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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
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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
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
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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
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]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
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xit =
1
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fi =
1
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i
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∑
t
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* ¶ ¶ · 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
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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
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hLdPJLdWUA PJ<?AWK[i:fOYM@ AA@xAiPUIBK[WUAPU<dASAfdI"Y.WAKMh<cdWJYOfOL:hÁP KMi?f©:i?K[S.PU<dA^hQYMWUWUAhÁPU.A
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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
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#<?AWUA
Q(x) = IEω
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],
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ω ∈ ΩK[WUAPU<dAc:YIJIJs?AIJhAi?KMWJY.I(NÇKMhhQY.WUfOidy&PJY¡IeY.@xA
c?WJY.s?K[sdPTSIJc?KMhA(Ω, IP )
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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@$hYMWUWJAIeclYMi?f?IPJYKg@"LdPUIePUKMyMAIePJYOh<?K.ITPUh=sdi?KMWJS ׿ÁÊ9Ä9»ÇÆ°Ù9ÈT»0»º¼»º#Å9Ô¼º¼ßTº ƹ9ÈTÀÇÈÁã
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]#hQPJL?KMSMPJ<dAPT«YIePJYOh<?K.ITPUhxidAK[W@BYOfOAIgc?WJAIeAi.PUAf¡K[WUAByMAidAWK[¦KPUY.i?IvY[PJ<dAhQLdWUWJAiP@BYOfOAIM#PJ<BPU<dARKMfd9KMiPUK[y.A=Y[YY.idyEYMW§@xY.WJA#WUYMs?L?ITPc?KMi?I.s:AhKML?IeA=PU<dARIJYMLOPJYMi:IK[WUA#@BYMWUA:AQNOs?A.vPJ<dAS<?K9MAIJAhYMi?fpITPK[y.A"K[WUK[sdAIrPJYhYMWUWJAhÁPEPJ<dAc?KMipfOAc:Ai?fOidy^YMipPU<dAA.AiPUIKMIAKMIr?WITPvITPK[[email protected]<dAIeA"LOPJL?WJAxc:YIJIJsdPJAI"ari^PJ<dABYMPJ<dAWr<?KMi?fPvIEAQNOclAhQPJAfPJ<?K[PrPU<dAxWJAIeLdPUIrYMsdPUK[idAf#PJ<PU<dAIJAx@xYOfOAI3@BAKMIJLdWUAf^Y.idS^i&PJ<dAxYMsOZTAhÁPU.AL?i?hÁPUY.iªPUAWU@I#slA^c:YY.WJAW«sdLOPPJ<:KPIs:AhKML?IeAPU<dA^K[LdAIBY.sOPUKMidAf i PJ<?IB=K9SK[[email protected][ITPUhrPJ<:K[iPJ<dAgY.idAI#hYM@BidyWUYM@PJ<dAgY.WJyMi?KMl@BYOfOAI;=<dAWJAK[WUA§K[IJYIeY.@xA«@BclYMWJPUKMi.PfOIUKMfd9KMiPUK[y.AIKMi?fgcdWUYMs?A@BI#PU<IJAQPePUi?yRLdcPJ<dAIeA§IePJYOh<?K.IT
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]Rs:Y.LOPgPJ<dABPUAh<di?YMYMyMhKMKMWJKPUY.i?I3PU<dAS¤#XidYMPvslAi?hQL?fOAfslAhK[L?IJAxPJ<dAWJABAQNOITPIEPJ<dAclYMhQSPJ<?K[PrPU<dAxWJAIeLdPUI1NǬrVX K[LdAI$P#@"L?ITPvslA"WUAK.h<?K[sdA#PJ<¤cdWJY.Ai NÇAQNOIePJidyP=PJAh<didY.Y.yMSMiYMWfOAWPUYxslAhQY.@Bc?K[WUAf#PU<AKMh<Y[PU<dAWI
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8 3 9 ².´ * ¶ X7 6 I
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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
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~ 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
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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
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¯yMLdWUA M YOh[\^K.hQWUYMsdYOh"@BYOfOAYMW#@BNAf@xididy?
