUA LdWJAIRWUAQàK[WUWUKMidyMAPJ<dABK[WU

K[sdAI$PÁ¤¯[email protected]∗tPJ<dA $) " "^hQY.i.PK[ididyPJ<dAK[WUKMsdAIgYM#PJ<dA

t0PU<

c:AWJYOftO#]#fdfPJ<dAIJYMLOPUY.iK[WUK[sdAIYMAW

t?K[i?fhKM

zPU<dAgWJAIeL?PYMPJ<?I#Y.c:AWUK[PJYMi

z =∑

t

z∗t .

ud#¬Y hQYMi:IefOAWPU<dAIUK[@BA=cdWJY.sdA@ sdLdP#PU<xY.idSYMidA=clAWUYOfK[i?fxfOA@BKMi?fdIKMi?fxhK[c:KMhQPJAIIeAP#PJYBPU<dAgPJYMPUKMfOA@KMi?fK[i?fhKMc?KMhPUAI=iPU<dAgYMWUy.i:K[cdWUYMsdA@§_Y..AgPRKMi?fi?K[@BAgPUIIeY.LdPJYMi

z∗

AhK[L?IJAzI«AKMIJsdAEPUYPU<dAgIeAhQYMi:fcdWUYMsdA@OP=YMYI§PJ<?K[P

z∗WUAK.h<dAI«Kxs:APePJAW#YMsOZTAhÁPU.A

KMLdA.=PJ<?K[PI#AK9.AIB@BYMWUAidYMPAQNPJWKMhQPJAfªY.WJA^i¾PJ<?A&@Bi?AM ài?fOAAf#K©c?KMWePUPUY.i Y[vPJ<dAANOcdY.PUAf¤YMWUA"sS

z∗i.PUYc:AWJYOfdIgIJK[PJITSidyPJ<dAfOA@K[i?fdIgKMi?fphK[c:KMhQPJAIEIEYMLdi:fPU<dAipPJ<?I

c:K[WJPJPJYMi#slAMdidYMP#YMidSAK.IesdAvPJYxPJ<?Av?WUIeP#cdWUYMsdA@OsdLOPrYMcOPU@K[°]¨PJAWPJ<dIK[WUyML?@xAiP=I=cdWUAIJAiPJAfPJ<?Av?WUIeP#c?K[WJP#Y[PJ<dAg<?ALdWUIePJhEIidY[PLdPJAIJLdWJc?WJIeidy:

F#AfdL?hQAEPJ<?AgcdWJY.sdA@ PJYK"YMidA°clAWUYfi:ITPK[i?hAEsSKMyMy.WJAy.KPUi?yfOA@K[i:fdI#K[i?fhKMc?KMhPUAIIeY..AvP?K[i?fhKM

z∗PUIIJYMLOPJYMi

qRAAPJAPJ<dAidY[PxAQNOcdYMPJAf¡c:K[WJP"Y[=PU<dA@BidAMPU<?KPxPK[§sdYOhI"K[i?f¡@K.hQWUYMsdYOhIv#PJ<KMIUIeYOhK[PJAfK[LdAI=YM

z∗s:Ai?y¦AWUY?

ari:hQAXPU<dA?WIePc?KMWePIfdYMidA.PUIIJYMLOPJYMi

z∗fdAQPJAWJ@BidAIPJ<dA«c:Y.WePUY.iY[PU<dA§@BidAPUYRslA«hQY.i?IJfOAWJAf

iPJ<dI#IePJAcYMPU<dAK[yMY.WJPJ<d@;=<dI#@BAK[i:IPJ<?K[P

;=<dAIJ¦AEYMPU<dAgcdWUYMsdA@I=WUAfOL:hQAfsSfOAAPJidy !+ !) /$ !K-c?K[WJPUI=YMPU<dAg@BidA.

