DEFINING THE PRODUCTION SCALE OF AN UNDERGROUND MINE

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1 The International Journal Of Mineral Resources Engineering, Vol. 12, No.1 (2007) 1-19 Atilim University Press DEFINING THE PRODUCTION SCALE OF AN UNDERGROUND MINE B. DING Delft University of Technology, the Netherlands C.W. PELLEY Queen s University, Canada J.J. deruiter Delft University of Technology, the Netherlands In the mining industry, the production scale of a mine operation has been defined as the production rate, or the mine capacity. The production level, another important criterion of the production scale, has been neglected. Such a misconception of the production scale may cause improper definition of the capacity, and therefore cause serious problems in the future mine operations. In fact both the production rate and level affect the mining economy in different ways and one affects the determination of the other. This paper demonstrates how the production rate and level influences the mining operation economically, examines the interrelationships between these two parameters, and illustrates how to determine the mine production scale by looking at both the production rate and production level definition. This paper is based on the researches on hardrock mining cases, but the application of the principles delivered may be extended to the other types of mines, provided that their special characteristics are considered. 1. Introduction At the preproduction mine planning stage, production scale definition is considered a key determinant for decisions on other planning parameters. Given a clearly delineated deposit, how the valuable minerals can be economically extracted becomes an important topic of study. This covers mining and processing methods (how the ore will be recovered and treated), the production scale (how much of the deposit should be exploited and how fast should it be depleted) and depletion scheduling and sequencing. When talking about the production scale, people usually refer either to the capacity of the mine or treatment plant, to the total mineral products to be produced over the mine life, or to the mineable reserve of the orebody together with grade information. In fact, neither the capacity nor the total production output is sufficient to give a complete picture of the mine production economics. Only when the production rate and total production information is provided together can a quick evaluation of the project be conducted, and hence the scale of production be known. The size of the reserve is not- 1

2 B.DING - C.W. PELLEY - J.J. deruiter hing more than the raw material, the input of the production, and therefore it should not be used as a measure of the production scale. Notably, however, this raw material is quite different from that of other manufacturing industries which can be obtained from the markets. The deposit is exhaustible and as production proceeds the quality and workability of the material may be decreasing and unavoidably the costs of production be increasing. The total production, or production level is closely related to the reserve parameters. Little existing literature has emphasized the production level. As early as in 1954 Carlisle stated 1 : in attempts to apply economic theory to production scale definition, confusion has been caused by the failure to distinguish clearly between the rate of recovery of the mineral product and the completeness of extraction of the total mineral occurring in the ground. The misconception of the production rate being the production scale still exists amongst the related publications. The neglect of production level in defining the mine production scale may cause serious future operating problems. This paper is based on research on hardrock mining cases, but the authors believe it may also serve as a guideline to the production scale determination for the other types of mines, such as coal and industry minerals operations. It is advised however that those evaluating different types of minerals follow only the principle provided in this paper but not necessarily the entire procedure described. 2. Production Rate and Production Level Concept and Relationships The production scale definition is eventually based on a detailed cost and benefits analysis and, by means of economic optimization techniques, must define two distinct variables concerning a mine, the production rate and the production level Concept of production rate The production rate refers to the capacity, the tonnage of ore to be recovered or total metal to be produced per unit of production time, usually per year. It determines the size of the mine operation and specifies the requirements of the size of the mining and mineral treatment facilities, equipment, and number of stopes working at the same time, etc. For economic review, the production rate determines the capital requirement, the cost of the production and, given the grade information, how much revenue can be expected within each unit of time. One should keep in mind that the production rate determines directly the size of all the development, mining and treatment facilities, accesses, and equipment. Once the production rate is determined and development starts, it will be physically difficult for the production rate to be changed significantly, unless an expansion with new capital investment is planned. Therefore, close attention has to be paid to how the production rate is defined at the preproduction planning stage when sufficient information may not yet be available. 2

