# Probabilistic practical optimization of production schedulling for multi-element deposits

CIM Bulletin, Vol. 1, No. 2, 2006

S. Ramazan and R. Dimitrakopoulos

The file is a zipped PDF document. Optimization of production schedules in open pit mines is important for managing cash flows. The risk of not meeting planned production targets is a critical aspect of optimization, however, it is not accounted for by traditional schedulling methods. This paper describes a new, probabilistic, mixed integer programming (MIP) formulation. The formulation has been developed so as to minimize the risk of not achieving planned production targets, both ore feed for the mill and metal production, while aiming to maximize the total discounted economic value. In addition, the new MIP formulation minimizes the movement of excavators and accommodates equipment access. In the proposed formulation, a probability is assigned to each ore block under consideration. This represents the desirability of the particular block being mined in a given period and expresses the chance that a block will contain the desired quality and quantity of ore, ore grades above given cutoffs, quantity of metal, and deleterious elements being within required ranges. The probability is calculated from jointly simulated geostatistical models representing the pertinent attributes of the mineral deposit under consideration. The probability is multiplied by the economic value, which is referred to as probabilistic value in the objective function of the schedulling formulation. The orebody risk discounting concept is also implemented to manage the risk distribution over production periods. Due to the orebody risk discounting effect, the low-grade and high-risk areas of the deposit will be scheduled for later periods of production. In addition, the proposed model considers the smoothness of the schedules internally during the optimization process. A smooth schedulling pattern is defined as the pattern generated by assigning the same periods to several blocks that are adjacent to each other so as to minimize the movement of equipment during mining operations. A case study is carried out using a multi-element nickel laterite orebody characterized by seven attributes: Ni, Co, Mg, and Al grades, volume of per cent rock, and the thicknesses of two layers (LS and RS). The seven attributes are simulated using a joint conditional simulation method on a dense grid of nodes. For the purpose of schedulling, the deposit is represented by 2,030 blocks, 40 by 40 m2, along east-west and north-south directions. The figure summarizes the comparison of the proposed probabilistic schedule (PS) with the traditional schedule (TS). It shows average deviations per mining period from expected optimal ore production targets (bars), and the probability of deviations in ore production per mining period (lines), each with respect to the 35 jointly simulated orebody models. In the first year, the TS model has a 91% chance of deviating by around 383,000 t on average.  However, the PS model has only a 57% chance of deviating by only about 85,000 t on average. Furthermore, in the second period, the PS has only a 34% chance of deviating by around 91,000 t, while the TS has a 74% chance of deviating by almost 270,000 t. Note that the PS model does not consider minimizing the risk of not meeting production targets for the last period. The risk in the earlier periods is considered more costly for the objective function than the risks in the later periods. The results indicate that the proposed probabilistic method is superior to traditional optimization methods in terms of minimizing the risks of not meeting production targets and maximizing the achievable net present value (NPV) of the operation.
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