Mark Kuchta

Using aggregation to optimize long-term production planning at an underground mine

Motivated by an underground mining operation at Kiruna, Sweden, we formulate a mixed integer program to schedule iron ore production over multiple time periods. Our optimization model determines an operationally feasible ore extraction sequence that minimizes deviations from planned production quantities. The number of binary decision variables in our model is large enough that directly solving the full, detailed problem for a three year time horizon requires hours, or even days. We therefore design a heuristic based on solving a smaller, more tractable, model in which we aggregate time periods, and then solving the original model using information gained from the aggregated model. We compute a bound on the worst case performance of this heuristic and demonstrate empirically that this procedure produces good quality solutions while substantially reducing computation time for problem instances from the Kiruna mine.

Implementing a Production Schedule at LKAB's Kiruna Mine

LKABs Kiruna Mine, located in northern Sweden, produces about 24 million tons of iron ore yearly using an underground mining method known as sublevel caving. To efficiently run the mills that process the iron ore, the mine must deliver planned quantities of three ore types. We used mixed-integer programming to schedule Kirunas operations, specifically, which production blocks to mine and when to mine them to minimize deviations from monthly planned production quantities while adhering to operational restrictions. These production schedules save costs compared to schedules produced manually by meeting desired production quantities more closely and reducing employee time spent on preparing schedules.

A REVISED FORM OF THE BERGMARK-ROOS EQUATION FOR DESCRIBING THE GRAVITY FLOW OF BROKEN ROCK

The proper design of underground mines using caving methods such as block caving or sublevel caving requires a good understanding of how broken rock flows under the influence of gravity. To date, the mathematical equation that best describes the shape of the extraction drawbody is given by the Bergmark-Roos Equation. The derivation of the Bergmark-Roos Equation is reviewed and a revised version of the equation is developed which accounts for a non-zero opening width. Equations for the area, volume, and maximum width of the drawbody for a given extraction height are developed for both the original and modified forms of the Bergmark-Roos Equation. The equations derived can be used to evaluate existing mine layouts as well as in the design of new mining geometries.

A Review of Long- and Short-Term Production Scheduling at Lkab’s Kiruna Mine

LKAB’s Kiruna Mine, located in northern Sweden, produces about 24 million tons of iron ore yearly using an underground mining method known as sublevel caving. To aid in its ore mining and processing system, Kiruna has adopted the use of several types of multi-period production scheduling models that have some distinguishing characteristics, for example: (i) specific rules governing the way in which the ore is extracted from the mine; (ii) lack of an inventory holding policy; and (iii) decisions that are not explicitly cost-based. In this chapter, we review two models in use at Kiruna and three techniques we have employed to expedite solution time, support the efficacy of these techniques with numerical results, and provide a corresponding discussion.

The Area Between a One-Dimensional Ordinary Brownian Curve and a Straight-Line Approximation

For a fractal curve, the measured perimeter length increases as the ruler length decreases. When the perimeter of a fractal curve is traced with a given ruler size, there are areas between the straight-line segments and the true curve. If the underlying geometric rule for creating the fractal curve is known, then the size of the areas can be calculated. An equation has been developed for calculating the absolute value of the area between a straight-line approximation and the true curve for the random function ordinary one-dimensional Brownian motion. One potential application of the equation developed is in estimating the amount of ore loss and waste rock dilution that would occur in a mining operation as a result of the errors in the geologic model of the boundaries of an orebody.