3 Defining The Production Scale 2.2. Concept of the production level The production level defines the total amount of minerals to be extracted from a given mineral deposit and sold to the market. In classic economic theory, the production level is, in reality, the total output of the operation. In the mining industry, it can be the total ore to be mined or total amount of mineral products to be produced over the mine life. A common mistake existing in the mineral industry is to confuse the production level with the total reserve. They are indeed different concepts although in some special cases they can be close to each other quantitatively. The production level is the output of specific production economics whereas the reserve is the fundamental input. The reserve is usually defined by a cutoff grade determined by the economic situation, but the production level, which eventually determines the cutoff grade, is determined strategically based on the operating objectives. To recover 100% of the reserve would normally be unjustifiable either technically or economically, hence the total production of ore has to be a portion of that estimated. On the other hand, however, the production level determination has to be based on the amount or reserve of the deposit. Normally a large reserve infers a high production level. Unlike the production rate, there are two reasons the production level must be modified as the operation progresses. One, the reserve is estimated on the basis of an earlier economic situation, and thus will vary with any economic changes in the future. If, for example, the prices of the contained minerals drop significantly, then part of the reserve may become marginal or sub-marginal. If the production level remains unmodified, to recover this part of the material will undoubtedly diminish the total profit of the project. Two, at different planning stages, changes in the recoverability of the orebody may be expected as progressively detailed studies are conducted. At the early pre-production planning stage, the production level is defined simply based on the orebody delineated with very limited information and usually it is assumed that all the materials inside the orebody are equally accessible and treatable. At a later development planning stage, when development layout and stope arrangement are under question, the grade distribution within the orebody dominates the problem. As a consequence, the production level is forced to change because some of the originally outlined reserve may become unmineable whereas some materials which were considered unmineable may have to be extracted or may become mineable if being extracted with the neighboring high grade material Relationships between the production rate and production level When determining the production scale, the inter-relationships between these two variables should be studied in accordance with different economic situations and production potentials. In the literature, the production level is frequently neglected with more emphasis being placed on production rate optimization. This treatment may be valid for open-pit mine planning because the total production of a pit operation is usually determined directly by the cutoff grade optimized using final pit optimization techniques and due to its low selectivity. But analysis of the level of production can be extremely im- 3

4 B.DING - C.W. PELLEY - J.J. deruiter portant for underground operations, especially when highly selective mining methods are adopted. In his 1954 publication, Carlisle elucidated 1 the relationships between the production rate and level expressed in terms of rate and level of recovery of the minerals contained in the orebody. He argued that, although decisions as to the rate and the level of recovery commonly influence each other, there is no inherent relationship between the variables. Based on this comment, he emphasized that possible relationships between the production rate and level shall depend on economic situations, extraction methods, orebody type and internal variables of the orebody as well as many other factors. His study has provided a basis of subsequent research in this field, although as an economist rather than a mining engineer, his neglecting the unique features and conventions of the mineral industry often made his points confusing. Generally speaking, given a reserve profile and the mining methods, and under the same economic situation, the production rate depends largely on the production level, although special cases may also be observed. If the deposit is small, the total production is limited. It obviously does not make sense to build a mine and treatment plant of high production rate with high capital expenditure. On the other hand, if a large deposit is under question and thus high mine output or production level is expected, a low production rate will undoubtedly overly increase the production life and reduce the present value of the production. One trend in the modern mining industry is that new technology has allowed the mine operation to recover lower grade ore. This makes the total production of ore sometimes extremely high when the so-called world class deposits are concerned (for example, large copper deposits in South America). Traditionally based on the present value consideration, the mine life is thought not to exceed 20 years. But for these very large deposits because the production rate is confined by many physical factors such as environment restrictions, limited size of available equipment and working sites, etc, the mine life may be decades, which simply do not follow the rules of how the production rate and level are related. 3. The Production Scale Definition Model In most manufacturing industries, it is usual that production optimization problems are solved by a multi-variable optimization model. But in the mineral industry this is difficult to achieve because there are too many uncertain factors involved in the decision making process. These complicated interrelated and interdependent factors make it virtually impossible to build a valid mathematical model to handle the optimization problems. Besides, due to the unique feature that mining production uses the highly changeable and exhaustible reserve as an input, the related cost and benefit functions are difficult to express as continuous mathematical functions as in simple input-output production cases. Therefore, the traditional mathematical solution cannot be applied to such optimization problems. The frequently adopted approaches to solve the problem are rule of thumb, trial and error, or pick and try methods. 4

