Integrated Parametric Graph Closure and Branch-and-Cut Algorithm for Open Pit Mine Scheduling under Uncertainty

Open pit mine production scheduling is a computationally expensive large-scale mixed-integer linear programming problem. This research develops a computationally efficient algorithm to solve open pit production scheduling problems under uncertain geological parameters. The proposed solution approach for production scheduling is a two-stage process. The stochastic production scheduling problem is iteratively solved in the first stage after relaxing resource constraints using a parametric graph closure algorithm. Finally, the branch-and-cut algorithm is applied to respect the resource constraints, which might be violated during the first stage of the algorithm. Six small-scale production scheduling problems from iron and copper mines were used to validate the proposed stochastic production scheduling model. The results demonstrated that the proposed method could significantly improve the computational time with a reasonable optimality gap (the maximum gap is 4%). In addition, the proposed stochastic method is tested using industrial-scale copper data and compared with its deterministic model. The results show that the net present value for the stochastic model improved by 6% compared to the deterministic model

Production scheduling and mine fleet assignment using integer programming

Production Scheduling, extraction sequence of mining blocks in different production periods to maximize profit over the life of the mine and subjected to different constraints, is an important aspect of any mining activity. Mine production scheduling problem can be solved using various approaches, but the best approach is one which can give an optimal result. Production scheduling solely cannot result in a proper planning thus, fleet assignment problem needs to be incorporated into production scheduling problem to have a realistic mine plan. Proper fleet assignment ensures that the fleet is not under or over utilized. Fleet assignment problem is integer type programming since, size of fleet cannot be a floating number. In this thesis, production scheduling and fleet assignment problem are solved using branch and cut algorithm. Production schedule for 4736 blocks from a case study of coal mine is done with a production period of 5 years. Solution time for solving the production scheduling problem was 48.14 hours with an NPV value of Rs 4.45938x1011. Short terms production scheduling is done for one year and the NPV value obtained was Rs 7.59796x1010 with a solution time of 57.539 minutes. Fleet assignment is done for first year and is observed that the size of dumper fleet can be reduced to 30 thus saving huge amount of initial capital investment

Otimização do teor de corte e do sequenciamento de minas subterrâneas

Orientador: Priscila Cristina Berbert RampazzoDissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação CientíficaResumo: Métodos de lavra subterrânea são aplicados na extração de vários metais e minerais. O planejamento de métodos subterrâneos difere do planejamento de métodos de superfície pelo fato de que não é necessário extrair todas as áreas de produção dentro dos limites econômicos finais para se ter uma sequência factível, ou seja, nos métodos subterrâneos é fisicamente possível que algumas áreas permaneçam não lavradas mesmo estando dentro do limites econômicos finais. O planejamento estratégico é a área central do planejamento de longo prazo de uma mina e visa definir estratégias de escala de produção, métodos de lavra e de beneficiamento mineral, selecionar as áreas que serão lavradas e otimizar a sequência de lavra destas áreas de produção. Para garantir a viabilidade econômica do empreendimento, o planejamento estratégico deve considerar as características-chave dos empreendimentos de mineração, que são: a necessidade de capital intensivo, o longo período de retorno do investimento e o ativo (reserva) limitado. Essas características devem ser consideradas durante o processo de valoração de um empreendimento mineiro, que normalmente é feito através do cálculo do VPL, valor presente líquido. Dentre as principais alavancas do planejamento estratégico, o teor de corte utilizado na seleção dos blocos que serão lavrados e o sequenciamento de mina são os que geram maior número de opções, fazendo com que avaliações de cenários demandem muito tempo e se tornem inviáveis na prática dada a necessidade de respostas rápidas para tomadas de decisão. Neste trabalho, três diferentes modelos matemáticos são propostos para abordar, de forma conjunta, o problema da seleção dos blocos de lavra de uma mina subterrânea e a otimização do sequenciamento destes blocos. Tais modelos consideram o VPL como principal objetivo a ser maximizado e resultam no uso do teor de corte como fator que equilibra as capacidades de produção dos diferentes estágios de um sistema de mineração. A abordagem matemática adapta a modelagem clássica de problemas de sequenciamento considerando os blocos de lavra como tarefas e as atividades de escavação de galerias (desenvolvimento de acessos) e de produção de minério (lavra) como máquinas. Os modelos propostos são testados com base em casos reais, utilizando-se métodos de solução exata e um algoritmo genético. Os resultados computacionais mostram que o algoritmo genético é mais eficiente do que os métodos exatos, sobretudo para instâncias maiores, mais próximas da realidadeAbstract: Underground mining methods are used at the extraction of many metals and minerals. Underground mining planning differs from surface mining planning mainly because, in the first case, it is not necessary to extract all mining blocks within the ultimate economic limits to have a feasible sequence, i.e., it is physically possible to an underground mine to have some areas left \textit{in situ} even if they are inside the ultimate economic limits. Strategic planning is the core area of long-term mining planning and aims to define the scale of production, mining and processing methods, to select areas that will be mined, and to optimize the mining sequence. To guarantee the economic feasibility of a mining asset, strategic planners must also consider the key aspects of mining businesses, which are: capital-intensive requirements, long-term payback, and limited asset (reserves) life. These characteristics must be considered during the valuation process of a mining asset, which is normally conducted through NPV, net present value, calculations. Among the main strategic planning levers, cut-off grades (used at the selection of blocks that will be mined) and the mine sequencing are the ones that generate the greatest number of options. As scheduling multiple scenarios requires a great deal of time, this is infeasible in real situations given the need for quick responses. In this dissertation, three mathematical models are proposed to tackle, at the same time, two problems: the selection of the mining blocks in an underground mine, and the optimization of their sequence. These models consider NPV as the main objective to be maximized and result in using cut-off grades as a factor that balances the main capabilities of a mining system. The mathematical approach adapts classical scheduling models considering mining blocks as jobs; and tunnels excavation (access development) and ore production (mining) activities as machines. The proposed models are tested, with real cases, using exact-solution methods and a genetic algorithm. Results show that the genetic algorithm is more efficient than the exact methods, especially for greater instances that are similar to real problemsMestradoMatematica Aplicada e ComputacionalMestre em Matemática Aplicada e Computaciona

