New integer programming models for tactical and strategicunderground production scheduling

Abstract

We consider an underground production scheduling problem which consists of determining the proper time interval(s) in which to complete each mining activity so as to maximize a mine’s discounted value, while adhering to precedence, activity durations, and production and processing limits. We present two different integer programming formulations for modeling this optimization problem. Both formulations possess a resource-constrained project scheduling problem structure. The first formulation uses a fine time discretization and is better suited for tactical mine scheduling applications. The second formulation, which uses a coarser time discretization, is better suited for strategic scheduling applications. We illustrate the strengths and weakness of each formulation with examples. Introduction: Project scheduling is an important aspect of underground mine planning that consists of determining the start dates for a given set of activities so as to maximize the value of a project, while adhering to operational and resourceavailability constraints. Important activities that require scheduling include development, drilling, stoping or other ore-extraction techniques, and backfilling. Precedence relationships impose an order in which activities can be carried out based on their location in the mine. For example, ``the activity a associated with development of an area must be completed before the activity a’ associated with extraction of that same area can begin.” Resources include attributes of the mining operation such as the amount of extraction and mill capacity available per time period, and are determined by capital and equipment availability, among other factors. Correspondingly, for our setting, resource-availability constraints consider the amount of material that can be extracted and sent to the mill (i.e., processed) per time period. We define the Underground Mine Project Scheduling Problem, or UG-PSP, as that of scheduling a set of mining activities in such a way as to maximize the net present value of the project, while adhering to precedence and resource-availability constraints; in general, optimization models for underground scheduling are more complex than their open pit counterparts (O'Sullivan, Brickey, and Newman, 2015). The UG-PSP is a particular case of the Resource-Constrained Project Scheduling Problem (RCPSP), a class of optimization problems known for their difficulty (Artigues et al., 2008). It should be noted, however, that the UG-PSP may have a multitude of feasible solutions. Many mine planning software packages typically rely on heuristics. In this article, we are concerned with using mixed-integer programming to determine a provably optimal schedule, i.e. the schedule with the highest net present value. Trout (1995) first proposed a mixed-integer program to solve a 55-stope UG-PSP over a two-year time horizon using multiple time fidelities. The detailed formulation did not gain widespread adoption due to slow solution times. Little et al. (2013) demonstrate the value of implementing scheduling optimization in the mine design process. Others have created case-specific formulations for a variety of underground mines (Carlyle and Eaves, 2001; Nehring et al., 2010; Martinez and Newman, 2011; Epstein et al., 2012). Newman and Kuchta (2007) provide a model for scheduling the Kiruna mine in which activity duration spans multiple time periods; see also Sarin and West-Hansen (2005), O'Sullivan and Newman (2014), and Brickey (2015) for similar models applied to different mines. Little et al. (2011) outline several aggregation techniques to reduce the number of variables a UG-PSP problem containts, while Salama et al. (2015) examine how changing the production rate changes the value of the UG-PSP solution. UG-PSP Formulations: We begin by introducing notation for our integer programming (IP) formulations of the UG-PSP, and by noting our assumptions. Our formulations are streamlined, generalized, and highly versatile. That is, they contain precedence and resource constraints, which can be tailored to a specific application, and which are the primary two types of