5 Defining The Production Scale The invalidity of the multi-variable optimization approach to the production scale definition problem arises also from the important fact, as stated previously, that for a mining project once the production rate is determined there is not much room for change, whereas the production level may keep changing throughout the mine life. If the multivariable optimization technique is applied, then the production rate and level are bounded. When one changes the other has to be changed accordingly to hold the production optimum. If in the future, when the production level is significantly modified as the reserve information changes for any particular reason, the production rate has to remain unchanged and the production may not meet operational goals. In order to satisfy the situation, the production rate and production level definitions are discussed in two separate optimization problems. One optimization problem is to define the production level assuming the production rate is determined, and the other is, under exactly the same economic situation, to determine the production rate given the production level. The production scale definition model is to combine these two optimization problems, iteratively using the result of one to modify the other. Basic assumptions As this paper aims only to study the definition of the production rate and level at an early pre-production planning stage when the feasibility study and initial mine design is conducted, some assumptions are applied. Should any significant economic change occur in the future stages such as the production planning stage, although the production rate may not be changed broadly, it may always be possible to modify the production level by using the production level optimization process described in this paper. The assumptions are: - The mine property is developed in a single phase, all the investment is placed at the starting point of the development - All the ores in the deposit are equally accessible - The ore inside the orebody is equally treatable - The price remains constant throughout the mine life - There is no inflation throughout the mine life. 4. Production Rate Definition 4.1. Principle Gordon 2 stated in 1985 that mineral extraction is different from the production of other commodities, and decision making for mineral production must consider the optimum life of the facility and the annual rates of production during each operating year. Exploitation of mineral deposits involves the special problem of depletion. That is the resource is exhaustible, and tradeoffs have to be made between the benefits and costs of speeding depletion using more immobile capital. Good references on production rate definition can be found from several different discussions and studies 3,4,5,6. The main benefit is that production occurs sooner and thus has a higher gross pre- 5

6 B.DING - C.W. PELLEY - J.J. deruiter sent value. However, this may cause other costs to increase, the most obvious being that to increase the rate of annual production more capital must be invested in fixed facilities. At some point, the cost of this investment increase will outweigh the benefit of the earlier sales revenues. This will certainly be true if the capacity costs rise at the same rate or a higher rate than the output increase. Even if the rise in costs is slower than the rise in exploitation rates, it is likely that the firm will still decide to spread exploitation over several years. This problem can be defined within two extremes. Operating at a low production rate, given the total mineable reserves, will cause the mine life to be long. The capital investment required to bring the mine into production may be quite low due to the use of small facilities and equipment, but the total present value will be low because of the limited annual output and the discounting of the future production. The other extreme is that the production rate is high. This will of course shorten the mine life and thus generate a high present value of production, but the high capital investment for the large mining and mineral processing facilities and complex management systems will reduce the overall project returns. The production rate optimization problem is to find out the specific production rate somewhere between the two extremes so that net present value, the total present value of the net cash flow, is maximized. Therefore, a net present value maximization problem will be introduced into the system for production rate definition Net Present Value Maximization Problem Net present value maximization is the most frequently adopted technique in the mining industry in determining the different parameters of the mining operation, such as cutoff grade, the mine capacity, mine life, etc. This method is to inspect the value of the annual net cash flow at the present time. As the net cash flow, the difference between the positive and the negative cash flows, is a function of many different variables, the net present value also varies with these variables. For simplicity purposes, usually only the variables that are to be determined by the NPV maximization problem are thought to be variable, whereas the other variables are assumed to be constant throughout the mine life. As discussed previously, under a determined production level, the main objective of this maximization problem is to adjust the production rate so that the mine operation can reach the highest net present value Cash flow function The basic elements of cash flow analysis are demonstrated in Figure 1. It is straightforward that all the elements involved in the yearly cash flow calculation, the revenue, cash and non-cash charges, are connected directly to the production rate. Therefore, based on the previous assumptions, the annual net cash flow of the mine operation can be expressed as a function of the production rate: 6