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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\]$^^
tMn " "
8 9 @ Q ² ² ~ J7 6 7|D7·O¤ 6 · GO~ ² ² ~|_OAMAWUKMfOA?idPJYMi?IEK[WUAWUA LdWJAfiY.WUfOAWPUY#WJPJAPJ<dAxhQY.i?ITPUWUKMiPIK[i?f^IJAQPELdc^PU<dA@BYOfOA°r]P:WUIePOAQP
Im = i |sdYOh
ii.PUAWIeAhÁ[email protected]
m]rIA0OPU<dAIJAQPI
Ni
KMi?fNm
YMPJ<dAi?AyM<s:Y.WUI=YMs?YOhiK[[email protected]
mWJAIeclAhQPJMAS
KMWJAvfOA?idAf]Rs:Y.LOPA.AAQNdhQL?IJY.i9PXhQY.i?hQAWJi:IslY[PJ<PJAh<didYMYMy.AI;=<[email protected]
mK !3AhQPUI
PU<dA«IJAQPUU
m
[email protected][i?fgPJ<dAIeAPUP
m
YMdsdYOhOIYOhK[PJAfi?IJfOA§PUIhQY.idA§YMdi :LdAi?hAM]rIL:IeL?KM0KrfOIUhQY.LdiPWKPUAY[
αK[i:fK[i<?YMWU¦YMi
TK[WUA§hYMi?IJfdAWUAfPJ<L?I
s, tWUAcdWUAIJAiPclAWUYOfdI
]RIJY?yMMAiBPT«YidAyM<slYMWUidy 1!'Y[s?YOhOI.PJ<?AEfO !3AWUAi?hARslAQPTAAiPJ<dAW<dAyM<PI<?K9.APUYBs:AslYML?i?fOAfsSKxclAWhQAiPK[yMA
δY[K[iLdidPUKMWJSs?YOh I=<?AyM<P%NsdYOhIRK[WUAEhYMi?IJfdAWUAf<dYM@BY[
y.AidAYML?IiPJ<dAWRfd@BAi:IeYMi?I$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
"!#$"%$&'!(&)*"!+,!+.-/!+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
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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]$PÁ
]s:APePUAWK[cdc?WJYKMh< N¨WUYM@ PJ<?APU<dAY.WJAPJhK[IefOAKP"AK.ITPPhK[i©slAfdYMidAPU<dAi?AQNPBIUh<dA@BKIhYMi?IJfOAWUAf
CM#;=<dAvLdIJA LdAi:hQAvY[AQNOcdYMPUK[PJYMiIdNOAftO#]PJ<?AgWUKMi?fOYM@9KMWJK[s?AIKMWJAvWUAKM¦Afud_YM@BAhYMWUWJAhÁPJMAEKMhQPJYMi?I#KMWJAvclAWJYMWU@xAf
ÐÏ×ШØ
"!#$"%$&'!(&)*"!+,!+.-/!+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
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) &)/".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]
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t
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S =∑
i
vi +∑
m
vm −
[∑
i,t
vi pit +∑
m,t
vm umt
]NeCD?P
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(∀i)∑
t
pit ≤ 1N0tMnP
(∀m)∑
t
umt ≤ 1.N0tdCP
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pit ≤ rit (∀i)(∀t),N0t.tP
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rit ≤∑
s≤t
pi−s (∀i)(∀t).N0tMuP
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(∀i)(∀t)(∀j ∈ Ni)∑
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i
(vi + wi)pit ≤ CBt ,
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s≥t
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ums ≤ 1 (∀m)(∀s ∈ UPm)(∀t).
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umt ≤ fm +∑
n∈Nm
∑
s<t
uns (∀m)(∀t).N0tKMP
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∑
m∈Sg
fm ≤ sg (∀g).N0t>P
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s≥t
ums +∑
s≤t
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(∀t)∑
m
umt ≤ K.NÇu.nP
¯i?K[SPJ<dAhK[c?K.hQPTShQY.i?ITPUWUKMiP=WJAKMfdI
(∀t)∑
m
wmtumt ≤ CUt .
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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
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(∀t)∑
i
vi pit +∑
m
vm umt ≥ Dt,N0u[oKP
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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°¿
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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