no. 6
Optimal pits and optimal transportation
Ekeland, Ivar1 ; Queyranne, Maurice2, 3
1 CEREMADE, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris, France.
2 CORE, Université Catholique de Louvain, Voie du Roman Pays 34, 1348 Louvain-la-Neuve, cedex 16, Belgium, France.
3 Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC V6T 1Z2, Canada.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 6, pp. 1659-1670.

In open pit mining, one must dig a pit, that is, excavate the upper layers of ground before reaching the ore. The walls of the pit must satisfy some geomechanical constraints, in order not to collapse. The question then arises how to mine the ore optimally, that is, how to find the optimal pit. We set up the problem in a continuous (as opposed to discrete) framework, and we show, under weak assumptions, the existence of an optimum pit. For this, we formulate an optimal transportation problem, where the criterion is lower semi-continuous and is allowed to take the value +∞. We show that this transportation problem is a strong dual to the optimum pit problem, and also yields optimality (complementarity slackness) conditions.

Reçu le :
DOI : 10.1051/m2an/2015026
Classification : 37A05, 49J20, 49J45, 90C26, 90C35, 90C48
Mots clés : Optimal transportation, optimal pit mine design, Kantorovich duality
Ekeland, Ivar 1 ; Queyranne, Maurice 2, 3

1 CEREMADE, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris, France.
2 CORE, Université Catholique de Louvain, Voie du Roman Pays 34, 1348 Louvain-la-Neuve, cedex 16, Belgium, France.
3 Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC V6T 1Z2, Canada. 
@article{M2AN_2015__49_6_1659_0,
     author = {Ekeland, Ivar and Queyranne, Maurice},
     title = {Optimal pits and optimal transportation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1659--1670},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {6},
     year = {2015},
     doi = {10.1051/m2an/2015026},
     zbl = {1357.37091},
     mrnumber = {3423270},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015026/}
}
TY  - JOUR
AU  - Ekeland, Ivar
AU  - Queyranne, Maurice
TI  - Optimal pits and optimal transportation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 1659
EP  - 1670
VL  - 49
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015026/
DO  - 10.1051/m2an/2015026
LA  - en
ID  - M2AN_2015__49_6_1659_0
ER  - 
%0 Journal Article
%A Ekeland, Ivar
%A Queyranne, Maurice
%T Optimal pits and optimal transportation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 1659-1670
%V 49
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015026/
%R 10.1051/m2an/2015026
%G en
%F M2AN_2015__49_6_1659_0
Ekeland, Ivar; Queyranne, Maurice. Optimal pits and optimal transportation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 6, pp. 1659-1670. doi : 10.1051/m2an/2015026. http://www.numdam.org/articles/10.1051/m2an/2015026/

F. Alvarez, A. Jorge G. Andreas and S. Nikolai, A continuous framework for open pit mine planning. Math. Methods Oper. Res. 73 (2011) 29–54 | DOI | MR | Zbl

D. Bienstock and Z. Mark, Solving LP Relaxations of Large-Scale Precedence constrained problems. Proc. of 14th Conference on Integer Programming and Combinatorial Optimization (IPCO 2010). Vol. 6080 of Lect. Note Comput. Sci. Springer (2010) 1–14. | MR | Zbl

G. Carlier, Duality and Existence for a Class of Mass Transportation Problems and Economic Applications, in Adv. Math. Econ. Springer, Japan (2003) 1–21 | MR | Zbl

I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types. Econ. Theory 42 (2010) 275–315 | DOI | MR | Zbl

D. Espinoza, M. Goycoolea, E. Moreno and A.N. Newman, MineLib: A library of open pit mining problems. Ann. Oper. Res. 206 (2012) 91–114 | MR | Zbl

A. Griewank and S. Nikolai, Duality results for stationary problems of open pit mine planning in a continuous function framework. Comput. Appl. Math. 30 (2011) 197–215. | MR | Zbl

J. Guzmán, Ultimate Pit Limit Determination: A New Formulation for an Old (and Poorly Specified) Problem Workshop on Operations Research in Mining, Viña del Mar, Chile (2008) 10–12.

P. Huttagosol and R.E. Cameron, A Computer Design of Ultimate Pit Limit by Using Transportation Algorithm, in Proc. of the 23rd International Symposium on Applications of Computers in Mining (1992) 443–460.

Th. B. Johnson, Optimum open pit mine production scheduling. Report ORC-68-11, Operations Research Center. University of California Berkeley (1968).

R. Khalokakaie, Computer-aided optimal open pit design with variable slope angles. Ph.D. thesis, University of Leeds (1999).

R. Khalokakaie, P.A. Dowd and R.J. Fowell, Lerchs-Grossmann algorithm with variable slope angles. Mining Technology 109 (2000) 77–85. | DOI

G. Matheron, Paramétrage de contours optimaux. Note géostatistique 128. Fontainebleau. Février (1975).

G. Matheron, Compléments sur le paramétrage de contours optimaux. Note géostatistique 129. Fontainebleau, Février (1975).

N. Morales, Modelos Matemáticos Para Planificación Minera. Engineering thesis. Universidad de Chile, Santiago (2002).

A.M. Newman, E. Rubio, R. Caro, A. Weintraub and K. Eurek, A review of operations research in mine planning. Interfaces 40 (2010) 222–245. | DOI

J.-C. Picard, Maximal closure of a graph and applications to combinatorial problems. Manag. Sci. 22 (1976) 1268–1272 | DOI | MR | Zbl

Strogies, Nikolai, and Andreas Griewank, A PDE constraint formulation of Open Pit Mine Planning Problems. Proc. Appl. Math. Mech. 13 (2013) 391–392 | DOI

D.M. Topkis, Minimizing a submodular function on a lattice. Oper. Res. 26 (1978) 305–321. | DOI | MR | Zbl

C. Villani, Topics in Optimal Transportation. In vol. 58 of Grad. Stud. Math. AMS (2003). | MR | Zbl

Cité par Sources :