7 Defining The Production Scale NCF = NCF PR (0,PL) (pr) (4.1) where, PL - production level pr - production rate Fig. 1. Cash flow structure of a mine operation 4.4. The NPV function The net present value of the mining project is the total of the annual cash flow of each production year discounted to the present time. For simplification purpose, the present time is set as the year immediately before the production starts. Based on the NCF estimation discussed above, the net present value function can be expressed as: 7

8 B.DING - C.W. PELLEY - J.J. deruiter T NCF (pr) NPV = - TCC(pr) (4.2) 0 (1+i) t where, i discount rate; T mine life t the number of production year TCC(pr) - total capital expenditure at present time, a function of production rate pr production rate As the ore reserve profile and the production level are given, the mine life T can be determined accordingly by a specific chosen production rate. Therefore, the NPV is a function of only the production rate NPV maximization Given the NPV function of the production rate as illustrated on the above section, we can now maximize the NPV with respect to the production rate, or find a specific production rate at which the operation can reach the highest net present value. The maximization problem can be expressed as the following equation. ( ) T NCF (pr) NPV max =Max pr - TCC(pr) (4.3) 0 (1+i) t Notably, the NCF and the NPV may not be mathematically expressed as a continuous function and so the problem is difficult to solve using mathematical approaches. In this case, a simple pick and try method may be adopted to solve this maximization problem. This will rely on the computer program which is to calculate and compare the NPV at different discrete production rates, find the NPV maximum and hence define the production rate optimum. Figure 2 illustrates the principle of defining the specific production rate that maximizes the net present value. 5. Production Level and Planning Cutoff Grade 5.1. Principles As discussed previously, the production level relies largely on the quality of the ore reserves. The lower the grade or the poorer the quality of ore, the higher will be the costs of recovery of the valuable products 1. The quality of ore is measured by many different factors, typically the ore grade and its distribution, orebody continuity, the workability of the natural deposition of the mineralized bodies, and the ore treatability. Although detailed study on the quality variability of the ore of the deposit is out of the scope of this paper, the underlying principle that the higher production rate can only be achieved by extracting more low quality material at higher costs is true. 8

9 Defining The Production Scale Fig. 2. Net present value vs. production rate Looking at the tonnage-grade curve, it is obvious that, for any deposit, the tonnage and average grade always go in inverse directions - the higher the tonnage, the lower the average grade. It implies that, although increasing the production level will increase the project revenue, the costs will also be increased to match the production requirement of extracting and treating the higher quantity of lower quality ore. If the production level is set too high, the added revenue generated by a production level increment may be totally offset or outweighed by the added costs. The key question of the production level selection is how far we should go when exploiting the deposit so that the operation will be more profitable. Profit and cost observations become the unique valid technique to answer this question. According to the classic production optimization theory, two project profitability indicators, the total unit profit and the total profit, are usually examined. This section attempts to explain how to define the optimal production level by solving the TUP and TP maximization problems. Given the reserve or resource profiles in terms of total tonnage and average grade, the production level optimization problem becomes how much of the reserve should be recovered at a given production rate, so that the mine operation can be best off by looking at the total profit and/or the total unit profit of the production. Given the orebody, each cutoff grade gives a unique set of tonnage and average grade, and hence the amount of mineral products that can be recovered. Inversely, if the production level which defines the amount of mineral products is determined, the tonnage and the associated average grade required to achieve such a production level can be used to make cutoff decision. This determined cutoff grade is the optimized planning cutoff grade Cost and profit For the definition of the production level optimization problem by using total profit and total unit profit maximization techniques, detailed cost and benefit analysis should be 9

