1. Proposal abstract 

A crucial problem in the mining industry is to determine the optimal sequence of  extraction of virtual blocks in which the mine has been structured for exploitation.

A Chilean mine, typically can be constituted by more than 1.000.000 blocks. The sequencing models for this structure are very complex, from an Optimization point of view. In fact, these are combinatorial linear models, having a huge number of variables, which means that the exact resolution by standard Optimization techniques is very frequently inefficient.

Two main concepts have to be considered: the optimality of the sequence of extraction and the robustness of the solution. This last aspect is very relevant, due to the variability of prices, minerals grades estimations and other sources of uncertainty, and it deals with a very active field of research in mathematical optimization nowadays. The search for the best sequence of exploitation, in the deterministic case, is our main objective. The sequence problem has been always studied by mathematicians and mine engineers, by discrete models, whose resolution is extremely expensive in terms of computer execution time. New ideas are related to the use of HPC (High Performance Computing) in order to parallelize the resolution of the numerous linear programming subproblems appearing in decomposition or Branch and Bound approaches.

In a first part, we propose to study the resolution of the combinatorial associated problem, in a given time horizon (typically, 20-30 years), by using heuristics and related techniques.

In the second part, our aim is to investigate and characterize the behavior of the sequence of nested pits representing the exploitation , by a continuous mathematical model.

In the third part, we study the relationship between discrete and continuous formulations of open pit mine planning problems: asymptotic analysis of discrete 0-1 models as the number of variables increases and the time step-size goes to 0 as well as direct discretization in time and space of continuous models.

We use different mathematical tools, like Combinatorial Optimization, Heuristics strategies, Optimal Control, Functional Analysis and Convex Optimization. Numerical algorithms will be derived from the theoretical models.

 

2. Project objectives and hypothesis

The aim of the project is to propose and validate discrete and continuous optimization models for the strategic mine planning problem, for the open pit case. For this study, we consider that:

1. Heuristics for the resolution of the combinatorial problem are a very promising approach to solve huge discrete mine planning problems.

2. Good representations of the long term mine planning can be found by Optimization models in continuous functional spaces.

3. These Optimization problems in functional spaces have optimal solutions, that is to say, sequences of feasible profiles in continuous or discrete time, nested in the vertical coordinate.

General Objectives:

1. To characterize and study the open pit mine planning problem, by continuous optimization models. Analytical properties of these models and the corresponding extensions to the real mine planning problem will be analyzed.

2. To propose qualitative interpretations of the solutions, in order to make numerical optimization algorithms applicable for the approximate resolution.

3. To find good heuristics approach (local search of a given feasible solution, for example) for the resolution of the combinatorial associated problem.

Specific Objectives:

1. To prove existence of the solutions for the continuous model and characterize

2. To prove that the final pit problem (unbounded capacity and only one period) contains the optimal solution of the sequenced problem.

3. To propose iterative procedures in functional spaces to find good approximations of the solutions.

4. To propose and implement heuristics procedures for the discrete version of the

 

3. Proposed Research

       See More Details

 

4. Methodology

This project deals with mathematical research applied to a relevant industrial problem: the long term planning of exploitation for open pit mines.

The main mathematical tools to be used in this research are:

•      Optimization theory
•      Functional and convex analysis
•       Convergence of numerical or iterative procedures
•       Combinatorial models and heuristics strategies for resolution

The specific methodology is essentially given in (the previous) Section: Proposed Research (See More Details), which contains the overall project description.

In general terms, our approach to the problem is the following:
For the discrete model

• To study the problem in a separate way, to propose and apply heuristics for an efficient approximate resolution.
• To take limit to derive continuous models.

For  the continuous model

• To study the problem in terms of existence, unicity, duality and sensibility.
• To discretize the problem and establish the connection with the integer traditional models.

 

5. Outreach To Society Activities

Industrial, scientific and social impacts

The optimal sequencing of blocks from a mine is strongly related to the net value of the mine, the more important part of the mine business.

From the point of view of modeling and the resolution by adapted complex algorithms, is a not well solved issue in this industry and this become a very relevant domain for applied research.

The goal of the project is to generate good mathematical models and practical algorithms for real applications to mine planning. The (mathematical) continuous version of the current models could be very useful in inducing and adapting continuous techniques to the discrete model.

Moreover, we have installed in our HPC cluster a web platform for planning services (sequencing) for open pit mines, based on combinatorial optimization models and algorithms based on linear programming and heuristics. Our final aim is to improve the practical performance of this decisional tool and the competiveness of the Chilean mine industry.

The principal investigator is an active member of the Chilean Operations Research and Applied Mathematics community.  He is currently involved in various projects in which the application of Mathematics becomes essential, due the complexity of the challenges. By participating in national and international congresses and seminars, he contribute to disseminate new techniques and to attract new researchers to the field.

 

6. Results 2013-2014

6.1. Secuencia óptima con reserva expuesta 

Determinación de secuencia óptima de pits anidados de extracción, manteniendo en cada etapa una cierta porción de reserva a la vista. Esto es, un conjunto de bloques factibles de extraer al comienzo del periodo, que garantizan un beneficio mínimo a obtener. Esta es una exigencia práctica muy importante, pues permite dar más regularidad al negocio minero, en el sentido del flujo de ingresos y beneficios.

Este trabajo generó un artículo (conjunto con un investigador de la Curtin University, Australia y un estudiante de doctorado de la Universidad de Chile):

Título: Optimizing Open Pit Block Scheduling with Exposed Ore Reserve (Abrir aquí).

Autores: José Saavedra-Rosas, Enrique Jélvez, Jorge Amaya y Nelson Morales,

Está sometido en segunda revisión (junio 2015) para publicación en la revista:

SAIMM, Journal of the Southern African Institute of Mining and Metallurgy (ISI).

6.2. Transición de la explotación de cielo abierto a subterráneo

Se ha trabajado en colaboración con una investigadora de la McGill University (Montreal) y una tesista de ingeniería de minas sobre el problema de encontrar la envolvente óptima de un amina a cielo abierto y una mina relacionada subterránea, de manera de evaluar económicamente los beneficios esperados de la decisión de transición. Esto se expresa en dos documentos:

Primera versión de la memoria de ingeniería de Cecilia Julio:

“Transición de explotación desde minería a cielo abierto a subterránea, bajo incertidumbre geológica”, codirigida por los investigadores de este proyecto.

Artículo aceptado para presentación y Proceedings del congreso APCOM 2015. Este evento se realiza regularmente cada dos años y es el más importante del mundo en materia de modelamiento y uso de cálculos computacionales en la industria minera.

Título: Optimal Economic Envelope of Joint Open-Pit and Underground Mines (abrir aquí)

Autores: Cecilia Julio, Jorge Amaya, Amina Lamghari y Nelson Morales.

 6.3. Diseño de rampas en minería a cielo abierto 

Se ha trabajado en la propuesta de un modelo matemático que permita decidir el diseño de rampas en una mina a cielo abierto. Este es un problema industrial muy relevante y que tiene gran impacto en los costos de acceso y transporte a largo plazo. El documento reportado contiene la primera versión del modelo y durante el año 2015 se extenderá a casos más generales. Las primeras pruebas computacionales muestran la potencia y la aplicabilidad de la idea. El trabajo está en el contexto de la memoria de ingeniero del tesista Emilio Molina.

Título: Modelo Matemático para Diseño de Rampas en Minería a Cielo Abierto (abrir aquí). Autores: Emilio Molina, Jorge Amaya y Nelson Morales.