Anderson Graduate School of Management – Decisions, Operations, and Technology Management UC Los Angeles Peer Reviewed Title: Optimizing Long�Term Production Plans in Underground and Open Pit Copper Mines Author: Epstein, R., University of Chile Goic, M., University of Chile Weintraub, A., University of Chile Catalán, J., Fundación para la Transferencia Tecnológica Santibáñez, P., Fundación para la Transferencia Tecnológica Urrutia, R., Fundación para la Transferencia Tecnológica Cancino, R., CODELCO Chile Gaete, S., CODELCO Chile Aguayo, A., CODELCO Chile Caro, F., UCLA Anderson School Publication Date: 04-01-2010 Publication Info: Decisions, Operations, and Technology Management, Anderson Graduate School of Management, UC Los Angeles Permalink: http://escholarship.org/uc/item/475353w1 Abstract: We present a methodology for long�term mine planning based on a general capacitated multicommodity network flow problem formulated as a comprehensive math programming model. The planning methodology considers underground and open pit ore deposits sharing multiple downstream processing plants over a long horizon. The purpose of the model is to optimize several mines in an integrated fashion, though the real size instances are hard to solve due to the combinatorial nature of the problem. We tackle this by solving the relaxed problem of a tight linear formulation and we round the resulting near�integer solution with a customized procedure. The outcome of this project has been implemented at Codelco, the largest copper producer in the world. Since 2001, the system is used on a regular basis and has increased the net present value of the production plan for a single mine by 5%. Moreover, the integrated approach for multiple mines provided an additional increase of 3%. The system has allowed the planners to evaluate more scenarios. In particular, the model was used to study the option of delaying by four years the conversion of Chiquicamata, Codelco's largest open pit mine, to underground operations.
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Optimizing Long‐Term Production Plans in Underground and Open Pit Copper Mines Rafael Epstein, Marcel Goic, Andrés Weintraub Industrial Engineering Department, University of Chile, Casilla 2777, Santiago, Chile {repstein, mgoic, aweintra}@dii.uchile.cl
Jaime Catalán, Pablo Santibáñez, Rodolfo Urrutia Fundación para la Transferencia Tecnológica, Avda. Beauchef 993, Santiago, Chile {jcatalan, psantiba, rurrutia}@dii.uchile.cl
Raúl Cancino, Sergio Gaete, Augusto Aguayo CODELCO Chile, División Norte – El Teniente – División Andina {rcancino, sgaete, aaguayo}@codelco.cl
Felipe Caro1 UCLA Anderson School of Management, Los Angeles, CA 90095
[email protected]
April 22, 2010 ABSTRACT We present a methodology for long‐term mine planning based on a general capacitated multi‐ commodity network flow problem formulated as a comprehensive math programming model. The planning methodology considers underground and open pit ore deposits sharing multiple downstream processing plants over a long horizon. The purpose of the model is to optimize several mines in an integrated fashion, though the real size instances are hard to solve due to the combinatorial nature of the problem. We tackle this by solving the relaxed problem of a tight linear formulation and we round the resulting near‐integer solution with a customized procedure. The outcome of this project has been implemented at Codelco, the largest copper producer in the world. Since 2001, the system is used on a regular basis and has increased the net present value of the production plan for a single mine by 5%. Moreover, the integrated approach for multiple mines provided an additional increase of 3%. The system has allowed the planners to evaluate more scenarios. In particular, the model was used to study the option of delaying by four years the conversion of Chiquicamata, Codelco's largest open pit mine, to underground operations.
1
Corresponding author.
