J. Cent. South Univ. Technol. (2010) 17: 492-497
DOI: 10.1007/s11771-010-0512-6 
Dynamic optimization of cutoff grade in underground metal mining
GU Xiao-wei(顾晓薇), WANG Qing(王青), CHU Dao-zhong(初道忠), ZHANG Bin(张斌)
College of Resource and Civil Engineering, Northeastern University, Shenyang 110004, China
? Central South University Press and Springer-Verlag Berlin Heidelberg 2010
Abstract:
In order to maximize the overall economic gain from a metal mine operation, selection of cutoff grades must consider two important aspects: the time value of money and the spatial variation of the grade distribution in the deposit. That is, cutoff grade selection must be dynamic with respect to both time and space. A newly developed method that fulfills these requirements is presented. In this method, the deposit or a portion of it under study is divided into “decision units” based on the mining method and sample data. The statistical grade distribution and the grade-tonnage relationship of each decision unit are then computed based on the samples falling in the unit. Each decision unit with its grade-tonnage relationship is considered as a stage in a dynamic programming scheme and the problem is solved by applying a forward dynamic programming based algorithm with an objective function of maximizing the overall net present value (NPV). A software package is developed for the method and applied to an underground copper mine in Africa.
Key words:
cutoff grade; optimization; dynamic programming; underground metal mining；
1 Introduction
Cutoff grade is one of the most important parameters in metal ore mining because of its influence on the overall economic gain of a mining operation. Choosing the best cutoff grades that maximize the overall economic outcome has been a major topic of research since 1960’s.
LANE’s work has been regarded as the landmark in cutoff grade optimization [1]. His model takes the maximization of present value as the objective function and is able to consider the capacity constraints of mining, concentrating and refining stages as well as the capacity balancing between pairs of the three stages. HALLS [2] introduced the concept of “dual cutoff grades”, one for determining whether to mine or to leave, the other for determining whether to treat the mined material as ore or waste. DAGDELEN and JOHNSON [3] established the relationship between the dual cutoffs and Lagrangian multipliers of reserve parameterization in their production scheduling model. YUN [4] applied genetic algorithm to cutoff grade selection relating cutoff grade to various economic and technical factors. ATAEI and OSANLOO [5] used a hybrid genetic algorithm and combined with the grid search to find the optimum cutoff grades of multiple metal deposits with the objective of maximizing net present value (NPV). NIETO and BASCETIN [6] incorporated an iterative process in Lane’s algorithm to calculate an “optimization factor” for each production year which dynamically adjusts the remaining reserves and thus the total life of the mine to maximize the project NPV. URBAEZ [7] investigated the cutoff grade problem in reserve classification for blend product ore, indicating that blend product ore cannot be defined by economic cutoff grade only and production schedule plays an important role. HALL [8] discussed the cutoff grade problem in the context of short and long term gains.
Cutoff grade affects pit design and production scheduling through its influence on the ore geometry and ore quantity. Therefore, some researchers formulated cutoff grade as an internal decision variable in their production scheduling schemes that are usually either a linear programming or a dynamic programming formulation [9-11]. Others researchers selected cutoff grades based on pit optimization either through optimizing pit for different cutoff grades or through back calculating cutoff grade from the tonnage-grade relationship of the optimum pit [12-13].
Although maximizing NPV is desired in most cases in selecting cutoff grades, maximizing total profit may still be valid from the viewpoint of fully utilizing economic resources. Therefore, break-even analyses are still used in determining cutoff grades, especially in China [14-15].
After many years of research, the cutoff grade selection remains by and large an unsolved problem. Two aspects must be considered in solving this problem. One is the dynamic nature of cutoff grade with respect to time, the other is the dynamic nature with respect to space. In other words, the optimum cutoff grade varies with both time and the place of mining in the deposit, and the two aspects are interrelated. The first aspect has been considered in Lane’s algorithm and other’s, which are capable of determining the cutoff grade in each mining period, and usually the resulting cutoff grade decreases as mining progresses when maximizing NPV is the objective function.
However, the second aspect has not been properly dealt with in existing methods. Mineral grade varies with location to a certain extent in almost all deposits, different zones having different statistical grade distributions and average grades. Ignoring such variation will result in unrealistic cutoff grades. For example, in Lane’s algorithm, the overall statistical grade distribution is used for all mining periods and, as a result, the cutoff grade decreases as mining progresses. In cases where the mineral grade is low in the upper part of the deposit and high in the lower part, using high cutoff grade in the early mining stage is obviously not appropriate and may not even be practical.