]jIeA LdAi?hAvY[AQNOcdYMPUKPUY.iIePJ@L?IePs:AvY.Ldi?f

¬YgPU<dAWUAgI=i?Y[PRIJAi?IJAvi@Bi?@B¦idyBPJ<dAvANPUWUK.hÁPUAfYMWUAM;=<dAK.ITPclYMiPyMMAIRKBK9SPUYWJAfOL?hA,NÇK[yK[i)P«PJ<?AIe¦AYMi?hYM@Bi?ycdWJY.sdA@YMANPJWKMhÁPUi?y

K9KMfIJA LdAi:hQA#WUYM@z∗PJYcdLdP«c?KMWeP§YMPJ<dAEhQY.i?ITPUWUKMiPIiPU<dARY.sOZTAhQPJMA#L?i?hÁPUY.i.clAi:K[¦idy

iOAKMIJsdPTS.]PAKMIePPTYBYMcOPUY.i?I#hKMislAhQY.i?IJfOAWJAfPUYfOYBPPJ<dAfdA@K[i?f?I"?hYMi?IePJWK[iPUI=K[i:fPU<dABhK[c:KMhQPTSYMidAIg;=<dAIeABh<dY.hAIRKMWJA.3ipIeY.@xABIJAi?IJAMhQY.@BcdA@BAiPUKMWJS3##<dAxcdWUAIJAWUidyPJ<dAhK[c?K.hQPTSWUAIePJWUhQPJYMi?I#<dA, &+/$PJ<?AAQNPJWKMhQPJYMiYMAW«PU<dAgclAWUYOfdI%NÇyMWK[iPJidyIJYM@BA !)ANOcdY.PKPUY.iK[@BYMLdiPIRKPEAK.h<YMidA PÁ:PJ<dABK[i:K[YMyMY.L?I#YOhhLdWIsSWJAK[NidyBPU<dAIJAxhYMi?IePJWK[iPUI?slAQhK[L?IJAPUYEIUKPUIeSgfOA@KMi?fdIIe<?YMLdfIJAc?KMWUK[PJA§ANOcdY.PKPUY.ii"c:AWJYOfdI[K9MYMfOidy<Ldy.A§AQNOcdYMPUK[PJYMi?I

××Ù"[U\]$^^

Page 33: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

u.n " "

KMi?fPJ<?Aid<?Acdi?yPU<dAghK[c:KMhQPTShQY.i?ITPUWUKMiPIPUY"slAv<dYMf«aRhYMLdWIeA.is:YMPJ<hKMIJAI§PJ<?AWUAEIidY[P=K[WUWK[iPTSBY[AKMIJs?PTS.?K[i:fPU<?KPI=#<SPJ<?I=I#YMi?SKx<dALdWUIePJh[¬RY[PJhQARPJ<?K[P§PU<dArhYM@BcdA@BAiPUKMWJidAIJI3ZTL:ITP@xAiPJYMidAfI§idY[PidAhAIUIJKMWJSK[fiPU<dARY.WJyMi?KM

c?WJY.sdA@ #<dAWUAxs:YMPJ<¡hQYMi:ITPUWUKMiPUIEK[WUAL?i?fdK[@BAi.PK[°"¯?YMWvi:ITPK[i?hAM«YMi?AxK9.YMfdIPU<dAhKMc?KMhPTShYMi?IePJWK[iPUIvi¡PJ<dAYMWUy.i?KM@xYOfOA0PU<dAi©PIgc:YIJIJsdABPU<?KPPJ<dAIJYMLOPUY.i¡K.IeOIEPUYAQNPJWKMhQPgPJ<[email protected]¦Y.idAIvYM§PJ<[email protected]^fOA@BKMi?f IghQY.i?ITPUWUKMiPIrsSANPJWKMhÁPUi?yKY[PgY[=¦[email protected]?IeSM#<dh<^Ie<:K[s:AgiOAK.IesdAvs:AhK[L:IeAgYMPJ<?AhQY.IePUI=iMY..Af

aRhQYML?WUIJAMPI=clY.IUIesdARPU<?KP%NÇAMAiK[¨PJAW=PJ<dAvWUAKNdKPUY.i)PXPU<dAvcdWUYMsdA@®slAhYM@BAI=iOAK.IesdAM

(i?IUKPUIe?AffOA@K[i:fdI§hYMLdfK[[email protected]?fOIJs?PTS33PU<dARY.WJAhYMiPUKMidAfi©Y.idAYMPU<dA@ IWUA LdWJAfPUY&IJK[PJITS&PT«Yfd !3AWUAiPclAWUYOfdIPJ<dA@BYOfOA§#«idY[P"KMYPr] PJWUhSIeY.LdPJYMi^#<dh<&hQY.Ldfs:AxL?IJAfPJYIeY..AgPJ<?IrIRPJY<?K9MAIJYM@BAIJKMhfOA@K[i:fKPvPJ<?A":WUIePvIePUKMyMA.IeYPhK[i&slAK9K[KMsdA"i¤PJ<?ABIJAhYMi?fBariPU<dABY[PU<dAWv<?KMi?fPgIvK[IJYidAhAIUIJKMWJSPJYWUA@BA@"s:AWgPJ<:KPi?fOIJs?PTSpY[[email protected]#ZTL?IeP"K@BYOfOAhQYMi:ITPUWUKMiPPJ<?K[PRfOYAI=idYMP<dYMfiWUAK[YMWUf