Development and evaluation of models for assessing geochemical pollution sources with multiple reactive chemical species for sustainable use of aquifer systems: source characterization and monitoring network design

Michael designed a groundwater flow and reactive transport optimization model. He applied this model to characterize contaminant sources in Australia's first large scale uranium mine site in the Northern Territory. He identified the contamination sources to the groundwater system in the area. His findings will assist planning actions and steps needed to implement the mitigation strategy of this contaminated aquifer

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

Modelling the tactical decisions for open-pit mines

Open-pit deposits are often characterized by a stack of layers of different geological nature. Some layers are worthless while the ore of the others is of a varying economic value depending on grade. To reach a layer, it is necessary to have first removed the upper layers above the extraction zone. This action results in uncovering the layer in this particular place and in facilitating access to the layers below. This process involves a series of 2 to 7 operations; each one is performed by a machine, some of which are able to perform up to 3 different operations. Ensuring the consistency of mining extraction scheduling over a few months, in order to meet known or forecast demand, is a challenging task. A mining extraction model based on mathematical programming has been proposed but it is hardly usable due to its size. A Discrete Events Simulator modelling is currently being tested to measure the impact of dynamic rules used to allocate the machines and select the target mining area

Stochastic-optimization of equipment productivity in multi-seam formations

Short and long range planning and execution for multi-seam coal formations (MSFs) are challenging with complex extraction mechanisms. Stripping equipment selection and scheduling are functions of the physical dynamics of the mine and the operational mechanisms of its components, thus its productivity is dependent on these parameters. Previous research studies did not incorporate quantitative relationships between equipment productivities and extraction dynamics in MSFs. The intrinsic variability of excavation and spoiling dynamics must also form part of existing models. This research formulates quantitative relationships of equipment productivities using Branch-and-Bound algorithms and Lagrange Parameterization approaches. The stochastic processes are resolved via Monte Carlo/Latin Hypercube simulation techniques within @RISK framework. The model was presented with a bituminous coal mining case in the Appalachian field. The simulated results showed a 3.51% improvement in mining cost and 0.19% increment in net present value. A 76.95yd³ drop in productivity per unit change in cycle time was recorded for sub-optimal equipment schedules. The geologic variability and equipment operational parameters restricted any possible change in the cost function. A 50.3% chance of the mining cost increasing above its current value was driven by the volume of material re-handled with 0.52 regression coefficient. The study advances the optimization process in mine planning and scheduling algorithms, to efficiently capture future uncertainties surrounding multivariate random functions. The main novelty includes the application of stochastic-optimization procedures to improve equipment productivity in MSFs --Abstract, page iii