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1. INTRODUCTION Basically there are two major forms of mining: underground and open pit. For both, the use of models to support extraction planning has a long practice in the mining industry. Traditionally, systems based on average values have been used. A fundamental concept has been that of a cutoff grade (Lane 1988) based on the idea of finding which is the minimum grade in a block (i.e., the basic geological planning unit) which is profitable to mine. Given the advances in software and hardware, lately the idea of modeling the mine in finer detail and reality as is possible using mixed‐integer programming (MIP) models has gained strength (Alford et al. 2007, Cacceta 2007). This approach is far more powerful, as it allows representing the mine in complete detail, where the block of rock is the minimum element that can be defined. In this form, a MIP can represent the extraction problem in a more exact, realistic way. These models can incorporate in a natural form constraints related to geomechanical, economic or extraction sequencing conditions. The size of such models though makes these formulations difficult to solve, and the literature so far presents few applications of large‐scale mines. This paper presents and solves a MIP model that has been successfully used in Chilean copper mines by state‐owned Codelco, the largest copper enterprise in the world, for both underground and open pit mines. The paper has several novelties: It integrates the mine extraction with downstream processes, showing the advantages of integrating planning along the whole production chain, until the final product, refined copper is obtained to be shipped. Traditionally, mining and plant processes have been solved separately, iterating between the two to match supply and demand. It considers, as is the case in Codelco, the interaction of multiple mines, multiple products and multiple downstream processes. In particular, it incorporates the notion of prime material competing for limited processing capacity as the mineral goes along the chain. Traditionally, planning has been carried out not considering this multiplicity. It develops a model that generalizes the concept of underground and open pit mining, as part of a capacitated network representation. This allows representing any form of mining as part of a process in the production chain, and in particular the change from open pit to underground becomes natural to introduce. The network representation allows representing directly: a) constraints related to maximum and minimal rates of extraction, b) smoothness of extraction and correct sequencing of extraction points in underground mines, c) correct breaking of rocks, geomechanical constraints and possible interactions between sectors, and d) gradually sloping walls in open pit mining to avoid sliding of waste material. Also, in the network, the use and investments in machinery, equipment and transportation are modeled in a natural 2
way. These models are very large MIP's, impossible to solve using standard commercial codes. Heuristic techniques were developed which allowed solving these problems with relatively small error bounds. The model to be shown in the next sections has been implemented at Codelco. First, a model was developed and used for extraction and investment planning on a 25 year horizon at El Teniente, a very large underground copper mine. Its use supported decisions on the timing, sequencing of blocks to be mined, as well as deciding which blocks did not yield enough value to be mined. Improvements due to the use of the model led to an increase in value of the mine of over $100 million. Then, the model was expanded to incorporate open pit mines. It is being used for planning in the northern groups of mines in Chile. In this case, several mines share common plant installations, which include processes of flotation, leaching, bioleaching and low grade sulfates. The need to have one model integrating into the production chain both underground and open pit mines is due to increasingly common case of mixed mines, where both types of mines feed material to the downstream processes, or an open pit mine is converted to an underground one. We show specific cases of such mines. An important result of the use of the model is in guiding the allocation of mineral to the processing plants, taking advantage of the comparative advantages of mineral from different mines to be processed in the different plants. In this case, many instances have been run of the model to test multiple scenarios. The solution of these runs is coordinated with mining analysis to determine the timing and sequencing of the operation in order to be integrated with downstream plant processes. We show an important impact of the introduction of our model into the decision process, with gains between 5% and 8% in net present value confirmed by three independent case studies here described. The reminder of the paper has the following structure. In the next section we present a brief literature review. In section 3 we describe the problem and we provide some background on mining operations and the planning process at Codelco. In section 4 we present our planning methodology based on a general capacitated multi‐commodity network flow problem. In section 5 we introduce the customized rounding heuristics used to generate approximate solutions. In section 6 we provide an overview of the model’s implementation and impact at Codelco. In the final section we conclude. The theoretical proofs are given in the appendix. 2. LITERATURE REVIEW Most optimization models available in the literature have been developed for open pit mines and they solve only a partial version of the long‐term planning problem (Newman et al. 2009). Historically, the problem has been divided in two, the ultimate pit problem, which determines 3
the economic mineral reserve in an open pit mine, and the production‐scheduling problem, which considers not only which blocks to remove, but also when to remove these blocks. There are two main approaches to solve the ultimate pit problem. One is based on cutoff grades and the other one is based on Operations Research (OR) techniques. The optimal cutoff grade methodology was made popular through the work of Lane (1988). Fytas et al. (1987), Kim and Zhao (1994), and Poniewierski et al (2003) have extensively discussed this process. The basic premise of this approach is that one can determine a cutoff grade policy to maximize net present value subject to capacity constraints, with higher cutoff grades in the initial years of the project leading to higher overall net present value. It has important operational advantages and it is embedded in the background of all mining practitioners. However, the assumption of a fixed cutoff grade, which depends on an arbitrary delineation between ore and waste, generates suboptimal solutions because it ignores that the value of a block is not inherent to the block but rather depends on the interaction with the rest of the blocks and the capacity of the downstream processes. The use of OR techniques to solve the ultimate pit problem started with the classic “moving cone” heuristic, see for example Kim (1978) or Laurich (1990). This approach takes a block as a reference point and expandes the pit upwards according to pit slope rules. This solution can be suboptimal but it is intuitively appealing. Among the algorithms