10 B.DING - C.W. PELLEY - J.J. deruiter conducted. To do so, the following terms of costs and benefits are adopted. Some of the cost items may be a function of both the production rate and level, but since in this optimization problem the production rate has been assumed to be determined, all the costs and profit will vary only with the production level. 1) Total unit revenue (TUR) Total unit revenue is the income of selling unit mineral product in the market, so it is simply the price of the product: TUR=P (5.1) where, P is the price of the mineral product. 2) Average total unit cost (ATUC) Average total unit cost is estimated by distributing all the cost items to each unit of mineral product extracted. This will include both the capital and operating costs averaged to each unit of production in some way. The cost can be estimated on a yearly basis and then distributed to each unit of production. (1)Capital cost This includes all the ongoing capital requirements (or termed the total capital costs, TCC) which are classified as fixed capital costs (FCC) and variable capital costs (VCC). The FCC are those items that have to be incurred as long as the project development proceeds, and therefore can be considered irrelevant to the production level. The VCC are the capital costs that are directly or indirectly affected by the production scale, either by production rate or level or both. Since the production rate is assumed to be fixed, the general relationship between the TCC and production level can be expressed as: TCC(PL) =FCC+VCC(PL) (5.2) The capital costs may not be simply averaged directly by the total units of production to calculate the average capital cost, but should depend upon the source of the capital funding and in which way they are recovered. For example, if the mining corporation uses a loan, then amortization may be applicable in loan payback. In this case the yearly equivalent capital expenditure should be calculated using the formula (5.3) below, rather than simply averaging the TCC to each production year as demonstrated in formula (5.4) which may or may not be suitable for other types of capital funding, such as internal capital. TCC n (PL) = TCC(PL) CRF(ML, R) (5.3) where, TCC n total capital expenditure in year n; R is the discount rate; ML is the mine life; CRF is the capital recovery factor. PL is the production level and rate. TCC n (PL) = TCC(PL) / ML (5.4) 10

11 Defining The Production Scale (2) Operating costs The operating costs are composed of fixed operating costs (FOC) and variable operating costs (VOC). FOC refers to those that have to be spent as long as the operation is running, typically the costs of administration, services, insurance, depreciation, interests, and taxation, etc. VOC are the variable or direct operating costs which vary solely with the production rate under the given production conditions. It seems that, in each production year, neither FOC nor VOC is relevant to the production level, but one should notice that as the production level goes up more amounts of lower grade ore may have to be recovered. Consequently, if the production rate remains the same in terms of tons per annum, the total units of products (if the ore is not the final product) may decrease and it brings the unit cost of the products up. Therefore when calculating the average total cost of the product, it has to be adjusted by the average grade associated with the production level. Based on the above discussion, the AUTC function can be expressed as: ATUC(PL)= TCC n (PL)+FOC +VOC / ḡ (PL) (5.5) PR where, ATUC is the average total unit cost, a function of PL; ḡ is the average grade of the reserve which is related to PL. 3) Total unit profit (TUP) Total unit profit is simply the difference of the total unit revenue and average total unit costs. Since ATUC is a function of PL, so is TUP. TUP(PL) = TUR ATUC(PL) (5.6) 4) Marginal capital cost (MC) Marginal cost is the cost increase when the production level is increased by 1 unit. It is expressed as the deviation of the total cost function with respect to the production level. Since operating costs are constant components in the total cost function, so marginal cost is only the derivative of the total capital cost: MC(PL)= d(tcc) (PL)) d(pl) (5.7) 5) Total profit (TP) and marginal profit (MP) Besides the above-mentioned cost and benefit items, there are total profit and marginal profit which can be generated as functions of production level. Total profit (TP) is the total gain of the mine venture when it is operated at a certain production rate and production level. Marginal profit (MP) refers to the increase of profit when the production 11

