Open Pit Mine Production Scheduling - Zuse Institute...

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

Outline

Open Pit Mining Production Scheduling

. 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.

Problem

satisfying

for each block.

Problem

satisfying

for each block.

Outline

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:

∑i∈X wi.

Outline

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

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

Objective function

NetPresentValue (y, z) =

T∑t=1

)t−1∑

aggs k

−mkyk,t +∑

blocks i

pizi,t

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

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

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

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

s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , Txi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T∑

Rizi,t ≤ Pt for t = 1, . . . , T

Outline

Standard LP relaxation

Rizi,t ≤ Pt for t = 1, . . . , T

0 ≤ xi,t ≤ 1 for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N

. typically: interior point outperforms simplex (when solving from scratch)

Standard LP relaxation

Rizi,t ≤ Pt for t = 1, . . . , T

0 ≤ xi,t ≤ 1 for i ∈ N , t = 1, . . . , Txi,0 = 0 for i ∈ N

. typically: interior point outperforms simplex (when solving from scratch)

Outline

Full relaxation of resource constraints

Ri(xi,t − xi,t−1) ≤Mt for 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

max NetPresentValue (x)

PresentValue (x)

xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T

≤ xp

for (i, p) ∈ S

, t = 1, . . . , T∑i

∈ {0, 1} for i ∈ N

, t = 1, . . . , Txi,0 = 0 for i ∈ N

PresentValue (x)

xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T

≤ xp

for (i, p) ∈ S

, t = 1, . . . , T∑i

∈ {0, 1} for i ∈ N

, t = 1, . . . , Txi,0 = 0 for i ∈ N

Outline

Lagrangean relaxation of resource constraints

Rizi,t ≤ Pt for t = 1, . . . , T

[Mt −

∑iRi(xi,t − xi,t−1)

[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

. 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)

[Mt −

[Pt −

∑iRizi,t

xi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T

[Mt −

[Pt −

∑iRizi,t

xi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T

[Mt −

[Pt −

∑iRizi,t

xi,t ≤ xp,t for (i, p) ∈ S, t = 1, . . . , T

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/.

Outline

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, . . .

. . . .

Outline

PCKs in open pit mining

Ri(xi,t − xi,t−1) ≤Mt for t = 1, . . . , T ← add up∑i

Rizi,t ≤ Pt for t = 1, . . . , T

. For each time period, the OPMPSP formulation contains oneprecedence constrained knapsack.

PCKs in open pit mining

s.t. xi,t−1 ≤ xi,t for i ∈ N , t = 2, . . . , T

xi,t ≤ xp,t for (i, p) ∈ S

, t = 1, . . . , T

− xi,t−1)

≤t∑

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

. For each time period, the OPMPSP formulation contains oneprecedence constrained knapsack.

Outline

Precedence Constrained Knapsack Polytope

PCKP := conv{x ∈ {0, 1}N |

∑i∈N

aixi 6 b, (i)

xi 6 xj for (i, j) ∈ S}

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]

. Nothing implemented in MIP solvers

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.

Variable fixing

Notation

. A(i) :=∑

j∈Pred(i) aj

. If A(i) > b, fix xi = 0.

Remarks

Variable fixing

Notation

. A(i) :=∑

j∈Pred(i) aj

. If A(i) > b, fix xi = 0.

Remarks

Variable fixing

Notation

. A(i) :=∑

j∈Pred(i) aj

. If A(i) > b, fix xi = 0.

Remarks

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

Pairwise conflicts

. A(i, j) :=∑

Pairwise conflicts

. A(i, j) :=∑

Clique inequalities

. 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)

Clique inequalities

xi 6 1 (1)

Clique inequalities

xi 6 1 (1)

Clique inequalities

xi 6 1 (1)

Clique inequalities

xi 6 1 (1)

Clique inequalities

xi 6 1 (1)

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),∑

xj 6 xi (2)

is valid for PCKP .

xj 6 xi (2)

is valid for PCKP .

xj 6 xi (2)

is valid for PCKP .

xj 6 xi (2)

is valid for PCKP .

xj + (1− xi) 6 1 (2)

is valid for PCKP .

xj + x′i 6 1 (2)

is valid for PCKP .

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.

j∈C xj > 1, or

j∈C xj > xi

j∈C xj > 1, or

j∈C xj > xi

Summary

Open Pit Mine Production SchedulingSome IP modelling and solution techniques

Ambros M. Gleixner

Zuse Institute Berlin

09/28/2009

Open Pit Mine Production Scheduling - Zuse Institute...

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