with a convergence proof to optimality, historically the most important are Lerchs and Grossmann (1965) and Picard (1976). The first one is based on graph theory but it can be shown that its structure is very similar to the dual simplex algorithm, see Underwood and Tolwinski (1998). The second paper shows the equivalence of the ultimate pit problem and the well‐studied problem of finding a maximum closure in a graph. Therefore, the ultimate pit problem reduces to a maximum flow problem that can be solved, for example, with a push‐relabel procedure. A comparison of both procedures together with a complete literature review of the ultimate pit problem can be found in Hochbaum and Chen (2000). Other relevant papers in this vein are Wright (1989), who suggests a dynamic programming approach, Frimpong et al. (2002), who use neural networks, and Jalali et al. (2006), who propose the use of Markov chains. Even though these algorithms are efficient, they do not consider what happens with the mineral beyond the extraction stage and, for instance, do not provide a production schedule that considers all the downstream capacities. OR techniques have also been used to solve the production‐scheduling problem for a single mine. Among the papers based on optimization methods, Gershon (1983) uses mixed integer programming, Dagdelen and Johnson (1986) suggest a Lagrangian relaxation, Caccetta and Hill (2003) use branch and cut, while Tolwinski and Underwood (1996) propose a dynamic programming approach. Given the complexity of the problem, several papers have resorted to heuristic methods such as simulation (Fytas et al. 1993, Erarslan and Celebi 2001), neural 4
networks (Denby et al. 1991), simulated annealing (Kumral and Dowd 2005) and genetic algorithms (Samanta et al. 2005). For other approaches that combine optimization and heuristic methods see also Sevim and Lei (1988), Sarin and West‐Hansen (2005), Busnach et al. (1985), Dowd and Onur (1993), Elevli (1995), Wang and Sun (2001), Smith et al. (2003), Ramazan (2007), and Chicoisne et al. (2009). The literature on underground mining is more recent, partially due to the complicated nature of underground operations. No parallel to the Lerchs‐Grossman algorithm exists for underground mines, so the early optimization work does not rely on network‐based methods. Strategic decisions in underground mining may consider how to geographically position various facilities, e.g., mills, on the mine site. Qinglin et al. (1996) use neural networks based on a deposit’s geotechnical and economic factors to optimally select an underground mining method. Yun et al. (1990) use a genetic algorithm to determine the number and spacing of openings given restrictions on their relative placement. Alford (1995) describes the floating stope method as a tool for analysis of mineral reserves and stope geometry. Barbaro and Ramani (1986) formulate a mixed integer programming model that can be used to determine whether to produce in a given time period. Kuchta et al. (2004) and Kuchta et al. (2003) describe the implementation of a production scheduling mixed integer program at LKAB's Kiruna underground mine. The objective is to minimize the deviation from a pre‐specified demand target per period, which contrasts with our model that maximizes net present value (NPV). The sizes of the instances are not comparable either but there are some similarities in terms the methods used to reduce the size of the model. As noted in Hochbaum and Chen (2000), a comprehensive model of mining operations has seldom been addressed in the literature. Even fewer implementations of such models have been reported in practice. The work by Carlyle and Eaves (2001) is a noteworthy attempt. Their mixed integer model was developed for an underground platinum and palladium mine and it considers several planning decisions, but it was applied to only one sector of the mine and near optimal integer solutions were obtained with a commercial software package. In our case, we consider all the sectors simultaneously and we even allow for multiple mines. In fact, we are unaware of other implementations that solve an integrated long‐term planning problem with the level of detail considered here. 3. PROBLEM DESCRIPTION We now provide background information and give more details of the problem tackled in the paper. In section 3.1 we introduce the long‐term planning problem in the mining industry. In section 3.2 we provide some background on underground and open pit mining. Finally, in section 3.3 we describe the planning process at Codelco and how our methodology fits in it. 5
3.1 Long‐Term Planning A long‐term mining plan consists of a detailed specification of each aspect related to investments and production in the mines and plants. The importance of the plan stems from the fact that most decisions are final and cannot be reversed, meaning that they can drastically affect the future of the mine. For example, poor capacity choices at the concentration plant or selecting an inappropriate extraction sequence can trigger a significant yield decrease. Since most mines have a long life cycle, the planning horizon is set somewhere between 10 and 30 years, with shorter periods at the beginning when more detail is required and deviations in the production have a major impact in the results. The plan not only must be economically profitable, but it also must be technically impeccable. This means, among other things, that is has to comply with the geotechnical rules that govern the mine, it must consider constraints due to the chemical processes at the plants, it must meet environmental regulations, and it should guarantee that downstream capacities will not be exceeded. The long‐term planner must solve three main problems taking into account the considerations mentioned above: (i) Investment: determine the selection and timing of investments; (ii) Extraction: determine the production in the mine; and (iii) Processing: determine the operation of the plants. Solving each one of these problems, even separately, is already a complex task. However, the need of an integrated planning approach cannot be dismissed. For instance, the real value of an investment project is only appreciated once its coherence is verified with respect to the extraction and processing decisions. Our methodology explicitly solves problems (ii) and (iii), i.e., extraction and processing. The investment decisions are usually restricted to a small set of options and can be evaluated with our model by running different scenarios. 3.2 Mining Operations 3.2.1 Underground mining We here present a brief description of the operations at an underground mine. The first step is the selection of reserves. This consists in defining the boundaries of the orebody, delimiting the material to be removed. This is done according to economic criteria that determine what is profitable based on the grade distribution. Large underground mines are typically subdivided in sectors by design, and the selection of reserves is done for each sector. For planning and operational purposes, the geological model of the whole deposit is expressed through a block model. Each block is uniquely identified together with its geologic characteristic, in particular the ore grades. These values are estimated using geostatistic procedures (kriging). A column (or drawpoint) is the vertical aggregation of blocks, and a mine or sector corresponds to a set of neighboring columns.