A dynamic programming based model for optimizing cutoff grade in underground metal mines is developed. The model simultaneously considers the time and space dynamic nature of the cutoff grade problem.
2 Dynamic programming formulation
In underground metal mines, mining progresses level by level and stope by stope (or section by section) at a given level. In most cases, mineral grade varies with location, each zone having its local grade distribution whose mean and variance are different to a certain extent from those of other zones. When the variation in local grade distribution is relatively high, using the overall grade distribution to determine the best cutoff grade in different mining periods will be unrealistic and the results will not be optimum. In such cases, the cutoff grade selection for a particular mining period, a year, for example, must be related to the local grade distribution of the zone to be mined in that period. Furthermore, the cutoff grade for each mining period/zone should not be optimized as an isolated case independent of other periods/zones because the cutoff grade in one period affects the cutoff decisions in later periods. Therefore, the problem must be solved in a dynamic fashion taking into account of the local grade distribution in the decision-making process. This nature of the problem fits itself to a dynamic programming scheme.
2.1 Stages and states
The ultimate objective of cutoff grade optimization is to determine the best cutoff grade that should be used in mining each section (or zone). So each section is a decision-making unit and is taken as a stage in dynamic programming. Such units are hereafter referred to as “decision units”.
The portion of the deposit that has been explored and will be mined in the future is divided into decision units, that is, stages. The delineation of the units, their geometries and locations, is based on the mining method, the grade variation characteristics, and the sample data availability. If the grade variation is high and the sample spacing is small, a decision unit can be as small as a stope, or a small section of a mining level where the mining is not carried out in clearly defined stopes as in sublevel caving. If the grade is relatively stable with respect to location, the decision units can be relatively big and are delineated in such a way that the statistical grade distribution of a unit is clearly different from the grade distributions of its neighboring units. When samples are sparse, a small unit will not contain enough samples to get a statistical grade distribution. In such cases, one way is to have big units and a better way is to have relatively small units and to use the statistical grade distribution of a bigger section for each unit within the section. The important thing is that each unit has a grade distribution from which the grade-tonnage relationship can be established. The grade-tonnage relationship is a continuous function or a table of discrete data that relate ore quantity and average ore grade to cutoff grade. From this relationship, the ore quantity and the average ore grade can be calculated when a cutoff grade is given, or the cutoff grade corresponding to a given ore quantity can be calculated.
Once the decision units are delineated, the number of stages in the dynamic programming scheme is the same as the number of decision units.
A state of the ith stage is defined as the cumulative quantity of ore mined from the beginning to the end of stage i at this state, that is, the total ore quantity mined from unit 1 through unit i following a path that ends at this state. A stage has a number of states, each of which corresponds to a cumulative ore quantity. The number of states and their associated ore quantities of each stage can be calculated from the grade-tonnage relationship of the stage (decision unit). Taking a copper mine as an example, suppose that the lowest and highest cutoff grades considered are 0.5% and 1.2%, respectively. From the grade-tonnage relationship of the first decision unit, the ore quantities corresponding to cutoff grades of 0.5% and 1.2% can be obtained, that is, they are 2.0×106 and 1.0×106 t, respectively. Using a step size of 5.0×104 t, there will be 21 states in the first stage. From the grade-tonnage relationship of the second decision unit, if the ore quantities corresponding to cutoff grades of 0.5% and 1.2% are 8.0×105 and 5.0×105 t, respectively, the cumulative ore quantity of the first two stages will range from 1.5×106 to 2.8×106 t. With the same step size, there will be 27 states in the second stage. In this way, the states of all the stages can be determined.
2.2 Dynamic programming model
Once the stages and states are set up, they can then be put into a forward dynamic programming network as shown in Fig.1. The sequence of the stages are the same as that of mining the corresponding decision units in the long-term production schedule, i.e., the first stage corresponds to the unit that will be mined first and the second stage to the unit mined after the first, and so forth. The states of each stage are arranged from the lowest cumulative ore quantity to the highest as represented by the circles in the figure. Each arrow represents a possible transition. For the purpose of clarity, not all transitions are drawn in Fig.1. Since the decision units may not be of the same size, the distances between two adjacent stages on the horizontal axis may not be the same.
