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: Underground mine production scheduling determines when, if ever, activities associated with the extraction of ore should be executed. The accumulation of heat in the mine where operators are working is a major concern. At the time of this writing, production scheduling and ventilation decisions are not made in concert. Correspondingly, heat limitations are largely ignored. Our mixed-integer program maximizes net present value subject to constraints on precedence, and mill and extraction capacities with the consideration of heat using thermodynamic principles, while affording the option of activating refrigeration to mitigate heat accumulation. In seconds to hours, depending on the problem size (up to thousands of activities and 900 daily time periods), a corresponding methodology that exploits the mathematical problem structure provides schedules that maintain a safe working environment for mine operators; optimality gaps are no more than 15% and average less than half that for otherwise-intractable instances.

Abstract: In continuous steel casting operations, heats of molten steel are alloyed and refined in ladles, continuously cast and cut into slabs, and hot-rolled into coils. We present a mixed-integer program that produces a daily casting schedule and that is solved using state-of-the-art software for a 100% direct-charge steel mill; two casters concurrently produce slabs, which are rolled into coils at a single hot rolling mill. This model minimizes penalties incurred by violating plant best practices while strictly adhering to safety and logical constraints to manage risk associated with manufacturing incidents. An efficient formulation, combined with variable reduction and cutting planes, expedites solutions for small instances containing hundreds of variables and thousands of constraints by factors of at least two or three (and sometimes even 100); instances an order of magnitude larger along both problem dimensions suggest solutions that reduce costs incurred using plant best practices by as much as 40%.