Optimal pits and optimal transportation

(1)

Optimal pits and optimal transportation

Ivar Ekeland

¹

Maurice Queyranne

²

1CEREMADE, Universit´e Paris-Dauphine

2CORE, U.C. Louvain, and Sauder School of Business at UBC

CESAME Seminar in Systems and Control, UCL November 18, 2014

(2)

Introduction: Open Pit Mining A Continuous Space Model

An Optimal Transportation Problem The Kantorovich Dual

Elements ofc-Convex Analysis Solving the Dual Problem

Solving the Optimum Pit Problem Perspectives

(3)

Open Pit Mining

To dig a hole in the ground and excavate valuable minerals

Super Pit gold mine, Kalgoorli, Western Australia

Chuquicamata copper mine, Chile (4.3 km×3 km×900 m)

(5)

Open Pit Mining

To dig a hole in the ground and excavate valuable minerals

Diavik diamond mine, Canada

(6)

Open Pit Mining

To dig a hole in the ground and excavate valuable minerals

(7)

Open Pit Mining

To dig a hole in the ground and excavate valuable minerals

(8)

To dig a hole in the ground and excavate valuable minerals

(9)

Mining Processes

(10)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

I Where,when and what to excavate, to process subject to capacity and other resource constraints, and the time value of money (cash flows)

I Process choices, major equipment decisions

Mine production planning problem (decisions over time)

3.

Detailed operations planning

I Detailed mine design: benches, routes, facilities

I Operations scheduling, flows of materials, etc.

4.

Execution. . .

(11)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

I Where to dig? How deep? What to process?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(12)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

subject to capacity and other resource constraints, and the time value of money (cash flows)

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(13)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(14)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

subject to capacity and other resource constraints, and the time value of money (cash flows)

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(15)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(16)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(17)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(18)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(19)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(20)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

I Operations scheduling, flows of materials, etc. 4.

Execution. . .

(21)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

I Operations scheduling, flows of materials, etc. 4.

Execution. . .

(22)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

(23)

Open Pit Mine Planning

1.

Project evaluation: is it worth investing?

Optimum open pit problem (determining ultimate pit limits)

2.

Rough-cut planning: take time into account

Mine production planning problem (decisions over time)

3.

Detailed operations planning

4.

Execution. . .

(24)

Optimum Open Pit: Slope Constraints

The pit walls cannot be too steep, else they may collapse

Angouran lead & zinc mine, Iran (25 million tons rock slide, 2006)

Bingham Canyon copper mine, Utah (massive landslide, 10 April 2013)

(25)

Optimum Open Pit: Slope Constraints

The pit walls cannot be too steep, else they may collapse

West Angelas iron ore mine, Western Australia

(26)

Optimum Open Pit: Slope Constraints

The pit walls cannot be too steep, else they may collapse

(27)

Optimum Open Pit: Slope Constraints

The pit walls cannot be too steep, else they may collapse

(28)

The pit walls cannot be too steep, else they may collapse

(29)

Discretization: Block Models

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I typically, 1:5 or 1:9 pattern

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut) discrete optimization problem

I

implemented in commercial software (Whittle, Geovia)

(30)

Discretization: Block Models

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut) discrete optimization problem

I

implemented in commercial software (Whittle, Geovia)

(31)

Discretization: Block Models

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut) discrete optimization problem

I

implemented in commercial software (Whittle, Geovia)

(32)

Discretization: Block Models

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut)

discrete optimization problem

(33)

Discretization: Block Models

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut) discrete optimization problem

I

implemented in commercial software (Whittle, Geovia)

(34)

Discretization: Block Models

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut)

discrete optimization problem

(35)

Discretization: Block Models

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut) discrete optimization problem

I

implemented in commercial software (Whittle, Geovia)

(36)

[Lerchs and Grossmann, 1965]

Divide the volume of interest into 3D blocks

I

typically rectangular, with vertical sides

I

the slope constraints are approximated by precedence constraints

I

it is easy to determine the net profit from excavating, and possibly processing, the block itself

Leads to a nicely structured (dual network flow, minimum cut) discrete optimization problem

I

implemented in commercial software (Whittle, Geovia)

(37)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I production planning models even larger (×number of periods)

I

the real problem is, to a large extent, continuous:

I ore density and rock properties tend to vary continuously

I their distributions are estimated (“smoothed”) from sample (drill hole) data and other geological information

I

block precedences only roughly model the slope constraints

Earlier continuous space models:

I

Matheron (1975)

(focus on “cutoff grade” parametrization)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

calculus of variations model in functional space

I determine optimum depthφ(y)under each surface pointy s.t. bounds on the derivative ofφ(wall slope constraints)

All these continuous space approaches suffer from lack of convexity

I how to deal withlocal optima?

.

(38)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

s.t. bounds on the derivative ofφ(wall slope constraints)

All these continuous space approaches suffer from lack of convexity

.

(39)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(40)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(41)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(42)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(43)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints

Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(44)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints

Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(45)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(46)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(47)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(48)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(49)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(50)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

.

(51)

Discretized vs Continuous Models?

Discretized (block) models:

I

are very large (100,000s to millions of blocks)

I

the real problem is, to a large extent, continuous:

I

block precedences only roughly model the slope constraints Earlier continuous space models:

I

Matheron (1975)

I

Morales (2002), Guzm´ an (2008)

I

Alvarez & al. (2011) (also, Griewank & Strogies, 2011, 2013):

All these continuous space approaches suffer from lack of convexity

.

