A Framework for Open-Pit Mine Production Scheduling under ...
Open Pit Mine Production Scheduling - Zuse Institute...
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Open Pit Mine Production SchedulingSome IP modelling and solution techniques
Ambros M. Gleixner
Zuse Institute Berlin
09/28/2009
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 1 / 30
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Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 2 / 30
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Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 3 / 30
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Open Pit Mining Production Scheduling
Input
. block model with precedences
. deterministic block content
. ore prices and production costs
. equipment capacities
. possibly aggregation of blocks
Problem
Find an order of excavationwith
. max. net present value
satisfying
. precedence constraints,
. resource constraints.
Simultaneously: Determine
. opt. processing decisions
for each block.
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 4 / 30
![Page 5: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/5.jpg)
Open Pit Mining Production Scheduling
Input
. block model with precedences
. deterministic block content
. ore prices and production costs
. equipment capacities
. possibly aggregation of blocks
Problem
Find an order of excavationwith
. max. net present value
satisfying
. precedence constraints,
. resource constraints.
Simultaneously: Determine
. opt. processing decisions
for each block.
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 4 / 30
![Page 6: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/6.jpg)
Open Pit Mining Production Scheduling
Input
. block model with precedences
. deterministic block content
. ore prices and production costs
. equipment capacities
. possibly aggregation of blocks
Problem
Find an order of excavationwith
. max. net present value
satisfying
. precedence constraints,
. resource constraints.
Simultaneously: Determine
. opt. processing decisions
for each block.
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 4 / 30
![Page 7: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/7.jpg)
Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 5 / 30
![Page 8: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/8.jpg)
Getting started: computing ultimate pit limits
Given: Profit/negative cost wi of mining each block i ∈ N = {1, . . . , N}.Seek: Precedence-feasible set of blocks X ⊆ N with max.
∑i∈X wi.
(If i ∈ X than all predecessors of i are in X.)
Observation: This is a max-weight closure problem in the precedence graph(N ,S = {(i, p) ∈ N ×N | p predecessor of i}
)with node weights wi.
. Solve e.g. by min-cutalgorithm in a slightlyextended graph:
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 6 / 30
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Getting started: computing ultimate pit limits
Given: Profit/negative cost wi of mining each block i ∈ N = {1, . . . , N}.Seek: Precedence-feasible set of blocks X ⊆ N with max.
∑i∈X wi.
(If i ∈ X than all predecessors of i are in X.)
Observation: This is a max-weight closure problem in the precedence graph(N ,S = {(i, p) ∈ N ×N | p predecessor of i}
)with node weights wi.
. Solve e.g. by min-cutalgorithm in a slightlyextended graph:
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 6 / 30
![Page 10: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/10.jpg)
Getting started: computing ultimate pit limits
Given: Profit/negative cost wi of mining each block i ∈ N = {1, . . . , N}.Seek: Precedence-feasible set of blocks X ⊆ N with max.
∑i∈X wi.
(If i ∈ X than all predecessors of i are in X.)
Observation: This is a max-weight closure problem in the precedence graph(N ,S = {(i, p) ∈ N ×N | p predecessor of i}
)with node weights wi.
