Open-pit mining is a surface mining operation whereby ore, or waste, is excavated from the surface of the land. The open-pit design problem is deciding on which blocks of an ore deposit to mine in order to maximize the total profit, while obeying digging constraints concerning pit slope and block precedence. The open-pit design problem can be formulated as a maximum flow problem in a certain capacitated network, as first shown by Picard in 1976. His derivation is based on a restatement of the problem as a quadratic binary program. We give an alternative derivation of the maximum flow formulation, which uses only linear programming duality.
We consider the problem of finding an optimal mining sequence for an open pit during a number of time periods subject to only spatial and temporal precedence constraints. This problem is of interest because such constraints are generic to any open-pit scheduling problem and, in particular, because it arises as a Lagrangean relaxation of an open-pit scheduling problem. We show that this multi-period open-pit mining problem can be solved as a maximum flow problem in a time-expanded mine graph. Further, the minimum cut in this graph will define an optimal sequence of pits. This result extends a well-known result of J.-C. Picard from 1976 for the open-pit mine design problem, that is, the single-period case, to the case of multiple time periods.
We consider the problem of finding an optimal mining schedule for an openpit during a number of time periods, subject to a mining capacity restriction for each time period. By applying Lagrangian relaxation to the capacities, a multi-parametric formulation is obtained. We show that this formulation can be restated as a maximum flow problem in a time-expanded network. This result extends a well-known result of Picard from 1976 for the open-pit design problem, that is, the single-period case, to the case of multiple time periods.
This paper discusses a Lagrangian relaxation interpretation of the Picard and Smith (2004) parametric approach to open-pit mining, which finds a sequence of intermediate contours leading to an ultimate one. This method is similar to the well known parametric approach of Lerchs and Grossmann (1965). We give examples of worst case performance, as well as best case performance of the Picard-Smith approach. The worst case behaviour can be very poor in that we might not obtain any intermediate contours at all. We also discuss alternative parametric methods for finding intermediate contours, but conclude that such methods seem to have inherent weaknesses.
The selection of a mine design is based on estimating net present values of all possible, technically feasible mine plans so as to select the one with the maximum value. It is a hard task to know with certainty the quantity and quality of ore in the ground. This geological uncertainty, and also the future market behaviour of metal prices and foreign exchange rates, which are impossible to be known with certainty, make mining a high risk business.
Value-at-Risk (VaR) is a measure that is used in financial decisions to minimize the loss caused by inadequate monitoring of risk. This measure does however have certain drawbacks such as lack of consistency, nonconvexity, and nondifferentiability. Rockafellar and Uryasev (2000) introduce the Conditional Value-at-Risk (CVaR) measure as an alternative to the VaR measure. The CVaR measure gives rise to a convex problem.
An optimization model that maximizes expected return while minimizing risk is important for the mining sector as this will help make better decisions on the blocks of ore to mine at a particular point in time. We present a CVaR approach to the uncertainty involved in open-pit mining. We formulate investment and design models for the open-pit mine and also give a nested pit scheduling model based on CVaR. Several numerical results based on our models are presented by using scenarios from simulated geological and price uncertainties.
The selection of a mine design is based on estimating net present values of all possible, technically feasible mine plans so as to select the one with the maximum value. It is a hard task to know with certainty the quantity and quality of ore in the ground. This geological uncertainty and also the future market behavior of metal prices and foreign exchange rates, which are always uncertain, make mining a high risk business. Value-at-Risk (VaR) is a measure that is used in financial decisions to minimize the loss caused by inadequate monitoring of risk. This measure does, however, have certain drawbacks such as lack of consistency, nonconvexity, and nondifferentiability. Rockafellar and Uryasev [J. Risk 2, 21-41 (2000)] introduce the Conditional Value-at-Risk (CVaR) measure as an alternative to the VaR measure. The CVaR measure gives rise to a convex optimization problem. An optimization model that maximizes expected return while minimizing risk is important for the mining sector as this will help make better decisions on the blocks of ore to mine at a particular point in time. We present a CVaR approach to the uncertainty involved in open-pit mining. We formulate investment and design models for the open-pit mine and also give a nested pit scheduling model based on CVaR. Several numerical results based on our models are presented by using scenarios from simulated geological and market uncertainties.
Open-pit production scheduling deals with the problem of deciding what and when to mine from an open-pit, given potential profits of the different fractions of the mining volume, pit-slope restrictions, and mining capacity restrictions for successive time periods. We give suggestions for Lagrangian dual heuristic approaches for the open-pit production scheduling problem. First, the case with a single mining capacity restriction per time period is considered. For this case, linear programming relaxations are solved to find values of the multipliers for the capacity restrictions, to be used in a Lagrangian relaxation of the constraints. The solution to the relaxed problem will not in general satisfy the capacity restrictions, but can be made feasible by adjusting the multiplier values for one time period at a time. Further, a time aggregation approach is suggested as a way of reducing the computational burden of solving linear programming relaxations, especially for largescale real-life mine problems. For the case with multiple capacity restrictions per time period we apply newly developed conditions for optimality and nearoptimality in general discrete optimization problems to construct a procedure for heuristically constructing near-optimal intermediate pits.
M.G.Kreins metod med riktande avbildning utvidgas till klassen av de kvasitätt definierade symmetriska operatorer i ett indefinit inre produktrum, som har självadjungerade definitiserbara utvidgningar i något Kreinrum.