`K[c?K.hQPTS^hQYMi:ITPUWUKMiPUIEhKMi¤KMIJYslAhYM@BALdi:IJK[PJIT?AffOLdAxPUYK[ipK[i?KMY.yMY.L?IWUAK.IeY.i=PJ<dAWJAhQY.Ldfs:[email protected]&WUh<¢AidYML?yM<©PUY¤idYMP"<?K9.AK¤c:AWJYOf©#<dAWUAPhK[is:AAQNOcdYMPJAf#PU<dYMLdPxsdWUAKMidy¤hK[c:KMhQPTS¡hQYMi:ITPUWUKMiPUI&RY«AMAWPJ<dIxI"idY[P LdPUAIJAWUY.L?IxPBI"idY[PAQNOclAhÁPUAf&PJYYOhhQL?WRWUA LdAi.PUSKMi?f#<?Ai&Pv<?K[cdclAiKMfdfOPJYMi:K[@KMh<didAWUS^K[i?f&@K[iL?K[KMs:Y.WhKMixslAhQY.iPJWKMhÁPUAf ài?fOAAfMPJ<dIXIKEyMYYOfWUAK.IeY.iPJYgWUAKNPJ<dAhK[c?K.hQPTShQYMi:ITPUWUKMiPUIKP#PU<dAgYMsOZTAhÁPU.ARLdi:hÁPJYMi?slAhKML?IJArPU<dAhQYITP#YMY.K[PJidyxPJ<dA@I=s:APePUAWidY#i

¯i?K[S.dPI=@Bc:Y.WePK[iP#PJYBidYMPJhQAEPJ<?K[P

_i?hQAPJ<?AIe@Bcd?Af.AWIeYMiªiªPJ<dA?WIePIePJAc KMi?fPJ<dAYMWUyMi?K[@BYfdA#<?K9.APU<dAIJKM@BAYMsOZTAhÁPU.ALdi?hQPJYMi§KMi?f¡A.AWUS¤KMf©IeA LdAi?hAfOA?idAIK[i¢AKMIJsdAIeY.LOPUY.i¢Y[#PU<dAY.idAQc:AWJYOfp@xYOfOA0PJ<dAY.sOZTAhQPJMAxK[LdAYMsOPK[idAf¤sS^PJ<?A<dALdWUITPUhBhKMi¤slAL?IJAf¡KMIvKslYMLdi:fyMAidAWKPUYMWYMW=PU<dAgyMAidAWK[3cdWUYMsdA@

;=<dA<?ALdWUIePJhKMIJYIeL?yMyMAITPIK[iKMy.YMWUPU<d@ Y.WrIJYMidyBPU<dAc?WJY.sdA@§PUYIJK[PJITSfOA@BKMi?fdIc:AWJYOf¢PUY¤clAWUYf§sSIJAc?KMWUK[PJidy^PJ<?A:WUIePYMidA.§K[i?f K[y.yMWUAyKPJidyPU<dAWJAITPiªK¡IeAhQY.i?fITPK[y.AM

_YBPU<dAK[slYMA°@BAi.PUY.idAfiOAKMIJs?PTScdWUYMsdA@I=K[WUAMOi?fOAAfdi?Y[PLdPUAIeAWJYML:I

2 !Q!