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Once the reserves are identified, the following step is to program the ore extraction on a time scale taking into account geological restrictions and the downstream capacities. There are several mining methods for underground mines (Newman et al. 2009). In our model, we consider the block caving method which is the prevalent underground mining method at Codelco. In a nutshell, the block caving method consists in creating a void at each drawpoint so that the rock breaks and falls due to gravity and the weight of the overburden material. For this to happen, the rock extraction pattern must follow specific rules: (i) the columns have to enter production in a particular sequence to generate a "wave" that breaks the rock; (ii) the wave has to advance smoothly, which requires regularity in heights among neighboring columns; and (iii) at each drawpoint there is maximum extraction rate to prevent the roof from collapsing and there is a minimum number of blocks that must be removed in order to avoid solid pillars. Figure 1 shows the typical flow in an underground mine. The broken rock is removed from the drawpoints which are arranged along parallel crosscuts. Each crosscut has enough space for an LHD machine that hauls the material to a dumping point.2 The material is then funneled through ore passes and reaches the internal crusher, where the rock is crushed to a size that can be transported by train to the processing stages outside the mine. For our purposes, the flow finishes at the concentration plants where the profitable minerals are obtained by eliminating waste. Each one of these intermediate processes can be characterized through technical coefficients such as its capacity and variable operational cost.
Figure 1: Description of underground mining operations. 2
LHD stands for Loading, Hauling & Dumping.
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3.2.2 Open pit mining Following the discussion of underground mining we review here the operations at an open pit mine. Open pit mining is conceptually similar to underground mining, except that the orebody is reached from above, which requires removing plenty of overburden material from the soil. A preliminary definition of the regions to be mined is done using geological models that take into account the geometry of the pit and technical requirements (e.g., the maximum slope to prevent the walls of the pit from collapsing). The regions are called expansions which are also known as pushbacks or phases. In figurative terms, an expansion can be viewed as a "slice" of the pit. Each expansion is subdivided in benches of a pre‐determined height that must be extracted in order, from top to bottom. Figure 2 illustrates the geometry of expansions and benches and Figure 3 shows a view from above.
Figure 2: Description of open pit mining operations. Within an expansion, benches are extracted from top to bottom. E x p a n s ió n 4 8 N E
E x p a n s ió n 4 1 E
E x p a n s ió n 5 5 S
Figure 3: Expansions of an open pit mine, viewed from above. The bench is the minimal extraction decision unit. When a bench is mined all its material needs to be removed from the pit including waste without economic value. There are three processes involved in removing a bench: Perforation and blasting, where the bench is separated from the soil using dynamite; loading, where all the material is loaded into large size trucks; and transportation, where the material is taken to either a metallurgic process or to a 8
waste dump. It is worth to mention that perforation and blasting imposes severe constraints on the sequence in which expansions are extracted. For example, there is a minimum distance requirement between two expansions in operation on the same wall of the pit to avoid rock spillage. Like in underground mining, the sequence of downstream processes involves rock size reduction and ore recovery. However, there are some differences that are worth mentioning. First, the existence of stockpiles plays an important role in balancing the plant inflow and they are held for several years before entering the metallurgic processes. Second, a significant fraction of the material extracted from the mine is taken to the waste dumps which might be located several miles away from the pit. This introduces an important trade‐off between sending low grade ore to the downstream processes or to the costly dumps. The existence of more than a single ore type, sulfides and oxides, is a key element of an integrated plan. Although in the geological reserves of a copper mine we typically find both types, in the traditional planning scheme, mining plans are defined considering only one main metallurgic process for a particular type. Thus, an integrated plan considers different ore types in alternative concentration lines that would be otherwise discarded or sent to processes with low yields. 3.3 Codelco's Production Planning Approach Recently Codelco has migrated from a traditional planning scheme based exclusively on cut‐off grades to a methodology based on the model presented in this paper. In order to compare the results we outline the planning approach. Note that overall the steps have not changed. What has been modified is the way steps d) and e) are carried out, as explained below. a) The available resources are determined considering the current geologic characteristics of the deposit and the mine’s history. b) A selection of resources is performed. This is done considering Macro Options (i.e. alternative technologies, economic indicators, management directives, etc) in order to delimit the floor, perimeter and ceiling of the resources to extract during the planning horizon. The outcome is known as the Model of Mining Reserve that contains the resources with positive benefit taking into account the processing costs and the projected recovery. c) The phases and benches of extraction are designed on the basis of physical y geomechanical principles. d) The Consumption Strategy defines a Mining Plan, which contains for every period: tons of mineral to extract, productive sectors involved, ore grades and level of uncertainty based on the drillings and samples.