Integer linear programs and heuristic solution approaches for different planning levels in underground mining

Natürlich vorkommende Mineralien werden seit Tausenden von Jahren aus der Erde gefördert. Im Bergbau wird Operations Research (OR) hauptsächlich angewendet, um die Materialgewinnung zu vereinfachen und die Ressourcen für die Gewinnung effizienter zuzuordnen. Optimierungsprobleme im Bergbau werden üblicherweise nach ihrem Planungshorizont eingeordnet. Dabei werden Layout- und Designprobleme auf strategischer, Produktions- und Planungsprobleme auf taktischer und Ressourcenzuordnungsprobleme auf operativer Planungsebene behandelt. In dieser kumulativen Dissertation betrachten wir eine der größten deutschen Kalibergwerke und befassen uns mit drei Optimierungsproblemen auf drei verschiedenen Planungsebenen. Zunächst betrachten wir eine sogenannte „Gewinnungsprogrammplanung“ für einen Planungshorizont von einem Monat auf taktischer Planungsebene. Die betrachtete qualitätsorientierte Zielfunktion zielt auf eine gleichmäßige Kalisalzgewinnung hinsichtlich des beinhalteten Kaliums ab. Da die Menge der Gesamtförderung a priori unbekannt ist, kann die in der Gesamtförderung enthaltene Kaliummenge mithilfe nicht-linearer Nebenbedingungen in der mathematischen Formulierung bestimmt werden. Die Herausforderung besteht in der Linearisierung der entsprechenden Nebenbedingungen, damit ein gemischt ganzzahliges lineares Programm eingeführt werden kann. Darüber hinaus schlagen wir eine Heuristik vor, welche mindestens eine zulässige Lösung für realitätsnahe Probleminstanzen innerhalb eines angemessenen Zeitraums findet. Die Performanceanalyse an 100 zufällig generierten Probleminstanzen zeigt, dass eine subtile Kombination des vorgeschlagenen mathematischen Programms mit der eingeführten Heuristik nahezu optimale Lösungen für praxisrelevante Probleme findet. Als Nächstes betrachten wir eine „Grobplanung des Maschineneinsatzes“ innerhalb eines Planungshorizonts von einer Woche, welche zwischen der taktischen und der operativen Planungsebene eingeordnet werden kann und untersucht, ob die Ergebnisse der Gewinnungsprogrammplanung für die erste Woche des folgenden Monats umgesetzt werden können. Hierzu wird ein Maschinenplanungsproblem zur Minimierung des maximalen Fertigstellungszeitpunkts berücksichtigt. Wir stellen ein gemischt ganzzahliges lineares Programm vor, welches bestimmte Umstände in einem untertägigen Bergwerk wie die Wiederholung der Erstfreigabe berücksichtigt. Die größte Herausforderung besteht darin, einen Lösungsansatz zu entwickeln, der nahezu optimale Lösungen für große Probleminstanzen findet. Also wird eine Heuristik vorgeschlagen, der absichtliche Verzögerungen von Jobs vor Bearbeitungsstufen einbezieht, d. h. sogenannte aktive Pläne erzeugt. Die Performanceanalyse zeigt, dass kleine Probleminstanzen mit CPLEX optimal gelöst werden können. Bei größeren Instanzen liefert die vorgeschlagene Heuristik die besten Ergebnisse. Schließlich wird auf der operativen Planungsebene eine „Feinplanung des Maschinen- und Personaleinsatzes“ berücksichtigt. Das betrachtete Problem verfolgt einen gleichmäßigen Fortschritt im untertägigen Bergwerk innerhalb einer Arbeitsschicht. Um realistische Lösungen zu erstellen, müssen verschiedene Arten von Rüstzeiten in Betracht gezogen werden, die abhängig von der Bearbeitungsreihenfolge der Operationen an Maschinen und Arbeitern entstehen. Die größte Herausforderung besteht darin, die spezifischen Umstände einer Arbeitsschicht mathematisch darzustellen, z. B. die Berücksichtigung der Pausen der Mitarbeiter für eine eventuelle Verzögerung der Bearbeitungszeit, das Bestimmen des bearbeiteten Prozentsatzes eines Jobs während einer Arbeitsschicht, die Berechnung der Entfernungs- und Umrüstzeiten usw. Wir stellen eine Heuristik vor, die aus zwei Schritten besteht. Im ersten Schritt wird eine Relaxation des Problems unter Einhaltung einen Teil der genannten Nebenbedingungen gelöst. Die gefundene, typischerweise unzulässige Lösung wird im zweiten Schritt durch Einfügen der vernachlässigten Zeiten repariert. Die Ergebnisse zeigen, dass die vorgeschlagene Heuristik für 70 Prozent der realitätsnahen Probleminstanzen eine bessere Lösung als eine bestehende Heuristik finden kann. Anschließend formulieren wir ein neues, kompaktes, gemischt ganzzahliges lineares Programm, das mithilfe von TSP-Variablen alle Problemspezifikationen berücksichtigt. Wir zeigen, dass das vorgeschlagene gemischt ganzzahlige lineare Programm die vorgeschlagene zweistufige