`«L?WJWUAiPr@BYOfOA IRKMhWUYMsdL:ITPUidAIUIRIEKcdWUYMsdA@ËPU<?KPvidAAfdIRPJYs:ABIJYMMAfE;=<dIE«Y.WJcdWUAIJAiPUIPTYB@KZTYMW#KMcdcdWUY.KMh<?AI§PJYKMh<?AMArPJ<?K[P

A ¶ Cd|D7 ~ ¶+I ~ J7 6 I _OYMLOPJYMiPJY"IePJYOh<?K.ITPUhRcdWUYMyMWK[@[email protected]?IeP§s:AhKML?IeAY[dPU<dAW ?AQNOsdPTS3?WITPXITPK[y.A«fOAhQIeYMiKMWJA«@KMfOA«YYMidyPUY $ &'" . g#PJ<PJ<dALOPULdWJAK[i?fPJ<dAxK9K[KMsdAc:YIJIJs?PUAIrPJYK !lY.WUfPU<dAxLdi?hQAWePK[iPTSM3IeYPT«Y@xYOfOAIEK[WUAfOAMAY.c:Af

ÐÏ×ШØ

Page 34: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ u?C

i:IefOAPU<dIgWK[@BAYMWU3KTàITPK[y.AIv@BNOAf©idAK[W@BYOfOA°XK[i?f©K¤t9àITPK[yMAIvYMidA. Y[PJ<©YM

PJ<dA@ K[WUAfOWUAhQPRyMAidAWK[¦KPUY.i?I=Y[PU<dA"hLdWJWUAiP@BYOfOAIs?LOPr@Bi?fOidyPJ<?ALdi?hAWJPUKMiPTSYMPJ<dAcdWJYOhAIUIsdLOPr#<dAvPJ<?Av?WUIePhQY.LdfslAfO hLdPPJYIJAQP#Ldc?PJ<dAIJAhYMi?fidAAfdIKBfOAAclAWITPUL?fOS.

Q ² ¡ 6 · S : 'G?7¨ ³ |dD7| F d[7 ¶ 6 ]¨PJAWK"fOAAc:AW«WUAIeYMiBY[PU<dArhLdWUWJAi.P«@[email protected]?IJfOAWUK[PJYMi?IK[[email protected] NivPJ<?A§<dY.c:AXPU<?KPidA¡@BYOfOAIslAhYM@BA§fOAMAY.c:AfYMWIePJL?fdSidyPJ<?A@¥cdWUYMclAWUSPQsdLOPg.AcOPg@xc?hQPJS&i?hQL?fOAf&KfOA@K[i:fdIvK[i?fphKMc?KMhPTShQY.i?ITPUWUKMiPI;=<dAWUAIJLdPJidyg@BYfdA?I§[email protected]?IePJidAIJIsS3YY..i?yBYMW#cdK[i?I#IUKPUIeSi?yBfdA@K[i?f?I#K[i?fcdWUAIJAWUidyBKMI=@"L?h<K.I=c:YIJIJsdAgY.WJArYMW=LdPJLdWUAM

AhK[L?IJAvPJ<?AIeAhQYMi:fKMcdcdWUY.K.h<IJAA@I#@BY.WJAgcdWUYM@BIeidy:dKB<dALdWUITPUhvI#cdWUYMclY.IJAfPJYIJYMMAvPUIi?IePUKMi?hQAIOsdLdPPI#ITPUidAhQAIJIUK[WUSBPJYBcdWUYfOAviLd@BAWUhK[3WUAIJLdPUI?IeclAhKMSY.AWWJAK[fdK[PUKd;=<dI«Y.WJ¾K[IJYAQNOclY.IJAIIeAMAWUKMRi.PUAWUAIePJidyfOAKMIPJ<?K[PhYMLdf slA¤iPJWUYfdL?hQAf£i @Bi?idy

IUh<dAfOLdidy:RWUYMsdL:ITPUidAIUIhYM@"sdi?Af@BididyKMi?fphQYMiPUiLdY.L?IRWUAs?YOhi?yKMWJAxIeY.@xAxAQNdKM@xc?AIrYMPU<dA@

××Ù"[U\]$^^

Page 35: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

ut " "

##!#

CMYM<di WUyMAxK[i?f¤¯dWK[iYMI Y.LdMAWUKMLON!U /". !J "$ &+/ -/1!K-M_c?WJidyMAWC??KMO

¼t qEK9f EY.fds:AWJy: ! "!# ," -!+.(" -K" ÁVXWUYhAAfOi?y.IrY[PU<dAPU<dWUf&iPJAWei?K[PJYMi?KMhYMiOAWJAi?hQAYMi EAidAPJh]Ry.YMWUPU<d@I\YMWUy.KMigK[LO@K[i?iªVXLds?Ie<?AWI ài:h[RCD?>??cdcU>Mn>.md

uqrYMWUPËYOh<s?K[L?@ KMi?f*]Ridi?K `«<dAi *L"0"! !+" $ !+ / &) 0!) "!U "!# !+ " -K" J $&'! &) !)!K- &+/ ".K[iL?K[WUS CD??>? "!#$ %!& '(*)+(,-./)102//3(//)1-!45 6)+7

o AWUh<¡K[i?f Á ¯«EWJYIJIJ@K[i98 &) 1 -! ,&! &+1!+Á`K[i:KMfài?IeP\i?idyLd 8)Y NTCD?=.mPÁoMOmo:

