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

Optimizing mining rates under financial uncertainty in global mining complexes

AbstractThis paper presents a distributed and dynamic programming framework to the mining production rate target tracking of multiple metal mines under financial uncertainty. A single mine׳s target tracking is stated as a stochastic optimization problem and the solution is obtained by solving the dynamic program which gives the optimal production rate schedule of each mine as a Markovian feedback control on the price process. The global solution is distributed on multiple mines by a policy iteration method, and this iterative method is shown to provide the unique equilibrium among Markovian strategies. Numerical results confirm the efficacy of the proposed global method when compared to individual optimization of mining rate target tracking

Conditional simulation of IRF-k in the petroleum industry and the expert system perspective

Geostatistical modeling of reservoir rock-properties -- Conditional simulation of intrinsic random functions of order k -- Geostatistical estimation of the effective permeability tensor in a three-dimensional petroleum reservoir -- Quantitative-numerical characterization of the crystal viking field, south-central Alberta : an integrated approach -- The expert system perspective : a theory of artificially intelligent geostatics

Generalized Laguerre expansions of multivariate probability densities with moments

AbstractWe generalize the well-known Laguerre series approach to approximate multivariate probability density functions (PDFs) using multidimensional Laguerre polynomials. The generalized Laguerre series, which is defined around a Gamma PDF, is suited for simulating high complex natural phenomena that deviate from Gaussianity. Combining the multivariate Laguerre approximation and Bayes theorem, an approximation to the conditional PDFs is derived. Numerical results first showed the superiority of the Gamma expansion over other numerical methods. The ability of the Gamma expansion to fit mixtures of Gaussian ans super Gaussian PDFs, univariate and multivariate Lognormal PDFs, and complex geologic media is shown through different examples

A new computational model of high-order stochastic simulation based on spatial Legendre moments

Multiple-point simulations have been introduced over the past decade to overcome the limitations of second-order stochastic simulations in dealing with geologic complexity, curvilinear patterns, and non-Gaussianity. However, a limitation is that they sometimes fail to generate results that comply with the statistics of the available data while maintaining the consistency of high-order spatial statistics. As an alternative, high-order stochastic simulations based on spatial cumulants or spatial moments have been proposed; however, they are also computationally demanding, which limits their applicability. The present work derives a new computational model to numerically approximate the conditional probability density function (cpdf) as a multivariate Legendre polynomial series based on the concept of spatial Legendre moments. The advantage of this method is that no explicit computations of moments (or cumulants) are needed in the model. The approximation of the cpdf is simplified to the computation of a unified empirical function. Moreover, the new computational model computes the cpdfs within a local neighborhood without storing the high-order spatial statistics through a predefined template. With this computational model, the algorithm for the estimation of the cpdf is developed in such a way that the conditional cumulative distribution function (ccdf) can be computed conveniently through another recursive algorithm. In addition to the significant reduction of computational cost, the new algorithm maintains higher numerical precision compared to the original version of the high-order simulation. A new method is also proposed to deal with the replicates in the simulation algorithm, reducing the impacts of conflicting statistics between the sample data and the training image (TI). A brief description of implementation is provided and, for comparison and verification, a set of case studies is conducted and compared with the results of the well-established multi-point simulation algorithm, filtersim. This comparison demonstrates that the proposed high-order simulation algorithm can generate spatially complex geological patterns while also reproducing the high-order spatial statistics from the sample data

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