. Solve e.g. by min-cutalgorithm in a slightlyextended graph:
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 6 / 30
![Page 11: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/11.jpg)
Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 7 / 30
![Page 12: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/12.jpg)
A new MIP formulation (Boland-Dumitrescu-Froyland-G’08)
Variables
xk,t =
1 if aggregate k may be minedduring periods t, . . . , T
0 otherwise
yk,t = fraction of aggregate k mined during period t
zi,t = fraction of block i processed during period t
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 8 / 30
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A new MIP formulation (Boland-Dumitrescu-Froyland-G’08)
Variables
xk,t =
1 if aggregate k may be minedduring periods t, . . . , T
0 otherwise
yk,t = fraction of aggregate k mined during period t
zi,t = fraction of block i processed during period t
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 8 / 30
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A new MIP formulation (Boland-Dumitrescu-Froyland-G’08)
Objective function
NetPresentValue (y, z) =
T∑t=1
(1
1 + q
)t−1∑
aggs k
−mkyk,t +∑
blocks i
pizi,t
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 8 / 30
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A new MIP formulation (Boland-Dumitrescu-Froyland-G’08)
max NetPresentValue (y, z)
s.t. (x, y) precedence feasible
K∑k=1
Rkyk,t ≤Mt for t = 1, . . . , T
N∑i=1
Rizi,t ≤ Pt for t = 1, . . . , T
xk,t ∈ {0, 1} for k = 1, . . . ,K, t = 1, . . . , T0 ≤ yk,t ≤ 1 for k = 1, . . . ,K, t = 1, . . . , T0 ≤ zi,t ≤ yk,t for k = 1, . . . ,K, i ∈ Bk, t = 1, . . . , T
. integrated optimisation of processing decisions
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 8 / 30
![Page 16: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/16.jpg)
A new MIP formulation (Boland-Dumitrescu-Froyland-G’08)
max NetPresentValue (y, z)
s.t. (x, y) precedence feasible
K∑k=1
Rkyk,t ≤Mt for t = 1, . . . , T
N∑i=1
Rizi,t ≤ Pt for t = 1, . . . , T
xk,t ∈ {0, 1} for k = 1, . . . ,K, t = 1, . . . , T0 ≤ yk,t ≤ 1 for k = 1, . . . ,K, t = 1, . . . , T0 ≤ zi,t ≤ yk,t for k = 1, . . . ,K, i ∈ Bk, t = 1, . . . , T
. integrated optimisation of processing decisions
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 8 / 30
![Page 17: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/17.jpg)
For this talk: no aggregates, no fractional mining
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T
∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T∑i
Rizi,t ≤ Pt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. NP-hard by reduction from Precedence-Constrained Knapsack
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 9 / 30
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For this talk: no aggregates, no fractional mining
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T∑
i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T∑i
Rizi,t ≤ Pt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. NP-hard by reduction from Precedence-Constrained Knapsack
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 9 / 30
![Page 19: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/19.jpg)
For this talk: no aggregates, no fractional mining
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T∑
i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T∑i
Rizi,t ≤ Pt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. NP-hard by reduction from Precedence-Constrained Knapsack
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 9 / 30
![Page 20: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/20.jpg)
Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 10 / 30
![Page 21: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/21.jpg)
Standard LP relaxation
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T∑
i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T∑i
Rizi,t ≤ Pt for t = 1, . . . , T
0 ≤ xi,t ≤ 1 for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. typically: interior point outperforms simplex (when solving from scratch)
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 11 / 30
![Page 22: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/22.jpg)
Standard LP relaxation
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T∑
i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T∑i
Rizi,t ≤ Pt for t = 1, . . . , T
0 ≤ xi,t ≤ 1 for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. typically: interior point outperforms simplex (when solving from scratch)
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 11 / 30
![Page 23: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/23.jpg)
Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 12 / 30
![Page 24: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/24.jpg)
Full relaxation of resource constraints
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T
∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. substitute z according to objective function coefficients
. costs and profits decrease over time ⇒ all blocks are mined in period 1(if at all) equivalent to computing the Ultimate Pit Limit
. solve efficiently as max-weight closure problem
. use as preprocessing: all blocks in an optimal OPMPSP schedule mustbe contained in the Ultimate Pit – remove the others from the model
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 13 / 30
![Page 25: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/25.jpg)
Full relaxation of resource constraints
max NetPresentValue (x)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T
∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. substitute z according to objective function coefficients
. costs and profits decrease over time ⇒ all blocks are mined in period 1(if at all) equivalent to computing the Ultimate Pit Limit
. solve efficiently as max-weight closure problem
. use as preprocessing: all blocks in an optimal OPMPSP schedule mustbe contained in the Ultimate Pit – remove the others from the model
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 13 / 30
![Page 26: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/26.jpg)
Full relaxation of resource constraints
max
Net
PresentValue (x)
s.t.
xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T
xi
,t
≤ xp
,t
for (i, p) ∈ S
, t = 1, . . . , T∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi
,t
∈ {0, 1} for i ∈ N
, t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. substitute z according to objective function coefficients
. costs and profits decrease over time ⇒ all blocks are mined in period 1(if at all) equivalent to computing the Ultimate Pit Limit
. solve efficiently as max-weight closure problem
. use as preprocessing: all blocks in an optimal OPMPSP schedule mustbe contained in the Ultimate Pit – remove the others from the model
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 13 / 30
![Page 27: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/27.jpg)
Full relaxation of resource constraints
max
Net
PresentValue (x)
s.t.
xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T
xi
,t
≤ xp
,t
for (i, p) ∈ S
, t = 1, . . . , T∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi
,t
∈ {0, 1} for i ∈ N
, t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. substitute z according to objective function coefficients
. costs and profits decrease over time ⇒ all blocks are mined in period 1(if at all) equivalent to computing the Ultimate Pit Limit
. solve efficiently as max-weight closure problem
. use as preprocessing: all blocks in an optimal OPMPSP schedule mustbe contained in the Ultimate Pit – remove the others from the model
Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 13 / 30
![Page 28: Open Pit Mine Production Scheduling - Zuse Institute Berlinco-at-work.zib.de/berlin2009/downloads/2009-09-28/2009-09-28-1500... · Open Pit Mine Production Scheduling Some IP modelling](https://reader031.fdocuments.in/reader031/viewer/2022021914/5c8848f809d3f291748c967c/html5/thumbnails/28.jpg)
Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
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Lagrangean relaxation of resource constraints
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T∑
i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T∑i
Rizi,t ≤ Pt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
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Lagrangean relaxation of resource constraints
max NetPresentValue (x, z)
+∑
tµt
[Mt −
∑iRi(xi,t − xi,t−1)
]+∑
tπt
[Pt −
∑iRizi,t
]s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T
xi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T
∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. feas. region is integral ⇒ opt. multipliers as in LP, best dual boundequal to LP bound (Geoffrion 1974)
. z-variables can be substituted according to obj. coefficient
. transformation to unconstrained project scheduling
. efficient solution by min-cut computations (Mohring et al. 2003)
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Lagrangean relaxation of resource constraints
max NetPresentValue (x, z)
+∑
tµt
[Mt −
∑iRi(xi,t − xi,t−1)
]+∑
tπt
[Pt −
∑iRizi,t
]s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T
xi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T
∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. feas. region is integral ⇒ opt. multipliers as in LP, best dual boundequal to LP bound (Geoffrion 1974)
. z-variables can be substituted according to obj. coefficient
. transformation to unconstrained project scheduling
. efficient solution by min-cut computations (Mohring et al. 2003)
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Lagrangean relaxation of resource constraints
max NetPresentValue (x, z)
+∑
tµt
[Mt −
∑iRi(xi,t − xi,t−1)
]+∑
tπt
[Pt −
∑iRizi,t
]s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T
xi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T
∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. feas. region is integral ⇒ opt. multipliers as in LP, best dual boundequal to LP bound (Geoffrion 1974)
. z-variables can be substituted according to obj. coefficient
. transformation to unconstrained project scheduling
. efficient solution by min-cut computations (Mohring et al. 2003)
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Lagrangean relaxation of resource constraints
max NetPresentValue (x, z)
+∑
tµt
[Mt −
∑iRi(xi,t − xi,t−1)
]+∑
tπt
[Pt −
∑iRizi,t
]s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T
xi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T
∑i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. feas. region is integral ⇒ opt. multipliers as in LP, best dual boundequal to LP bound (Geoffrion 1974)
. z-variables can be substituted according to obj. coefficient
. transformation to unconstrained project scheduling
. efficient solution by min-cut computations (Mohring et al. 2003)
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Lagrangean dual (with Helmberg’s ConicBundle 0.2d)vs. LP-relaxation (with CPLEX 11.0 Barrier, no crossover)
Problem instance LP-relaxation Lagrangean dual
time obj. val. time obj. val. rel. rel.[s] [108] [s] [108] time obj. val.
ma-115-8513 11.2 7.230097 1.8 7.230097 0.158787 1.000000ma-296-8513 27.2 7.276624 2.6 7.276631 0.094993 1.000001ma-1038-8513 259.5 7.306851 9.5 7.306854 0.036415 1.000001
ca-121-29266 (avg) 34.9 59.898146 6.1 59.898190 0.175164 1.000001ca-121-29266 (05) 64.7 60.899745 6.6 60.899800 0.101763 1.000001
wa-125-96821 324.4 0.508814 21.8 0.508815 0.067139 1.000001
Realistic problem instances provided by industry partner BHP Billiton, http://www.bhpbilliton.com/.
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Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
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Precedence Constrained Knapsack Problem
Given:
. Items i ∈ N with sizes ai and values ci
. Knapsack capacity b
. Precedence order S ⊆ N ×N (transitively reduced)
Feasible Solution:
. set X ⊆ N with∑
i∈X ai 6 b and
. X is closed under S (If i ∈ X and (i, j) ∈ S, then j ∈ X.)