Heuristik erheblich übertrifft.Humans have been extracting naturally occurring minerals from the earth for thousands of years. In mining, operations research (OR) has been mainly used to help the mine planners decide how the material can be extracted and what to do with the material removed, what kind of resources to use for the extraction, and how to allocate the resources. It is very widespread to classify decision problems according to their time horizons, where 1. layout and design problems, 2. production and scheduling problems, and 3. operational equipment allocation problems are considered on strategic, tactical, and operational planning levels, respectively. In this cumulative dissertation thesis, we consider one of the biggest German potash mines and address three optimization problems on three different planning levels. First, we consider a so-called “extraction program planning” for a time horizon of one month on the tactical planning level. The related quality-oriented objective function aims at an even extraction of potash regarding the potassium content. For mathematically formulating the objective function, the amount of potassium contained in the output must be determined. Since the amount of total output is a priori unknown, the potassium amount can be determined primarily using non-linear constraints. The principal challenge is the linearization of the corresponding constraints to introduce a mixedinteger linear program with a quality-related objective function. We also propose a heuristic solution procedure that finds for realistically-sized problem instances at least one feasible solution within a reasonable amount of time. The performance analysis conducted on 100 randomly generated problem instances shows that a sophisticated combination of the proposed mixed-integer linear program and the introduced heuristic approach finds high-quality, near-optimal solutions for practice-relevant problems. Next, we deal with a “preliminary (conceptual) planning of machines” within a time horizon of one week. That problem can be classified between the tactical and operational planning levels and investigates whether the results of the extraction program planning can be implemented for the first week of the following month. For this purpose, a machine scheduling problem to minimize the makespan is taking into account. We propose a mixed-integer linear program considering particular circumstances in an underground mine, e.g., reentry. The main challenge is to provide a solution approach that can find near-optimal solutions for large-sized problem instances. For this purpose, we suggest a heuristic approach considering conscious delays of jobs in front of production stages, i.e., active scheduling is applied. The performance analysis shows that small problem instances can be optimally solved with CPLEX-solver. For larger problem instances, the best performance is achieved by the suggested advanced multi-start heuristic. Finally, a “detailed shift planning” considering a simultaneous assignment of machines and workers is taken into account on the operational planning level. That problem pursues an even progress in the underground mine within a work shift. During a work shift, in addition to a machine scheduling problem, a personnel allocation problem must be considered. Moreover, to provide realistic solutions, different kinds of setup times must be observed, depending on the processing sequence of the operations on machines and workers. The major challenge is to express the specific circumstances of a work shift mathematically, e.g., considering workers' breaks for a possible delay in the processing time of a job, determining the processed percentage of a job during a work shift, observing removal and changeover times, etc. A part of real constraints is formulated in a relaxed program as part of a heuristic solution approach. The proposed heuristic procedure consists of two steps. In the first step, a relaxed program neglecting some setup times is solved, and the typically unfeasible solution achieved is repaired in the second step by inserting the neglected times. The results show that the proposed heuristic can find for 70 percent of the realistic problem instances a better solution than an existing heuristic approach. Subsequently, we introduce a new, compact mixed-integer linear program using TSPvariables considering all the problem specifications. We show that the proposed mixed-integer linear program outperforms the proposed two-stage heuristic considerably