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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

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

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

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

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

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

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

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.

iX 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

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.

iX 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

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.

iX 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

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

A new MIP formulation (Boland-Dumitrescu-Froyland-G08)

Variables

xk,t =

1 if aggregate k may be minedduring periods t, . . . , T0 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

A new MIP formulation (Boland-Dumitrescu-Froyland-G08)

Variables

xk,t =

1 if aggregate k may be minedduring periods t, . . . , T0 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

A new MIP formulation (Boland-Dumitrescu-Froyland-G08)

Objective function

NetPresentValue (y, z) =

Tt=1

(1

1 + q

)t1aggs k

mkyk,t +

blocks i

pizi,t

Ambros M. Gleixner (ZIB) Open Pit Mine Production Scheduling 09/28/2009 8 / 30

A new MIP formulation (Boland-Dumitrescu-Froyland-G08)

max NetPresentValue (y, z)

s.t. (x, y) precedence feasible

Kk=1

Rkyk,t Mt for t = 1, . . . , T

Ni=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

A new MIP formulation (Boland-Dumitrescu-Froyland-G08)

max NetPresentValue (y, z)

s.t. (x, y) precedence feasibleK

k=1

Rkyk,t Mt for t = 1, . . . , T

Ni=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

For this talk: no aggregates, no fractional mining

max NetPresentValue (x, z)

s.t. xi,t1 xi,t for i N , t = 2, . . . , Txi,t xp,t for (i, p) S, t = 1, . . . , T

i

Ri(xi,t xi,t1) Mt for t = 1, . . . , Ti

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,t1 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

For this talk: no aggregates, no fractional mining

max NetPresentValue (x, z)

s.t. xi,t1 xi,t for i N , t = 2, . . . , Txi,t xp,t for (i, p) S, t = 1, . . . , T

i

Ri(xi,t xi,t1) Mt for t = 1, . . . , Ti

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,t1 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

For this talk: no aggregates, no fractional mining

max NetPresentValue (x, z)

s.t. xi,t1 xi,t for i N , t = 2, . . . , Txi,t xp,t for (i, p) S, t = 1, . . . , T

i

Ri(xi,t xi,t1) Mt for t = 1, . . . , Ti

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,t1 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

Outline

Open Pit Mining Production SchedulingProblem descriptionUltimate pit limitsTime-indexed MIP formulations

OPMPSP relaxationsStandard LP relaxationFull relaxation of resource c