Progressive hedging applied as a metaheuristic to schedule production in open-pit mines accounting for reserve uncertainty

AbstractScheduling production in open-pit mines is characterized by uncertainty about the metal content of the orebody (the reserve) and leads to a complex large-scale mixed-integer stochastic optimization problem. In this paper, a two-phase solution approach based on Rockafellar and Wets’ progressive hedging algorithm (PH) is proposed. PH is used in phase I where the problem is first decomposed by partitioning the set of scenarios modeling metal uncertainty into groups, and then the sub-problems associated with each group are solved iteratively to drive their solutions to a common solution. In phase II, a strategy exploiting information obtained during the PH iterations and the structure of the problem under study is used to reduce the size of the original problem, and the resulting smaller problem is solved using a sliding time window heuristic based on a fix-and-optimize scheme. Numerical results show that this approach is efficient in finding near-optimal solutions and that it outperforms existing heuristics for the problem under study

Méthode heuristique d’optimisation stochastique de la planification minière et positionnement des résidus miniers dans la fosse

RÉSUMÉ : La planification minière à long terme est essentielle afin d’estimer la viabilité d’un projet, d’obtenir les investissements nécessaires et d’optimiser les ressources disponibles. La recherche opérationnelle est en mesure de répondre efficacement à ce problème, plusieurs modèles mathématiques de programmation linéaire mixte ont été développés. La principale source d’incertitude, encore très peu considérée conventionnellement, est géologique. Pour la prendre en compte, des simulations conditionnelles, représentations équiprobables du gisement, peuvent être utilisées comme données d’entrée à un modèle stochastique en nombres entiers. Ainsi, l’objectif est de maximiser la valeur présente nette moyenne tout en proposant un ordonnancement de la production robuste à l’incertitude. En ajoutant un nombre conséquent de blocs, plusieurs périodes et destinations ainsi que de nombreuses contraintes opérationnelles, les modèles deviennent trop complexes à résoudre de manière exacte avec un solveur. Des méthodes heuristiques doivent alors être envisagées pour obtenir la meilleure solution possible en un temps réduit. Le travail exposé dans ce mémoire est composé de deux parties correspondant à deux différents articles. La première partie présente la résolution d’un modèle stochastique d’optimisation d’une mine à ciel ouvert à l’aide d’une nouvelle méthode heuristique. Sont tout d’abord proposées deux méthodes d’accélération : une relaxation partielle de la binarité des variables d’extraction en utilisant la structure particulière du modèle et les fortes relations entre variables et une convergence du modèle relaxé vers une solution quasi-binaire. Ensuite, un algorithme de tri topologique stochastique est proposé afin d’obtenir rapidement une solution complètement binaire à partir des résultats issus des stratégies d’accélération précédentes. Les résultats obtenus, testés sur un cas réel, sont concluants par leur rapidité et gap par rapport à la solution optimale. La deuxième partie modélise le stockage des résidus miniers et stériles au sein de la fosse au fur et à mesure de l’exploitation. Cette idée, souhaitée par le partenaire industriel lors de l’étude de faisabilité, permet de s’affranchir d’un espace de stockage limité autour de la fosse, de réduire l’impact environnemental de l’exploitation et de diminuer les coûts de remaniement lors de la réhabilitation finale du site. Cette fois, une méthode heuristique de fenêtre de temps sur un horizon fuyant a été utilisée pour résoudre le modèle. Les résultats sont prometteurs puisque l’impact de la disposition de matériel dans la fosse lors de l’exploitation ne dégrade la solution initiale que de 1.77%.----------ABSTRACT : Long-term mine planning is an essential step in order to estimate the viability of a project, to attract investments and to optimize available resources. Operations research is well suited to assess this kind of problem, several mixed integer programming models have been developed over the last decades. Even if still not conventionally considered, the geology is the main source of uncertainty in such a model. To consider it properly, a set of equiprobable conditional simulations of the deposit are used as input in a stochastic integer programming model. The objective is then to maximize the expected net present value while having a robust production schedule to the geological uncertainty. When many blocks are considered but also several destinations and operational constraints, the problem becomes too complex to solve by a general purpose solver. If an exact method is not conceivable, heuristic methods must be used to obtain the best solution as possible in a limited time. The study presented in the thesis is composed of two parts corresponding to two articles. The first one presents a new heuristic method to solve a stochastic open pit mine production scheduling problem. First, two acceleration strategies are proposed: a partial relaxation of the binarity of the extraction variables using the special structure of the model and the strong inter-correlations of the variables; a convergence of the fully relaxed model toward a quasi-binary solution. Then, a stochastic topological sorting algorithm is proposed to quickly obtain a binary solution from the result of the previous acceleration strategies. Applied on a real case study, the results are interesting for their rapidity and their gap to the optimality. The second part establishes a model to store tailings and waste materials directly inside the pit during the operations. This idea was raised in the feasibility of the industrial partner to overpass a limited space for eternal stockpiles, to reduce the environmental impact and the re-handling costs of the rehabilitation phase. This time, to solve the problem, a sliding window time heuristic method was used and the results are promising: the Cplex objective function is only 1.77% lower while considering the in-pit storage and the heuristic method than the initial model solved with an exact method

