یک مدل برنامه ریزی خطی برای برنامه ریزی معدن طولانی مدت در حضور عدم قطعیت درجه و انبار

The complexity of an open pit production scheduling problem is increased by grade uncertainty. A method is presented to calculate the cost of uncertainty in a production schedule based on deviations from the target production. A mixed integer linear programming algorithm is formulated to find the mining sequence of blocks from a predefined pit shell and their respective destinations, with two objectives: to maximize the net present value of the operation and to minimize the cost of uncertainty. An efficient clustering technique reduces the number of variables to make the problem tractable. Also, the parameters that control the importance of uncertainty in the optimization problem are studied. The minimum annual mining capacity in presence of grade uncertainty is assessed. The method is illustrated with an oil sand deposit in northern Alberta.

Mine planning is an important process in mining engineering that aims to find a feasible block extraction schedule that maximizes net present value (NPV). In the case of open pit mines, Whittle defines mine planning as: “Specifying the sequence of blocks extraction from the mine to give the highest NPV, subject to variety of production, grade blending and pit slope constraints” [1]. Technical, financial and environmental constraints must be considered. The uncertainty of the ore grade may cause discrepancies between planning expectations and actual production [2], [3] and [4]. Various authors present methodologies to account for grade uncertainty, and demonstrate its impact. Dowd proposed a risk-based algorithm for surface mine planning [5]. A predefined distribution function is used for some variables such as commodity price, mining costs, processing cost, investment required, grade and tonnages. Different schedules are generated for a number of realizations of the grades. The proposed method leads to multiple schedules reflecting the grade uncertainty. Ravenscroft and Koushavand and Askari-Nasab used simulated orebodies to show the impact of grade uncertainty on production scheduling [4] and [6]. They used simulated orebody models one at a time in traditional optimization methods; however, this sequential process does not optimize accounting for uncertainty. Ramazan and Dimitrakopoulos suggested a mixed integer linear programming (MILP) model to maximize NPV for each realization. Then, the probability of extraction of a block at each period is calculated. These probabilities are used in a second stage of optimization to arrive at one schedule. The uncertainty is not used directly in the optimization process [7]. Godoy and Dimitrakopoulos and Leite and Dimitrakopoulos presented a new risk-inclusive long term production plan (LTPP) approach based on simulated annealing [8] and [9]. A multistage heuristic framework was presented to generate a schedule that minimizes the risk of deviations from production targets. The authors reported a significant improvement in NPV in the presence of uncertainty; however heuristic methods do not guarantee the optimality of the results. Also, these techniques can be difficult to implement, and many parameters may need to be chosen in order to get reasonable results. Dimitrakopoulos and Ramazan presented a linear integer programming (LIP) model to generate optimal production schedules [10]. Multiple realizations of the block model are considered. This model has a penalty function that is the cost of deviations from the target production and is calculated based on the geological risk discount rate (GRD), which is the discounted unit cost of deviation from target production. They use linear programming to maximize a new function that is NPV less penalty costs. It is not clear how to define the GRD parameter. The shortcomings of the current mine planning methods include: (1) most of the methods show the effect of uncertainty on the mine plan, but do not suggest a method to minimize the risk of uncertainty, (2) the methods minimize the risk or maximize NPV without using uncertainty explicitly, (3) the methods are not suitable for real-size mining problems, (4) there is no methodology to easily calculate the cost of uncertainty, and (5) none of the presented methods generate an optimum plan in presence of grade uncertainty. In this paper, a mathematical programming formulism for long term mine planning in presence of grade uncertainty is proposed. The cost of uncertainty is quantified and used in a mixed integer linear programming model. A stockpile is considered in this new model. The cost of uncertainty is needed to determine the optimal trade-off between maximizing the NPV and minimizing the risk of grade uncertainty. The relationship between mining capacity and processing capacity and the cost of uncertainty is shown in this paper.