Goal: maximize∑
i∈X ci
Important substructure in many applications:
. Resource constrained scheduling
. Multi-level facility location, access & distribution network design
. Routing problems: confluent flows, Internet routing, . . .
. . . .
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Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
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PCKs in open pit mining
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T∑
i
Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T ← add up∑i
Rizi,t ≤ Pt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. For each time period, the OPMPSP formulation contains oneprecedence constrained knapsack.
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PCKs in open pit mining
max NetPresentValue (x, z)
s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T
xi,t ≤ xp,t for (i, p) ∈ S
, t = 1, . . . , T
∑i
Ri
(
xi,t
− xi,t−1)
≤t∑
s=1
Ms
for t = 1, . . . , T ← add up∑i
Rizi,t ≤ Pt for t = 1, . . . , T
xi,t ∈ {0, 1} for i ∈ N
, t = 1, . . . , Txi,0 = 0 for i ∈ N
0 ≤ zi,t ≤ xi,t − xi,t−1 for i ∈ N , t = 1, . . . , T
. For each time period, the OPMPSP formulation contains oneprecedence constrained knapsack.
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Outline
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
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Precedence Constrained Knapsack Polytope
PCKP := conv{x ∈ {0, 1}N |
∑i∈N
aixi 6 b, (i)
xi 6 xj for (i, j) ∈ S}
(ii)
Polyhedral structure of PCKP well studied:
. Dimension, Conditions for (i) and (ii) to be facet-defining
. Knapsack-based inequalities [Boyd’93, Park-Park’97]
. Induced cover inequalities
. (Induced) k-cover inequalities
. (Induced) (1,k)-configuration inequalities
. Sequential lifting results[v.d.Leensel-v.Hoesel-v.d.Klundert’99]
But:
. Nothing implemented in MIP solvers
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Variable fixing
Notation
. Pred(i) := {j : j predecessor of i} ∪ {i}(Items that must be included with i.)
. A(i) :=∑
j∈Pred(i) aj
(Total size induced by i.)
Observation: If A(i) > b, then xi = 0 for all x ∈ PCKP .
Variable elimination scheme
. If A(i) > b, fix xi = 0.
Remarks
. Can compute all A(i) in O(S).
. Fixing is very efficient in practice (reduces LP gaps substantially).
. CPLEX does not find these fixings even with most aggressiveprobing.
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Variable fixing
Notation
. Pred(i) := {j : j predecessor of i} ∪ {i}(Items that must be included with i.)
. A(i) :=∑
j∈Pred(i) aj
(Total size induced by i.)
Observation: If A(i) > b, then xi = 0 for all x ∈ PCKP .
Variable elimination scheme
. If A(i) > b, fix xi = 0.
Remarks
. Can compute all A(i) in O(S).
. Fixing is very efficient in practice (reduces LP gaps substantially).
. CPLEX does not find these fixings even with most aggressiveprobing.
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Variable fixing
Notation
. Pred(i) := {j : j predecessor of i} ∪ {i}(Items that must be included with i.)
. A(i) :=∑
j∈Pred(i) aj
(Total size induced by i.)
Observation: If A(i) > b, then xi = 0 for all x ∈ PCKP .
Variable elimination scheme
. If A(i) > b, fix xi = 0.
Remarks
. Can compute all A(i) in O(S).
. Fixing is very efficient in practice (reduces LP gaps substantially).
. CPLEX does not find these fixings even with most aggressiveprobing.
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Variable fixing
Notation
. Pred(i) := {j : j predecessor of i} ∪ {i}(Items that must be included with i.)
. A(i) :=∑
j∈Pred(i) aj
(Total size induced by i.)
Observation: If A(i) > b, then xi = 0 for all x ∈ PCKP .
Variable elimination scheme
. If A(i) > b, fix xi = 0.
Remarks
. Can compute all A(i) in O(S).
. Fixing is very efficient in practice (reduces LP gaps substantially).
. CPLEX does not find these fixings even with most aggressiveprobing.
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Pairwise conflicts
Apply same idea to pairs of items:
. A(i, j) :=∑
k∈Pred(i)∪Pred(j) ak
(Total size of i, j and predecessors.)
Observation: If A(i, j) > b, then xi + xj 6 1 for all x ∈ PCKP .
