Abstract: For the purpose of production scheduling, open-pit mines are discretized into three-dimensional arrays known as block models. Production scheduling consists of deciding which blocks should be extracted, when they should be extracted, and what to do with the blocks once they are extracted. Blocks that are close to the surface should be extracted first, and capacity constraints limit the production in each time period. Since the 1960s, it has been known that this problem can be cast as an integer programming model. However, the large size of some real instances (3-10 million blocks, 15-20 time periods) has made these models impractical for use in real planning applications, thus leading to the use of numerous heuristic methods. In this article we study a well-known integer programming formulation of the problem that we refer to as C-PIT. We propose a new decomposition method for solving the linear programming relaxation (LP) of C-PIT when there is a single capacity constraint per time period. This algorithm is based on exploiting the structure of the precedence-constrained knapsack problem and runs in O(mn log n) in which n is the number of blocks and m a function of the precedence relationships in the mine. Our computations show that we can solve, in minutes, the LP relaxation of real-sized mine-planning applications with up to five million blocks and 20 time periods. Combining this with a quick rounding algorithm based on topological sorting, we obtain integer feasible solutions to the more general problem where multiple capacity constraints per time period are considered. Our implementation obtains solutions within 6% of optimality in seconds. A second heuristic step, based on local search, allows us to find solutions within 3% in one hour on all instances considered. For most instances, we obtain solutions within 1-2% of optimality if we let this heuristic run longer. Previous methods have been able to tackle only instances with up to 150,000 blocks and 15 time periods.

Abstract: Similar to the mixed-integer programming library (MIPLIB), we present a library of publicly available test problem instances for three classical types of open pit mining problems: the ultimate pit limit problem and two variants of open pit production scheduling problems. The ultimate pit limit problem determines a set of notional three-dimensional blocks containing ore and/or waste material to extract to maximize value subject to geospatial precedence constraints. Open pit production scheduling problems seek to determine when, if ever, a block is extracted from an open pit mine. A typical objective is to maximize the net present value of the extracted ore; constraints include precedence and upper bounds on operational resource usage. Extensions of this problem can include (i) lower bounds on operational resource usage, (ii) the determination of whether a block is sent to a waste dump, i.e., discarded, or to a processing plant, i.e., to a facility that derives salable mineral from the block, (iii) average grade constraints at the processing plant, and (iv) inventories of extracted but unprocessed material. Although open pit mining problems have appeared in academic literature dating back to the 1960s, no standard representations exist, and there are no commonly available corresponding data sets. We describe some representative open pit mining problems, briefly mention related literature, and provide a library consisting of mathematical models and sets of instances, available on the Internet. We conclude with directions for use of this newly established mining library. The library serves not only as a suggestion of standard expressions of and available data for open pit mining problems, but also as encouragement for the development of increasingly sophisticated algorithms.

Abstract: We study a Lagrangian decomposition algorithm recently proposed by Dan Bienstock and Mark Zuckerberg for solving the LP relaxation of a class of open pit mine project scheduling problems. In this study we show that the Bienstock-Zuckerberg (BZ) algorithm can be used to solve LP relaxations corresponding to a much broader class of scheduling problems, including the well-known Resource Constrained Project Scheduling Problem (RCPSP), and multi-modal variants of the RCPSP that consider batch processing of jobs. We present a new, intuitive proof of correctness for the BZ algorithm that works by casting the BZ algorithm as a column generation algorithm. This analysis allows us to draw parallels with the well-known Dantzig-Wolfe decomposition (DW) algorithm. We discuss practical computational techniques for speeding up the performance of the BZ and DW algorithms on project scheduling problems. Finally, we present computational experiments independently testing the effectiveness of the BZ and DW algorithms on different sets of publicly available test instances. Our computational experiments confirm that the BZ algorithm significantly outperforms the DW algorithm for the problems considered. Our computational experiments also show that the proposed speed-up techniques can have a significant impact on the solve time. We provide some insights on what might be explaining this significant difference in performance.

Abstract: For decades, Out-of-Stock (OOS) events have been a problem for retailers and manufacturers. In grocery retailing, an OOS event is used to characterize the condition in which customers do not find a certain commodity while attempting to buy it. This paper focuses on addressing this problem from a manufacturer’s perspective, conducting a case study in a retail packaged foods manufacturing company located in Latin America. We developed two machine learning based systems to detect OOS events automatically. The first is based on a single Random Forest classifier with balanced data, and the second is an ensemble of six different classification algorithms. We used transactional data from the manufacturer information system and physical audits. The novelty of this work is our use of new predictor variables of OOS events. The system was successfully implemented and tested in a retail packaged foods manufacturer company. By incorporating the new predictive variables in our Random Forest and Ensemble classifier, we were able to improve their system’s predictive power. In particular, the Random Forest classifier presented the best performance in a real-world setting, achieving a detection precision of 72% and identifying 68% of the total OOS events. Finally, the incorporation of our new predictor variables allowed us to improve the performance of the Random Forest by 0.24 points in the F-measure.