12 B.DING - C.W. PELLEY - J.J. deruiter level is increased by a unit which can be expressed as the derivative of TP with respect to the production level: TP(PL) = TUP(PL)_PL (5.8) MP(PL)= d(tp) (PL)) d(pl) (5.9) 5.3. The TUP and TP maximization problems and solution As the relationships of the production level to the total unit profit and total profit are established, the two maximization problems that determine the best production level can be defined as Maximize TUP = TUR(PL)-ATUC(PL) subject to: PL Π(0, total metal content), and Maximize TP = TUP(PL)*PL subject to: PL Π(0, total metal content). Theoretically, these two maximization problems can be solved by using mathematical approaches, but the unique complexity of the mining industry make it difficult to express mathematically either TUP or TP as a continuous function of the production level (PL). It means the traditional methods are not valid in reality. In this case, with the assistance of computer programming, a simple pick-and-try method can be used. This method is to calculate the TUP (or TP) values corresponding to different values of PL which are plotted on a graph for cost-benefit analysis. Based on classic production theory, the problem can be solved by reviewing the relationships of cost and profit to the scale of production on the basis of a fixed production rate. Figure 3 7 illustrates a typical production cost and benefit economy. In this graph, the ATUC, TUR and MC curves intersect each other and present 4 intersections, A, B, C and D 1,7. According to the cost-benefit analysis, point A and D where the average total unit cost is balanced by the total unit profit define the breakeven production; Point B where the ATUC equals the marginal costs represents the lowest average total unit costs or the highest total unit profit; At point C the marginal cost is balanced by marginal profit which means the profit created by further production beyond this point will be overweighed by the underlying added costs and thus the total profit will be diminished. Therefore point C defines the maximal total profit Production level determination From Figure 3, we find that the maximum TUP and TP are different. The problem becomes which of them defines the optimal production level. Carlisle argued that the optimal production level should be at the maximal TUP po- 12

13 Defining The Production Scale int 1, but he is an economist rather than a mining economist. His point is true for simple input and output production where raw material from the market is the main input of the production. Mineral resources are different from other raw materials not only because it is an exhaustible special material, but more importantly because the cost of achieving every unit of such material as input to the production is different. While studying the total unit profit of the mine production, we have to remember that many cost items of ore extraction are related to the whole deposit rather than to a single ton like the raw materials purchased directly from the market. It implies that, the fewer the units you extract from the deposit, the higher the unit cost will be expected on the ore you recover. To treat an orebody in the same way as the production of other raw materials may result in serious mistakes. Moreover, a lower production level (as at the maximal TUP point) sacrifices a large quantity of valuable and (actually also profitable) nonrenewable resources although it may result in a lower production rate, and lower development capital input. Fig. 3. Optimization of mineral production - after Von Wahl, 1983 (modified) 13

14 B.DING - C.W. PELLEY - J.J. deruiter In fact, the maximal TUP and TP points define not only these two selectable production levels, but more importantly defined an optimal range in which a better optimal production level should be defined. Theoretically, the maximal net present value point should also be located in the range (refer to Figure 3). What production level should be defined for future production really depends on what the profit goal of the corporation is and what discount rate is selected. If the corporation seeks the highest total profit, then the higher production level should be set (to the right most of the range) and if the maximum unit profit of production is the objective of the mining operation, the left hand boundary of the range should be chosen. Some mine operations look more at the net present value by assigning a discount rate, then the production level should be somewhere between the two points. In this paper, the maximal TP point (point C in Figure 3) is basically considered the initial optimal production level point. This treatment not only simplifies the problem, but also, because this point defines the maximum use of the resources which generates the greatest project profit, it makes sense for the mining project. Where a high discount rate is applicable, however, this production level optimal point may have to be adjusted towards the TUP maximal point. Such an adjustment will be made during the production rate definition process, which uses the NPV maximization techniques as discussed in the previous sections, and therefore the final production level point should be somewhere close to the NPV maximization point (refer to Figure 3) The tonnage-based production level and planning cutoff grade It is more widely accepted and convenient to use tonnage for calculations in the mining industry. The production level as discussed so far defines how much metal is to be extracted from the orebody. To the mine superintendent, however, how many tonnes of ore and what average grade should be exploited to match the defined total metal production will be of more interest. If we call the production level defined in the above section the metal-based production level which reveals the overall economy of the operation, then the tonnage required from the outlined resource (or reserve) may be termed the tonnage-based production level. The tonnage-based production level is not only needed by the mine site but also, together with the corresponding average grade, it is the fundamental input of the production rate definition process. To define the tonnage-based production level, what needs to be done is merely to calculate inversely how many tonnes of ore should be mined from the metal-based production level. The original reserve information is the basis of this calculation. Knowing the total amount of metal to be produced and sold to the market, the amount of metal contained in the ore mined can be calculated with consideration of the metal losses in each processing step (mine, mill, smelter and refinery). Based on the tonnage-grade curves of the reserve and the mining recovery estimation, and applying the best ore mined first principle, the amount of metal which should come from how many tonnes of ore can now be calculated. The relationship of the metal-based production level to the tonnage and average grade requirement from a given reserve profile can be plotted. Figure 4 provides the curves of such a relationship generated for a gold deposit. It can be seen 14

