Search

At the time of this writing, the U.S. Geological Survey estimates that the average American-born human will need millions of pounds of fuels, minerals, and other extracted resources in his or her lifetime. Mining is a critical global industry, spanning all but one continent (Antarctica), with the highest-producing countries being China, the United States, Russia, Australia, and India. Increasingly, this demand is driving mining companies to explore and pursue deeper mineral deposits as near-surface deposits deplete. Correspondingly, there has been a significant rise in industry interest in applying operations research techniques to improve underground mine planning. Newman et al. [Newman AM, Rubio E, Caro R, Weintraub A, Eurek K (2010) A review of operations research in mine planning. Interfaces 40(3):222–245] present a review of such techniques, applied to both open pit and underground mining operations. We focus here on the advancements since that publication and concentrate on underground applications in metalliferous deposits, such as copper, iron, and gold.

Mining operations determine a long-term production schedule, often to maximize net present value. For a time horizon of between years and decades, optimization models seek the extraction times—with monthly or yearly fidelity—of three-dimensional, notional blocks of ore and waste within a deposit to satisfy spatial precedence constraints, as well as resource constraints on the amount of material extracted and sent to the mill. With algorithmic advances, as well as those in mine planning software and in hardware, we are able to solve instances with a decade-long horizon at daily fidelity. The resulting objective, repeatable, and defensible schedules inform production and maintenance supervisory decisions based on resource availability, that is, loaders, shovels, haul trucks, and mineral processors. We implement our solutions at the Turquoise Ridge underground gold mine in Nevada, United States. These solutions indicate more than a 2% increase in total ounces extracted over a decade while decreasing development footage by as much as 11% over the same time horizon. Furthermore, we are able to incorporate rules governing a shared resource and to evaluate binding versus nonbinding capacity constraints.

The National Basketball Association (NBA) is divided into two conferences, each of which comprises 15 teams. At the end of the regular season, the top eight teams from each conference, based on winning percentage, compete in the playoffs. Mixed-integer-programming (MIP) models determine when a team has guaranteed its position in the playoffs (clinched) or, conversely, when it has been eliminated before the completion of the regular season. Our models incorporate a series of complex two-way tiebreaking criteria used by the NBA to determine how many more games are needed either to clinch or to avoid elimination. We compare the time at which a given team has clinched or been eliminated, in terms of the number of games played in the season to date, as posted in the NBA official standings, against results from our mixed-integer program. For the 2017–2018 season, when our models outperform those of the NBA, they do so by an average of 4.1 games. We also describe a scenario in which the NBA erroneously reported that the Boston Celtics had clinched a playoff spot and, conversely, show that the Golden State Warriors had clinched a playoff spot before the official announcement by the NBA.