Fig.1 Dynamic propramming network for cutoff grade optimization
Let N denote the total number of stages and Mi the number of states of stage i (i=1, 2, …, N).
In general term, consider state j of stage i, denoted by Si,j. When Si,j is reached from state Si-1, k of previous stage (i-1), the transition function is given by:
qi, j(i-1, k)=Qi, j-Qi-1,k (1)
where qi,j(i-1, k) is the in-situ ore quantity mined in decision unit i when such a transition is made; Qi, j and Qi-1, k are the cumulative ore quantities associated with states Si,j and Si-1,k, respectively. This equation establishes the links between the states of two consecutive stages.
The cutoff grade, gi,j(i-1, k), corresponding to the ore amount qi,j(i-1, k) can be calculated from the grade-tonnage relationship, g, for decision unit i. That is, gi,j(i-1, k) is the cutoff grade that satisfies the following equation:
(2)
where Q is the total tonnage in decision unit i when cutoff grade is 0; fi(g) is the probability density function of mineral grade, g, for decision unit i, which can be obtained from the samples falling in the unit.
The average grade of the ore quantity qi,j(i-1, k) mined in such a transition with a cutoff grade of gi,j(i-1, k), is denoted by Gi,j(i-1, k) and calculated as
(3)
Assume that the final product of the operation is concentrated ore. A simplified calculation for the profit, Pi,j(i-1, k), made when state Si,j is reached from state Si-1,k, is given by
(4)
where Rm is the ore recovery rate of mining; Rp is the metal recovery rate of ore processing; Gp is the grade of ore concentrate; pi is the price of ore concentrate in stage i and could be a constant; Y is the dilution rate; Cm and Cp are the unit costs of mining and ore processing, respectively.
The time length, ti,j(i-1, k), required to mine qi,j(i-1, k) quantity of ore in decision unit i is
(5)
where mi is the ore production rate as planned for stage i (when mining decision unit i), which is the ore amount sent to the processing plant per year and could be a constant.
The cumulative time length to arrive at state Si,j, denoted by Ti,j(i-1, k), when Si,j is reached from Si-1,k, is calculated as
Ti,j(i-1, k)=Ti-1, k+ti, j(i-1, k) (6)
where Ti-1,k is the cumulative time length required to arrive at state Si-1,k from the beginning, following the best route.
A simplified calculation for the cumulative NPV, denoted by NPVi,j(i-1, k), realized at state Si,j is given by
(7)
where NPVi-1,k is the cumulative NPV achieved when state Si-1,k is reached, following the best route; d is the discount rate.
State Si,j may be reached from different states of the previous stage i-1 as shown in Fig.1. It is not hard to understand that, when state Si,j is reached from a different state of stage i-1, the ore quantity mined in stage i (Eq.(1)) will be different, and the cutoff grade, the time length, and the average ore grade corresponding to the new ore quantity will be different, too. Consequently, the cumulative NPV at state Si,j may also be different. Therefore, for state Si,j, each transition (an arrow in Fig.1) has a cutoff grade and a cumulative NPV. The one with the highest cumulative NPV is the best transition and the associated cutoff grade is the best cutoff grade, gi,j, for state Si,j. Thus, the recursive function is:
(8)
where Ki,j is the number of states in stage i-1 from which state Si,j of stage i can be reached. Eq.(8) implies the selection of the associated best cutoff grade, gi,j, through Eqs.(1)-(6) that relate the cutoff grade to the profit Pi,j(i-1, k) and time Ti,j(i-1, k).
The boundary conditions at time 0 are:
(9)
Using the above equations and starting from the first stage, the states are evaluated forward stage by stage, until all the states of all stages in Fig.1 are evaluated. The best transitions, the associated best cutoff grades and NPVs are obtained for all the states of all stages. Then, starting from the state of the last stage that has the highest cumulative NPV of all states of the same stage, tracing the best transitions back to the first stage, the optimum route can be found, which is called the optimum policy in dynamic programming. This optimum policy indicates the best cutoff grade to be used, and the corresponding ore quantity and economic gain in mining each of the decision units.
3 Application
Based on the above formulation, a detailed algorithm was devised and a software package developed. The package was applied to studying the cutoff grade of a copper mine in Africa (the mine is not named for confidentiality of data).