15 Defining The Production Scale in this graph, that as PL increases the tonnage required or the tonnage-base production level increases much more quickly. This is true because, based on the best ore first taken principle, the increment of metal production must be achieved from higher tonnage of lower grade materials. As far as the cutoff grade definition is concerned, Lane 8 pioneered many of the principles based on experience gained during many years working in this field and provided a series publications on this topic. In his book, the approach of cutoff grade definition considers the economic limits and capacity balancing amongst the three different mining stages. It can be argued that he looked only at the production rates of these stages while performing the NPV estimation, which is the objective of the cutoff grade optimization in his studies, but ignores the production level and how it might be affected by the grade distribution of a deposit. In certain cases, the grade curve may be quite flat as the cutoff grade is lowered but at a certain cutoff it suddenly turns sharply upwards. If a cutoff grade is set, by only looking at the production rate, at this inflection point or above, then a large amount of ore could become unmineable resulting in a project which does not have sufficient reserves to support the optimal production rate This paper is not intended to provide a detailed discussion on cutoff grade definition, but a valid approach is examined together with the production level optimization process. Once the tonnage-based production level, with the consideration of comprehensive recovery factors at different processing stages, is defined using the PL-tonnage curve, this determined tonnage can be applied back to the reserve tonnage-grade curves to determine what suitable cutoff grade should be selected. This is the planning cutoff grade as depicted in Figure 5. This cutoff grade is in fact defined indirectly by solving the total (unit) profit maximization problem and justified by the NPV considerations in the iterative process being discussed in the following section. Therefore this cutoff grade represents a more reasonable cutoff policy. Fig. 4. Production level vs. tonnage and grade 15

16 B.DING - C.W. PELLEY - J.J. deruiter Fig. 5. Defining planning cutoff grade using tonnage-grade curves 6. Optimization of Both Production Level and Production Rate Based on the above discussion, either the production rate or production level is defined under the condition that the other is fixed. There is a problem with such a treatment, i.e., the fixed production rate or level is not necessarily the best choice, and thus the resultant optimum may not really be the optimal result for the operation scale definition. To solve this problem, further examination of the pairs of production level and rate (one of them is the fixed input and the other is the output of the optimization problem) in both the optimization processes has to be conducted. To do so, both the optimization processes have to be iteratively run. This iterative process is illustrated in Figure 6. Fig. 6. The iterative optimization process of production rate and level 16

