Analyzing solutions of the openpit

n A common decision in openpit mining is to determine the extraction sequence of notional three-dimensional production blocks so as to maximize the net present value of the extracted orebody, while adhering to precedence and operational resource constraints. This openpit block sequencing (OPBS) problem is commonly formulated as an integer program, with binary variables representing if and when each block is extracted. In practical applications, the number of blocks can be large and the time horizon can be long; therefore, instances of this problem can be difficult to solve using the exact approach of optimization. The problem is even more challenging to solve when it includes explicit minimum operational resource constraints. Our maximum value feasible pit (MVFP) algorithm generates an initial integer feasible solution for OPBS problems, in which minimum operational resource constraints are strictly enforced. As an exact approach, we present a Tailored Lagrangian Relaxation (TLR), in which the selection of constraints to dualize is guided by information provided by the MVFP algorithm. We present results and graphics to demonstrate the utility of our techniques for instances containing up to 25,000 blocks and 10 time periods. Mining Engineering, 2013, Vol. 65, No. 2, pp. Official publication of the Society for Mining, Metallurgy and Exploration, Inc. Introduction Mining is a risky business, in which scheduling the efficient extraction of material is critical to enhancing profitability. Firms seek an extraction sequence of three-dimensional production blocks that maximizes net present value (NPV), while satisfying geospatial requirements and operational resource limitations (e.g., maximum per-period production capacity). Solving an optimization problem called the openpit block sequencing (OPBS) problem provides this sequence, specifying which blocks to remove, and when. 1. Our specification of the openpit block sequencing (OPBS) problem Openpit mine scheduling problems range in scope from the strategic ultimate pit limit (UPL) problem, first efficiently solved by Lerchs and Grossmann (1965), to tactical, precedenceconstrained production scheduling problems. This paper employs techniques to expedite solutions for the constrained pit limit problem (CPIT), the solution of which identifies those blocks to extract and when. Our specification of (CPIT) is a simplified version of the model first presented in Johnson (1968); we enforce geospatial and operational resource constraints. In the literature, many solution strategies for variants of (CPIT) (e.g., Moreno et al., 2010; Bienstock and Zuckerberg, 2010) consider only maximum operational resource constraints (e.g., maximum production capacity). However, in openpit mining, it is not always practical to stop and restart production and processing operations. Therefore, our specification of (CPIT) supplements these maximum operational resource capacities with positive minimum operational resource requirements. 2. Models The techniques we present rely on two mathematical formulations: (i) the ultimate pit limit problem (UPIT) from the seminal work of Lerchs and Grossmann (1965), and (ii) the constrained pit limit (CPIT) problem, as in Espinoza et al. (2011). Our mathematical notation for, and formulations of, these two problems follow. Sets, indices, data and decision variables: • b ∈ B: set of all blocks b • b'∈ Bb: set of blocks which W.B. Lambert, member SME, and A.M. Newman are PhD and associate professor in the Division of Economics and Business, Colorado School of Mines, Golden, CO. Email [email protected] and [email protected]. Paper number TP-12-007. Original manuscript submitted February 2012. Revised manuscript accepted for publication September 2012. Discussion of this peer-reviewed and approved paper is invited and must be submitted to SME Publications by Mar. 31, 2013.