¼m ¬RAIeY.i \^YMWK[AI 0" ;:.$ 0&'" &) "!U $6<"! !# bXidy.i?AAWUidy¢;=<dAIJI(Rid.AWIefdK.ffOA"`«<dAM4MLdSt[n.n.tO

=?¼`rVXhK[Wf 0" "/J O- $& "!# &&+$" "!'O $",!#" " " &+/ Á\^KMi?K[y.A@BAiP_OhQAi:hQA 9+9 NTCD?M"=KPÁ3Ct=>=3CtKM[tO

M Fg ;v F#YOh9K[AKMW 8 &) ?>" "! !# ! $ !U AhQPJLdWUA ¬YMPJAI((A@B C(@

> #_dK[i?K.IDd#]RWUK.hQAi:KdRKMi?f b# EY.AIE $" !K-O&+/ " $&'! &+,1!+gt[n.nMu?cdWUAc?WJiP

? E9;Y.#i?IJK[i:fgFg (i?fOAWJYYOf $" !K- -K" "*$&'!&+X1!+ e\]FMYMLdWUi?K[YM@KPU<dA@KPUhIK[c?cdAfis?L?IeidAIJI#KMi?fi?fdL?ITPUWJS V NTC??=PÁdt[oM*dtM[nd

ÐÏ×ШØ

Page 36: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

"!#$"%$&'!(&)*"!+,!+.-/!+01!+ u.u

%%¤! %%¤!

AQPhQY.i?IJfOAW=PJ<dAgi?AQNPAQNdK[@BcdAMX¯?YMWKxYMidA°@BY.i.PU<ZTY.sOPTY"c:[email protected]#Y.cOPJYMi?I#KMWJAvy..Ai ¯dY.LdW#c?K9S@BAiPUIOYMi?hAvc:AW«AA3OY[[tMnMnd [L:ITPY.idAE?i?K[c:[email protected]#YM?MnMn?]rIJIJLd@Bi?y©PJ<?K[PslY[PU<£YMcOPUY.i?IKMWJAAKMIJs?A^PU<dA LdAITPUY.i I <dh< IPJ<dA¤s:AITPh<dYMhQAM

(RiOYMWJPJL?i?KPUASMPIgidYMPKMIgAKMIJS&KMIhYM@Bc?KMWJidy ?MnMn#PJ<200 × 4 = 800

s:AhKML?IeAxPU<dAWUAIKes?KMidPU<?KPY !3AWICn jYMiPUAWUAIeP§c:AW«AA[ P«@BAKMi?IPU<?KP=IJK9idyvPJ<?A?WITP=IJKMKMWJS.PJ<?AWUAR#slA

200× (1 + 10%) = 220KP#PU<dAAi:fYMPJ<dAIJAhYMi?fAA3?KMi?fKP#PU<dIR@BYM@BAiPPRIc:YIJIJs?A

PUYB@BKMMAvK"IJAhYMi?ffOAclY[P=Y.W220 + 200

OIeYBKP=PJ<?AEAi?fY[PU<dAEPJ<dWfAAPJ<dAvs?KMid#WUAQPULdWJi440 × 1.1 = 484

aR=hYMLdWIeA.K[yK[iY.idAIe<?KMIUK9MAxi¤PU<dAs?KMidPU<dIK[@BYMLdiPgcdL?IvPU<dABPJ<dWfIUK[K[WUSMy.AQPePUi?ypKPJY[PK[#IJLd@ Y[M[m.tOPJ<L?IBKPxPJ<?AAi?f¢YMPJ<?A@BYMiPU<XPJ<?A?i:K[#K[@BYML?i.PB#hYMi?IJITP#Y["?mMtOPJ<:KPPI MmMtYMAW§PU<dAIeAhQY.i?fY.cOPJYMi;=<dAxcdWUAhAfOidyc?WJYOhQAfOLdWUA"hYM@Bc?K[WUAIPJ<dABK[@BY.LdiPEKPEPJ<?A"Ai?fYMXPJ<dAB@BYMiPJ<ls?LOPrPU<dAIJKM@BA

hK[[email protected] 0"!# àiPJ<dI=hKMIJARP=IidAhQAIUIUK[WUSPJYBhQY.iMAWJP§slY[PU<Y.cOPJYMi?IPUYxPJ<dAW*&+/!U U.