The cost of uncertainty in long term mine production plan was introduced. A mixed integer linear programming model considering stockpiles was presented to generate long term production schedules. The NPV is maximized with the Kriged block model. The second objective is to minimize cost of uncertainty. The concept of a stockpile used in proposed model helps to reduce the shortfalls and deviations from target production. The impact on the cost of uncertainty is significant. This model is superior to the previous models because the stockpile is considered in the optimization process. The asymmetric penalty function, where the cost of underproduction is higher than cost of overproduction. However the presence of a stockpile allows optimization to extract extra ore at early stages of the mine life. These extra ore reduce the chance of short falls at later years. The optimum mining capacity can be estimated considering the grade uncertainty. The objective function presented in Eq. (10) allows the mine planner to consider grade uncertainty. By changing mining capacity, different optimum solutions are generated. The cost of grade uncertainty is calculated for each case. A synthetic case demonstrates this procedure. Fig. 13 shows the relationship between cost of uncertainty and mining and processing capacities. The mining and processing capacities are incremented 10 times each with 5 and 10 unit intervals, respectively. Therefore the optimization process was run for each case with different mining and processing capacities. The vertical axis is the cost of the uncertainty. The following points can be concluded from this graph: • For a chosen processing capacity (each line), by increasing the mining capacity, the cost of uncertainty reduces until a certain point. • With a larger processing capacity, it is required to set a much larger mining capacity in order to get the minimum cost of uncertainty. This can be well understood by comparing two processing capacities of 95 and 140. In these two lines, the mining capacities resulting in minimum costs of uncertainty are 290 and 350, respectively. • Larger mines with high processing and mining capacities have higher cost of uncertainty. Full-size image (17 K) Fig. 13. Cost of uncertainty versus different mining and processing capacities in a synthetic case. Figure options A higher mining capacity reduces the cost of grade uncertainty because there is flexibility to makeup an unexpected shortfall. Also, any possible extra ore can be handled easily by stockpiling. As shown in this graph, as the mining capacity increases, the cost of uncertainty decreases. This decrement is not significant after a certain amount of mining capacity. For each processing capacity, there is an optimum mining capacity. After this point, the cost of uncertainty does not decrease further by increasing the mining capacity. The mining capacity estimated this way is higher than the capacity found by the traditional method with an estimated model. The best practice to get the optimum value of mining limit is to increase the mining capacity and calculate the cost of uncertainty for each case. The optimum mining capacity is the capacity that the higher values (than that capacity) do not change the cost of uncertainty, because after some point the processing capacity limits the total ore that is mined and higher mining capacity does not have any impact on the schedule. On the other hand if the stockpile capacity is high enough, the cost of uncertainty can be reach to zero with higher mining capacity. The size of the optimization problem with the original block model is too large for current commercial solvers such as CPLEX. A clustering technique was used to aggregate similar blocks into groups called mining-cuts. This reduces the number of variables and smoothen the generated schedule. To investigate the sensitivity of the project to the number of clusters or mining cuts, different numbers of mining cuts are generated. The proposed model is run with 100, 500, 1000, 2000 and 3000 mining cuts. Fig. 14 shows the number of mining cuts versus the NPV of the project. As it is shown in this graph, 2000 mining cut can be used for this case as the sufficient number of mining cuts. Using more mining cuts does not have significant improvement on the NPV maximization. Fig. 15 shows the run time for the case of 10 years mine life with different number of mining cuts. A polynomial function has been used to capture the trend which has been shown with dashed line in the graph. High R2 value shows that the number of mining cuts has polynomial effect on the run time of the optimization. From this line it can be predict that the run time for the case with 5000 mining cuts is about 90,000 s or 25 h with 8 CPUs. Full-size image (9 K) Fig. 14. NPV versus the number of mining cuts. Figure options Full-size image (11 K) Fig. 15. Number of mining cuts versus run time of the optimization.