The portion of the deposit selected for the cutoff grade study extends 2 130 m in length on the horizontal plane from 500 to 700 m below the ground surface in the vertical direction. Most parts of this portion were well explored through fan-patterned holes drilled from drives and drifts. There were 5 316 samples falling in the portion. The portion was divided into 12 decision units. The sample grade distribution of each unit was obtained and only two distributions, one for unit 2 and the other for unit 6, are shown in Fig.2. The ore quantities and the average ore grades corresponding to a set of cutoff grades for these two units are shown in Fig.3.
Fig.2 Grade distribution in unit 2 (a) and unit 6 (b)
The cost data were determined by a comprehensive analysis of all the cost items involved in mining and ore concentrating. Data used in the economic evaluation are: mining cost is 14 US$/t; concentrating cost is 4.8 US$/t; concentration grade is 40%; concentration price is 1 300 US$/t; mining recovery is 87.54%; mill recovery is 96%; dilution is 32.13%; production rate is 1.03×106 t/a; and discount rate is 10%.
Fig.3 Grade-tonnage relationship for unit 2 (a) and unit 6 (b)
By applying the developed optimization algorithm, the resulting optimum cutoff grades were obtained for the 12 decision units as shown in Table 1.
Table 1 Cutoff grade optimization results
The optimum cutoff grade exhibits a relatively small variation from 0.8% to 1.0%. This is expected because the grade distribution varies to a relatively small extent from unit to unit as can be seen from Fig.2. For a deposit whose grade distribution exhibits higher variation with location, the resulting optimum cutoff grade will vary to a greater extent.
It can be seen that the cutoff grade selected for decision unit 6 is somewhat higher than that for decision unit 2, mainly because the former has a higher average ore grade than the latter for a given cutoff grade (Fig.3).
The mine has been using two grades to delineate ore, a cutoff grade for single samples and a minimum average sample grade for a continuous area of ore. The former is 1.0% and the latter is 1.2%, for the entire mine. Therefore, the cutoff grade being practised is close to the optimum.
The cutoff grade results using Lane’s algorithm are: 0.9% for the first 7 units, 0.8% for units 8 and 9, 0.7% for unit 10, and 0.6% for the last two units. The total NPV obtained using the dynamic-programming based model is 1.884×107 US$, or 7.55%, higher than the NPV using Lane’s algorithm. The benefit of using the new model will be even higher if the local grade distribution exhibits a more pronounced variation among the decision units.
4 Conclusions
(1) A dynamic-programming based model is developed for cutoff grade optimization in underground metal mines. The merits of the model lie in that the local grade variation in a deposit is fully accounted for, and it is particularly advantageous to use the model in cases where the spatial grade variation is high. The main shortcoming is that the method has a high requirement for exploration samples. For those mines with few samples that cannot reflect the real local grade distribution, the advantage of the method diminishes.
(2) The model is applied to a copper mine in Africa and proved that the model is capable of handling real-life case studies. The application also demonstrates that the model is robust in that relevant constraints and comprehensive economic evaluations can easily be incorporated. Sensitivity analyses on relevant parameters can be easily done using the software. The model can also be adapted to optimize cutoff grade in open-pit metal mines with relatively small changes (delineation of the decision units will be different).
References
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Foundation item: Project(50974041) supported by the National Natural Science Foundation of China; Project(20090450112) supported by the Postdoctoral Foundation of China; Project(20093910) supported by the Natural Science Foundation of Liaoning Province, China
Received date: 2009-10-20; Accepted date: 2009-12-29
Corresponding author: WANG Qing, Professor; Tel: +86-24-83678400; Fax: +86-24-83680388; E-mail: qingwangedu@163.com
(Edited by YANG You-ping)
Abstract: In order to maximize the overall economic gain from a metal mine operation, selection of cutoff grades must consider two important aspects: the time value of money and the spatial variation of the grade distribution in the deposit. That is, cutoff grade selection must be dynamic with respect to both time and space. A newly developed method that fulfills these requirements is presented. In this method, the deposit or a portion of it under study is divided into “decision units” based on the mining method and sample data. The statistical grade distribution and the grade-tonnage relationship of each decision unit are then computed based on the samples falling in the unit. Each decision unit with its grade-tonnage relationship is considered as a stage in a dynamic programming scheme and the problem is solved by applying a forward dynamic programming based algorithm with an objective function of maximizing the overall net present value (NPV). A software package is developed for the method and applied to an underground copper mine in Africa.