A stochastic optimization method with in-pit waste and tailings disposal for open pit life-of-mine production planning

Environmental responsibility and the sustainable development of mineral resources are a topic of critical importance to the mining industry and at the same time relate to operational and rehabilitation costs to be considered in technical studies. Open pit mining operations impact their local environment in terms of their modification of the landscape and local ecosystems. Many of these impacts are the result of the transportation of large volumes of materials mined and shifted from and to different locations. External stockpiles and waste dumps occupy considerable space as well as involve substantial transportation costs to move materials from open pits to stockpiles and then move them back to the pit for rehabilitation after the end of exploitation. Depending on the shape of the deposit and the intended design of the pit, a desirable option may be to place it directly in the free spaces within the pit, instead of storing all waste and tailings materials in stockpiles and/or waste/tailings dumps. This paper presents a new mathematical formulation integrating to life-of-mine planning and the maximization of net present value, with the related waste and tailings disposal kept within the mined-out parts of a pit, using a stochastic integer program that manages geological uncertainty including metal grades, material types and related chemical compositions. In addition to the traditional variables related to the materials being extracted from the ground in the form of mining blocks, strips of ground following the dip of a pit are considered within the pit as decision variables, and the optimization process aims to optimally define both the sequence of extraction of mining blocks and the reservation of strips needed to store waste materials. An application at an iron ore mine demonstrates the feasibility, applied aspects and advantages of the proposed method

Um comparativo de metodologias no planejamento de lavra : sequenciamento direto de blocos vs. planejamento tradicional.

Programa de P?s-Gradua??o em Engenharia Mineral. Departamento de Engenharia de Minas. Escola de Minas, Universidade Federal de Ouro Preto.Desde o desenvolvimento do algoritmo de Lerchs-Grossmann em 1965, consolidou-se o processo do planejamento de lavra tradicional, que consiste nas seguintes etapas: determina??o dos limites da cava final, sele??o de pushbacks intermedi?rios e agendamento da produ??o. Apesar desse m?todo ser muito utilizado na ind?stria e aceito pela comunidade cient?fica, o algoritmo n?o considera o fator do custo de oportunidade, ao se basear em blocos valorados com o pressuposto de que s?o lavrados todos no per?odo de tempo atual. Essa metodologia consiste de etapas que, mesmo otimizadas separadamente, podem n?o o ser quando em conjunto. Com os avan?os tecnol?gicos computacionais, a t?cnica do sequenciamento direto de blocos vem crescendo. Esta t?cnica integra todas as fases e soluciona o problema do planejamento de lavra como um todo, melhorando os resultados econ?micos do projeto. BOS2M ? uma dessas ferramentas, que define ao mesmo tempo, em um modelo de blocos, quais deles extrair, quando e como process?-los, respeitando as restri??es existentes. Um estudo comparativo das duas metodologias aplicadas a uma mina real brasileira de ferro ? apresentado, com as an?lises das vantagens e limita??es de cada uma. Tamb?m ? apresentado a operacionaliza??o dos quinqu?nios e um plano anual do primeiro per?odo quinquenal de cada projeto. De modo geral, a dificuldade de implementa??o de todas as restri??es operacionais no sequenciamento direto de blocos consiste, atualmente, no maior obst?culo para a maior inser??o dessa metodologia na ind?stria. O sequenciamento anual do primeiro quinqu?nio, realizado como tentativa para tornar esse planejamento operacional, comprova que os resultados econ?micos provenientes dessa t?cnica s?o bem superiores ao do m?todo convencional de planejamento.Since the development of the Lerchs-Grossmann algorithm in 1965, the process of traditional mining planning has been consolidated, and consists of the following steps: determination of the ultimate pit limit, intermediate pushback selection and production scheduling. Although this method is widely used in the industry and accepted by the scientific community, the algorithm does not consider the opportunity cost, as it is based on blocks evaluated on the assumption of being mined in the current period of time. This methodology consists of steps that, even if considered individually optimal, may not be when put together. With the technological advances in computation, the technique of direct block sequencing has been growing. This technique integrates all phases and solves the problem of mine planning as a whole, improving the economic results of the project. BOS2M is one of these tools, which defines simultaneously, in a block model, which ones to extract, when and how to process them, respecting the existing constraints. A comparative study of the two methodologies applied to a real Brazilian iron ore mine is presented in this study, with analyzes of the advantages and limitations of each one. It is also presented the operationalization of the quinquennials and an annual plan of the first five-year period of each project. In general, the difficulty of implementing all operational restrictions in direct block sequencing is currently the biggest obstacle to the greater insertion of this methodology in the industry. The annual sequencing of the first five-year period, performed as an attempt to make this planning operational, proves that the economic results from this technique are far superior to the conventional planning method

Optimised decision-making under grade uncertainty in surface mining

Mining schedule optimisation often ignores geological and economic risks in favour of simplistic deterministic methods. In this thesis a scenario optimisation approach is developed which uses MILP optimisation results from multiple conditional simulations of geological data to derive a unique solution. The research also generated an interpretive framework which incorporates the use of the Coefficient of Variation allowing the assessment of various optimisation results in order to find the solution with the most attractive risk-return ratio