. {i, j} with A(i, j) > b: Induced cover of size 2
. Induced cover: X ⊂ N with A(X) > b
. [Boyd’93, Park-Park’97] Conditions when induced cover inequalitiesare facet-defining (for low-dimensional faces of PCKP)
. [v.d.Leensel-v.Hoesel-v.d.Klundert’99] Lifting facet-defininginequalities for low-dimensional faces of PCKP to facets of PCKP
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Pairwise conflicts
Apply same idea to pairs of items:
. A(i, j) :=∑
k∈Pred(i)∪Pred(j) ak
(Total size of i, j and predecessors.)
Observation: If A(i, j) > b, then xi + xj 6 1 for all x ∈ PCKP .
. {i, j} with A(i, j) > b: Induced cover of size 2
. Induced cover: X ⊂ N with A(X) > b
. [Boyd’93, Park-Park’97] Conditions when induced cover inequalitiesare facet-defining (for low-dimensional faces of PCKP)
. [v.d.Leensel-v.Hoesel-v.d.Klundert’99] Lifting facet-defininginequalities for low-dimensional faces of PCKP to facets of PCKP
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Pairwise conflicts
Apply same idea to pairs of items:
. A(i, j) :=∑
k∈Pred(i)∪Pred(j) ak
(Total size of i, j and predecessors.)
Observation: If A(i, j) > b, then xi + xj 6 1 for all x ∈ PCKP .
. {i, j} with A(i, j) > b: Induced cover of size 2
. Induced cover: X ⊂ N with A(X) > b
. [Boyd’93, Park-Park’97] Conditions when induced cover inequalitiesare facet-defining (for low-dimensional faces of PCKP)
. [v.d.Leensel-v.Hoesel-v.d.Klundert’99] Lifting facet-defininginequalities for low-dimensional faces of PCKP to facets of PCKP
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Clique inequalities
Idea
. Consider conflict graph CG = (N , E) withE := {ij | A(i, j) > b}
Precedence order Conflict graph CG
Observation: Any valid inequality for STAB(CG) is valid forPCKP .
Corollary: Let C be a clique in CG. Then (1) is valid for PCKP .∑i∈C
xi 6 1 (1)
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Clique inequalities
Idea
. Consider conflict graph CG = (N , E) withE := {ij | A(i, j) > b}
Precedence order Conflict graph CG
Observation: Any valid inequality for STAB(CG) is valid forPCKP .
Corollary: Let C be a clique in CG. Then (1) is valid for PCKP .∑i∈C
xi 6 1 (1)
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Clique inequalities
Idea
. Consider conflict graph CG = (N , E) withE := {ij | A(i, j) > b}
Precedence order Conflict graph CG
Observation: Any valid inequality for STAB(CG) is valid forPCKP .
Corollary: Let C be a clique in CG. Then (1) is valid for PCKP .∑i∈C
xi 6 1 (1)
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Clique inequalities
Idea
. Consider conflict graph CG = (N , E) withE := {ij | A(i, j) > b}
Precedence order Conflict graph CG
Observation: Any valid inequality for STAB(CG) is valid forPCKP .
Corollary: Let C be a clique in CG. Then (1) is valid for PCKP .∑i∈C
xi 6 1 (1)
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Clique inequalities
Idea
. Consider conflict graph CG = (N , E) withE := {ij | A(i, j) > b}
Precedence order Conflict graph CG
Observation: Any valid inequality for STAB(CG) is valid forPCKP .
Corollary: Let C be a clique in CG. Then (1) is valid for PCKP .∑i∈C
xi 6 1 (1)
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Clique inequalities
Idea
. Consider conflict graph CG = (N , E) withE := {ij | A(i, j) > b}
Precedence order Conflict graph CG
Observation: Any valid inequality for STAB(CG) is valid forPCKP .
Corollary: Let C be a clique in CG. Then (1) is valid for PCKP .∑i∈C
xi 6 1 (1)
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Cliques with common predecessors
Precedence order Extended conflict graphCG′
Observation: Let C be a clique in CG withP (C) =
⋂i∈C Pred(i) 6= ∅. Then, for any i ∈ P (C),∑
j∈C
xj 6 xi (2)
is valid for PCKP .
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Cliques with common predecessors
Precedence order Extended conflict graphCG′
Observation: Let C be a clique in CG withP (C) =
⋂i∈C Pred(i) 6= ∅. Then, for any i ∈ P (C),∑
j∈C
xj 6 xi (2)
is valid for PCKP .