<?KPIBPU<dA^PUYf?K9S IBA LdK[AiPK[@BY.LdiPYMWBPJ<?A"?.nMnpKPPU<dA^Ai?f¾YMrPU<dA@BYMiPU< PhQY.WJWUAIJc:Y.i?fdIvPUY^PJ<?AK[@BYML?i.P"Y[#@BYMidAS

xIeL?h<¡PU<?KPPIcdLOPxi¢PJ<dAs:K[id¤YMWPJ<dAo

c:AWJYOfdIdPI=yMY.i?yPUYs:[email protected]"?Mn.ndPJ<:KPP

(1.1) × (1.1) × (1.1) × (1.1) × x2︸ ︷︷ ︸ !"

︸ ︷︷ ︸#$!!" ︸ ︷︷ ︸%'& ()!"*+,(,& +'-

︸ ︷︷ ︸+,(/.#0%'213.4+5%

= 900

PJ<dAix .6+,87':9;13+ =

900

(1.1)4= 614.71

¯dY.Y#i?y"PU<dAIUK[@BAvWJLdAMOPI=c:YIJIJs?AEPJYhYMiMAWeP=AK.h<<Mt[n.nIUK[K[WUSBPJYI#hQYMWUWUAIJc:Y.i?fOidyPJYOfdK9S I#K[@BYMLdiP=PJYBy.AQP

x .6=, 7':9;13+ =200

(1.1)+

200

(1.1)2+

200

(1.1)3+

200

(1.1)4= 635.35

;=<dAgWUAIJLdPdYMhYMLdWIeA.OI#PJ<dAIUK[@BAMPU<dAgYML?W#c?K9S@xAiPUI OYMcOPUY.iI=PJ<?Ags:AITP§YA.AW.PU<?KPI§PJ<dAvhQL?IePJY.@K[WUS"=K9SPUYxhYM@Bc?KMWJA#PU<dI«i?fYMiLd@slAWINÇYMWcdWUY[ZTAhQPUI.Y.W«s?L?IeidAIJI$PÁK[i?fiPJ<dAKMLdA"YMsOPK[idAf&IvhK[AfPJ<dA !+ &) !U""U N0¬RVX­ PE#<dh<¤hKMislAxKMIJYLdi:fOAWITPUYYf&KMIrPJ<dA@Bid@Ld@ cdWJhQAvY.idAgIe<dY.LdfK.hhAcOP=YMWJIJAidy rPJ<dIefOAK rY.WTc?WJYMZTAhÁP > ØÙ9ÉvÆ°¹9ÃUƺ¼»½¹9ÈTÀÇÈ«Æ0¹9ÈÙ9à #È ?:@;@A?CBDFEAGIHDKJML:?N:DßT¿ #ÈT» À0¿ Oß°¹9¿¿Á»0º¼ÙÆ°¹9ȧ¿ÁÙ9È«ÅÃeÌ #ÈTÙÆ¿ÁÅÆ°º¼¿ÁÙ #ÈeÃÙ9»Æ°¿

ÅÃeÌrÃßT¿Á»0Æ#O?Æ°¹9È ÃÔ¼Ä9ÈX¿OÆ0¹9Èß¹ÃÙ9ßTÈXÆ°¿ ÃÂÁÈ»°¿ #ÈÊÄ9»°º¼Ù9ÈT»0»½º Æ0¹RÆ0¹9È #¿ÁÙ9ÈàÌÉ9Ä9ÀǺ¼ÙÆ0¹9ÈXÆ°º#È¿ÁÙ9ȹûư¿=½Ãº ÆÆ°¿=ÀÇÈTßTÈeº QÈ¿ÁÄ9ÀÅÃTÌ #ÈTÙQÆJã

××Ù"[U\]$^^

Page 37: Robust models for simultaneous open pit and underground mines · Robust models for simultaneous open pit and underground mines Nelson Morales To cite this version: Nelson Morales

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399