17 Defining The Production Scale To avoid overly extensive calculations and improve the model efficiency, it is wise to avoid choosing an obviously unreasonable production rate or level to start the iteration. Initially, a production rate can be estimated by using one of the available empirical equations 6,9 (say, Taylor s equation T/d = 0.014(Reserve) 0.75 a. Using this fixed production rate as the input of the production level optimization process, an initial optimal production range can be defined in which a production level can be selected according to the discount rate set by the corporation. This selected optimal production level in turn works as the fixed input in the production rate definition problem to choose the production rate. If the defined production rate is quite different from the initial one, then this production rate is applied to the production level optimization problem to modify the level of production. This process keeps going until the production level and rate combinations defined by the two optimization problems justify each other. Notably, the production level optimum is actually defined by a range of production levels. In usual cases, either the total profit maximum point or the total unit profit maximum point can be taken as the optimal production level based on the discount rate the corporation chooses. The lower the discount rate is set, the more likely the total profit maximizing production level is favorable. This range allows flexibility in production rate determination, i.e. any production level lying in this range can be considered optimal. Ideally, the optimal pair of production level and rate should both be optimized when one works as the input of either of the optimization processes to define the other. But in reality this ideal case can hardly happen. In this case, the optimal production range gives a more flexible concept of the optimal production level, i.e., any total production falling in this range can be thought of as an optimal production level. Hence, if the production rate defined by a specific production level from this range is close enough to the original production rate that defines this range, then the combination of this production level and production rate defines the optimal production scale. 7. Program Structure This production scale definition model is implemented by Visual Basic programs. The programs are divided into two major program groups that function for production rate and production level optimization respectively. The functions of these two groups of program are illustrated in Figure 7. The two groups of program should be running iteratively until the two sets of production rate and level are close to each other. a This equation calculates the daily capacity which has to be converted to yearly production rate. 17

18 B.DING - C.W. PELLEY - J.J. deruiter Fig.7. The structure of the VB program for the preproduction planning 8. Summary The production scale is defined by two quantitative variables, the production rate (capacity) and the production level (the total production of mineral products). This paper addresses fundamental discussions about the production scale definition at the preproduction planning stage. Based on a review of the previous studies on similar topics, a persistent misunderstanding of the production scale has been found. This misunderstanding is that the production scale is indicated by the production rate alone. In order to achieve valid definition of the production scale, the relationships of the two variables and other factors are studied. An optimization model is introduced to present a valid approach for both the production rate and production level definition. Due the special difficulties in the application of mathematical approaches to handle the optimization problems with the mining industry and because of the fact that the production level is subject to change as the project development progresses, this model is composed of two separate optimization problems, rather than a single multi-variable 18

19 Defining The Production Scale optimization problem, defining the production rate and level respectively assuming the other variable is known. 10 One of the optimization problem locates the optimal production rate that maximizes the NPV of an operation at a given total production requirement; the other, at a given capacity, defines the optimal production level that maximizes the project profit criteria. The final goal of this model is to provide a pair of production rate and level that solves both the optimization problems. As initially it uses an estimated production rate as input to optimize the production level or the other way around, such pair may not give a correct definition of the production scale. Therefore continuous modification is needed iteratively in both the optimization processes so that the goal can be reached. Although cutoff grade is not focused in the discussion in this paper, it is well known as one of the most important elements of the preproduction planning stage. A different approach to the definition of the cutoff grade is introduced. References: 1. D.Carlisle, 1954, The Economics of a Fund Resource with Particular Reference to Mining, The American Economic Review, 44, (1954), R.L.Gordon, The Production of Mineral Commodities, Economics of the Mineral Industry (A Series of Articles by Specialists), 4 th edition, 1985 AIME, M. T. Gentry, and T.J. O Neil, Mine Investment Analysis, Society of Mining Engineers, L.D. Smith, A Practical Guideline to Selecting an Optimum Production Rate, CIM Bulletin, January L.D. Smith, A Critical Examination of the Methods and Factors Affecting the Selection of Optimum Production Rate, CIM Bulletin, February H. K. Taylor, Rates of Working of Mines - A Simple Rule of Thumb, Technical Note, Transactions of the Institute of Mining and Metallurgy (Section A: Mining Industry), 95, Oct., (1986), A S. Von Wahl, Investment Appraisal and Economic Evaluation of Mining Enterprise, Trans Tech Publications, Clausthal-Zellerfeld, Federal Republic of Germany, K. F. Lane, Economic Definition of Ore Cutoff Grades in Theory and Practice, Mining Journal books, E. Arioglu, Examination of Empirical Formulae for Predicting Optimum, Tran. Instn. Min. Metall. (sect. A: Min. industry), 97, January 1988, A B. Ding, Examining the Planning Stages in Underground Metal Mines, Ph.D. Thesis, Mining Dept., Queen s University, Kingston, Canada,

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