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Cliques with common predecessors
Precedence order Extended conflict graphCG′
Observation: Let C be a clique in CG withP (C) =
⋂i∈C Pred(i) 6= ∅. Then, for any i ∈ P (C),∑
j∈C
xj 6 xi (2)
is valid for PCKP .
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Cliques with common predecessors
Precedence order Extended conflict graphCG′
Observation: Let C be a clique in CG withP (C) =
⋂i∈C Pred(i) 6= ∅. Then, for any i ∈ P (C),∑
j∈C
xj 6 xi (2)
is valid for PCKP .
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Cliques with common predecessors
Precedence order Extended conflict graphCG′
Observation: Let C be a clique in CG withP (C) =
⋂i∈C Pred(i) 6= ∅. Then, for any i ∈ P (C),∑
j∈C
xj + (1− xi) 6 1 (2)
is valid for PCKP .
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Cliques with common predecessors
Precedence order Extended conflict graphCG′
Observation: Let C be a clique in CG withP (C) =
⋂i∈C Pred(i) 6= ∅. Then, for any i ∈ P (C),∑
j∈C
xj + x′i 6 1 (2)
is valid for PCKP .
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Separation of clique-based inequalities
Given: x ∈ [0, 1]N fractionalSeek:
(1) clique C in CG such that∑
j∈C xj > 1, or
(2) clique C in CG and i ∈ P (C) such that∑
j∈C xj > xi
. Separation of (1) and (2) is equivalent to finding a maximum-weightclique in CG′ with node weights wk := xk and wk′ := 1− xk for allk ∈ N. w(C∗) 6 1: all inequalities (1) and (2) satisfied
. w(C∗) > 1 and C∗ ⊆ N : inequality (1) violated
. w(C∗) > 1 and C∗ = C ∪ {i′}: inequality (2) violated
. NP-hard, but: Efficient MWC implementations exist in anystate-of-the-art MIP solver.
. Ideally: extend the conflict graph of your MIP solver.
More details on facets and computational results: see e.g.
Boland-Froyland-Fricke-Sotirov’06, Bley-Boland-Fricke-Froyland’09.
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Separation of clique-based inequalities
Given: x ∈ [0, 1]N fractionalSeek:
(1) clique C in CG such that∑
j∈C xj > 1, or
(2) clique C in CG and i ∈ P (C) such that∑
j∈C xj > xi
. Separation of (1) and (2) is equivalent to finding a maximum-weightclique in CG′ with node weights wk := xk and wk′ := 1− xk for allk ∈ N. w(C∗) 6 1: all inequalities (1) and (2) satisfied
. w(C∗) > 1 and C∗ ⊆ N : inequality (1) violated
. w(C∗) > 1 and C∗ = C ∪ {i′}: inequality (2) violated
. NP-hard, but: Efficient MWC implementations exist in anystate-of-the-art MIP solver.
. Ideally: extend the conflict graph of your MIP solver.
More details on facets and computational results: see e.g.
Boland-Froyland-Fricke-Sotirov’06, Bley-Boland-Fricke-Froyland’09.
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Separation of clique-based inequalities
Given: x ∈ [0, 1]N fractionalSeek:
(1) clique C in CG such that∑
j∈C xj > 1, or
(2) clique C in CG and i ∈ P (C) such that∑
j∈C xj > xi
. Separation of (1) and (2) is equivalent to finding a maximum-weightclique in CG′ with node weights wk := xk and wk′ := 1− xk for allk ∈ N. w(C∗) 6 1: all inequalities (1) and (2) satisfied
. w(C∗) > 1 and C∗ ⊆ N : inequality (1) violated
. w(C∗) > 1 and C∗ = C ∪ {i′}: inequality (2) violated
. NP-hard, but: Efficient MWC implementations exist in anystate-of-the-art MIP solver.
. Ideally: extend the conflict graph of your MIP solver.
More details on facets and computational results: see e.g.
Boland-Froyland-Fricke-Sotirov’06, Bley-Boland-Fricke-Froyland’09.
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Summary
Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations
OPMPSP relaxationsStandard LP relaxationFull relaxation of resource constraintsLagrangean relaxation of resource constraints
Precedence Constrained Knapsack SubstructuresPrecedence Constrained Knapsack ProblemPCKs in open pit miningValid inequalities for the Precedence Constrained Knapsack Polytope
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Open Pit Mine Production SchedulingSome IP modelling and solution techniques
Ambros M. Gleixner
Zuse Institute Berlin
09/28/2009
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