(52)

Introduction: Open Pit Mining

A Continuous Space Model

An Optimal Transportation Problem The Kantorovich Dual

Elements of

c-Convex Analysis

Solving the Dual Problem

Solving the Optimum Pit Problem

Perspectives

(53)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

e.g.,E=A×[h1, h2], whereA⊂R² is theclaim

[h1, h2]is the elevation or depth range

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

x⁰∈Γ(x)andx⁰⁰∈Γ (x⁰)

=⇒ x⁰⁰∈Γ(x)

I reflexive: x∈Γ(x)

I closed graph: {(x, y) :x∈E, y∈Γ(x)}is closed

a pit F is a measurable subset of E closed under Γ: Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(x)dxnet profit from volume elementdx=dx1dx2dx3 atx

I g(F):=R

Fg(x)dx total net profit from pitF

I assumeR

Emax{0, g(x)}dx >0(there is some profit to be made)

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(54)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

I closed graph:{(x, y) :x∈E, y∈Γ(x)}is closed

a pit F is a measurable subset of E closed under Γ: Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):= _Fg(x)dx total net profit from pitF

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(55)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ: Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(56)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ: Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(57)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ: Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(58)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ: Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(59)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ: Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(60)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ:

Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(61)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ:

Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(62)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ:

Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I assume _Emax{0, g(x)}dx >0(there is some profit to be made)

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(63)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ:

Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(64)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ:

Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

(65)

Open Pit Problem: a Continuous Space Model

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ:

Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(66)

A general model [Matheron 1975]: Given

I

compact

E

⊂

R³

: the domain to be mined

I

map

Γ

: E E: extracting x requires extracting all of Γ(x)

I transitive:

=⇒ x⁰⁰∈Γ(x)

a pit F is a measurable subset of E closed under Γ:

Γ(F ) = F where

Γ (F)

:= ∪

_x∈F

Γ(x)

I

continuous function

g

: E →

R

I g(F):=R

I assumeR

Optimum pit problem: find

F^∗

∈ arg max{g(F ) : F is a pit}

(67)

A Profit Allocation Model

I

Let

E⁺

:= {g(x) > 0} and

E⁻

:= {g(x) ≤ 0} (compact sets)

I

Add a sink

ω

and a source

α

I unallocated costs of unexcavated points will be paid byα

I

Let

X

:= E

⁺

∪ {α} and

Y

:= E

⁻

∪ {ω} (also compact)

I

endowed with non-negative measures

µ

and

ν

defined by µ ({α}) =

R

E⁻

|g(z)| dz µ|

_E+

= g(z)dz ν ({ω}) =

R

E⁺

g(z)dz ν|

_E⁻

= |g(z)| dz

I

Profit allocations are allowed

I from every profitablex∈E⁺ to everyy∈Γ(x)∩E⁻

I from sourceαto ally∈E⁻ (unpaid costs)

I from allx∈E⁺ to sinkω (unallocated, or “excess” profits)

These restrictions will be modelled by a “transportation” (or

allocation) cost function

c

: X × Y −→

R

(69)

A Profit Allocation Model

I

Let

E⁺

:= {g(x) > 0} and

E⁻

:= {g(x) ≤ 0} (compact sets)

I

Add a sink

ω

I unallocated profits from excavated points will be sent toω

and a source

α

I

Let

X

:= E

⁺

∪ {α} and

Y

:= E

⁻

∪ {ω} (also compact)

I

endowed with non-negative measures

µ

and

ν

defined by µ ({α}) =

R

E⁻

|g(z)| dz µ|

_E+

= g(z)dz ν ({ω}) =

R

E⁺

g(z)dz ν|

_E⁻

= |g(z)| dz

I

Profit allocations are allowed

These restrictions will be modelled by a “transportation” (or

allocation) cost function

c

: X × Y −→

R

(70)

A Profit Allocation Model

I

Let

E⁺

:= {g(x) > 0} and

E⁻

:= {g(x) ≤ 0} (compact sets)

I

Add a sink

ω

and a source

α

I

Let

X

:= E

⁺

∪ {α} and

Y

:= E

⁻

∪ {ω} (also compact)

I

endowed with non-negative measures

µ

and

ν

defined by µ ({α}) =

R

E⁻

|g(z)| dz µ|

_E+

= g(z)dz ν ({ω}) =

R

E⁺

g(z)dz ν|

_E⁻

= |g(z)| dz

I

Profit allocations are allowed

These restrictions will be modelled by a “transportation” (or

allocation) cost function

c

: X × Y −→

R

(71)

A Profit Allocation Model

I

Let

E⁺

:= {g(x) > 0} and

E⁻

:= {g(x) ≤ 0} (compact sets)

I

Add a sink

ω

and a source

α

I

Let

X

:= E

⁺

∪ {α} and

Y

:= E

⁻

∪ {ω} (also compact)

I

endowed with non-negative measures

µ

and

ν

defined by µ ({α}) =

R

E⁻

|g(z)| dz µ|

_E+

= g(z)dz ν ({ω}) =

R

E⁺

g(z)dz ν|

_E⁻

= |g(z)| dz

I

Profit allocations are allowed

These restrictions will be modelled by a “transportation” (or

allocation) cost function

c

: X × Y −→

R

(72)

A Profit Allocation Model

I

Let

E⁺

:= {g(x) > 0} and

E⁻

:= {g(x) ≤ 0} (compact sets)

I

Add a sink

ω

and a source

α

I

Let

X

:= E

⁺

∪ {α} and

Y

:= E

⁻

∪ {ω} (also compact)

I

endowed with non-negative measures

µ

and

ν

defined by µ ({α}) =

R

E⁻

|g(z)| dz µ|

_E+

= g(z)dz ν ({ω}) =

R

E⁺

g(z)dz ν|

_E⁻

= |g(z)| dz

I

Profit allocations are allowed

These restrictions will be modelled by a “transportation” (or

allocation) cost function

c

: X × Y −→

R

Optimal pits and optimal transportation