The present study is focused on mining planning with an emphasis on the graph theory model proposed by Lerchs-Grossmann. The original paper published by Lerchs-Grossmann about determination of optimum final pit does not report the computational algorithm to solve the problem. This paper discusses and presents an algorithm based on the maximum flow graph computational work from Ford Fulkerson. The main steps for solving the problem and the results of the two-dimensional models are discussed. ResumoO presente trabalho tem como foco o planejamento de lavra com ênfase no modelo da teoria dos grafos de Lerchs-Grossmann. O trabalho publicado pelo autor em que se discute o teorema para a determinação da cava final ótima não apresenta o algoritmo computacional para resolver o problema. Esse trabalho apresenta um algoritmo baseado no fluxo máximo dos grafos como discutido no trabalho computacional de Ford Fulkerson. Serão apresentados os passos principais para resolução do problema e os resultados dos testes realizados para modelos bidimensionais. Palavras-chave: Lerchs-Grossmann; cava final; grafo; Ford Fulkerson.A proposal to find the ultimate pit using Ford Fulkerson algorithm Uma proposta para determinar cava final utilizando Ford Fulkerson

Mineral projects are composed of geological, operational and market uncertainties, and reducing these uncertainties is one of the objectives of engineering. Most surveys assess the impact of geological and operational uncertainties on the mining planning. The objective of this work is to study the impact of market uncertainty on the mineral activity. The influence of iron ore price simulation on mining sequencing will be evaluated. The price of iron ore has random behavior that is best represented by the Geometric Brownian Movement system. This study analyzed the historical series of iron ore in order to determine the percentage volatility and drift. Traditionally, a constant and deterministic price is used for the ore mined in all periods of a mineral project. The direct block scheduling methodology was adopted because it is able to apply the appropriate financial discount factor to the simulated probabilistic price. The proposed methodology was able to quantify the market uncertainty.

This study is focused on Direct Block Scheduling testing (Direct Multi-Period Scheduling methodology) which schedules mine production considering the correct discount factor of each mining block, resulting in the final pit. Each block is analyzed individually in order to define the best target period. This methodology presents an improvement of the classical methodology derived from Lerchs-Grossmann's initial proposition improved by Whittle. This paper presents the differences between these methodologies, specially focused on the algorithms' avidity. Avidity is classically defined by the voracious search algorithms, whereupon some of the most famous greedy algorithms are Branch and Bound, Brutal Force and Randomized. Strategies based on heuristics can accentuate the voracity of the optimizer system. The applied algorithm use simulated annealing combined with Tabu Search. The most avid algorithm can select the most profitable blocks in early periods, leading to higher present value in the first periods of mine operation. The application of discount factors to blocks on the Lerchs-Grossmann's final pit has an accentuated effect with time, and this effect may make blocks scheduled for the end of the mine life unfeasible, representing a trend to a decrease in reported reserves.

The transport distance in a mining operation strongly influences a mine operation revenue and its operational cycle because it is a fundamental part of the total mining costs. Generally, the transport route is determined based on an engineer's practical knowledge, which does not consider any mechanism to optimize the possible routes to be taken. In an attempt to establish a methodology for calculating the path that results in minimum costs to transport the mined block to its destination, the Dijkstra methodology is applied to a tree graph analysis, where the mining blocks are analysed as nodes of the tree. The transport cost is reflected as the arc of the graphs, which can use the Euclidean distance or the transport time for the calculation of the minimum path. The result obtained from the Dijkstra algorithm provided a non-operational route; to overcome this problem, an adjustment was performed through non-parametric equations. In this manner, it was possible to determine the transport costs for each block of the model. The paths based on Euclidean distance and transport time showed a tendency to increase for deeper mining regions. Identifying areas of largest growth and correctly quantifying their values increase the efficiency of mining planning.

Geologic modeling is an important step in determining the benefits and final pit dimensions for mining operations. Geostatistical models and distance-based functions are the main methods used to estimate the grade behavior. However, these two methods, despite their similar mean values, differ in spatial variability. The objective of this article is to prove, by comparing the two methodologies, that models with different spatial variability using the Lerchs-Grossmann algorithm will output subtly different final pit dimensions and scheduling. Furthermore, with the direct block schedule (DBS), these differences can be considerable. The tests compared the methodologies using the following three models: inverse distance (ID), ordinary kriging (OK) and turning bands simulation (TBS). The results demonstrate that the Lerchs-Grossmann algorithm is only slightly sensitive to the spatial variability of the grade; however, DBS requires the model populations to be better defined because of its greater sensitivity to spatial variability.