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e) The Mining Plan is economically evaluated, providing Profit Indicators, and measures of Technical Risk and Vulnerability. If the result is not satisfactory, the Planning Parameters are modified and the process is repeated from point c). f) If an external change occurs affecting the Macro Options, these are incorporated into the model in point b), and the process is repeated from that stage. g) The final result is a plan that provides the relevant information for operating each sector, including the starting period, operation rates and termination period. In the legacy planning approach steps d) and e) where carried out by looking separately at the different parts of the mine. This scheme allowed evaluating different options by means of alternative plans, but not in a unified (global) approach, as each part was treated separately. In addition, due to the high complexity of the procedure and the involved calculations, several weeks (if not months) were spent in obtaining one particular plan, preventing the planner from evaluating more options, going back and forth between the different steps. Using the model in steps d) and e) allows planning in an automated fast way. It also leads to better solutions as the whole system is planned in an integrated fashion, and the planner can evaluate multiple scenarios and options. 4. MODEL FORMULATION AND DISCUSSION In this section we present a math programming approach to optimize long‐term production plans in open pit and underground mines that share downstream processes. A graphical representation of the different stages in the model is given in Figure 4 where material flows from left to right.
Figure 4: Network flow representation for mining operations where several open pit and underground mines share downstream processes. 10
The network of mining operations begins with the production or rock extraction phase that takes places at sectors and expansions for underground and open pit mines respectively. This phase corresponds to the first stage on the left side of Figure 4. Then the extracted rock is fed into the network of downstream processes that involves size reduction and chemical reactions. The last stage is the concentration phase which yields the three final products: copper, molybdenum and arsenic. The first two have commercial value while the third one is a contaminant on which environmental restrictions are imposed. In what follows, we present the model with its essential components. When necessary we use the sub‐indices O and U to denote the two type of mines open pit and underground respectively. In section 4.1 we first introduce a unified model for the network of downstream processes. This part of the model is shared by all the mines considered regardless of their type. In contrast, we present a different model of the production phase for each type of mine in order to capture the unique characteristic of the respective mining methods. This is done in sections 4.2 and 4.3 for underground and open pit respectively. 4.1 Unified Model Formulation for Downstream Processes As shown in Figure 4, the downstream processes are modeled as a capacitated multi‐ commodity network flow problem. The starting nodes s ∈ S = SU ∪ SO represent the different sectors and expansions for underground ( SU ) and open pit mines ( SO ) respectively. The intermediate nodes u , v ∈ V represent stocking points and processing stages that precede the concentration plants, which are the final nodes ( F ) in the network. Several different products flow through the network. A given product k ∈ K represents material (i.e., rock) with an average rock size and a grade of copper, molybdenum, and arsenic within a certain range. Depending on the characteristics of the ore deposits, the product definition might also include other relevant attributes such as the chemical nature (sulfide or oxide) or certain geological properties (e.g., rock hardness). The set of commercial products is denoted COM , whereas the set of contaminants is denoted CONT .
The flow of product k ∈ K measured in tons and going from node u to node v in period
t ∈ Τ = {1,..., T } is denoted fuvkt . As the material flows through the downstream processes, the average rock size is reduced and the valuable minerals are separated from waste. Let TCvkk ' denote the transformation coefficient at node v ∈ V . That is, the amount of output k ' obtained for each ton of input k , with ∑ k '∈K TCvkk ' = 1 ∀v ∈ V , k ∈ K . Let cvkt denote the cost of processing a ton of product k at node v in period t ∈ Τ . The maximum processing capacity at node v in period t is given by CPvt . For nodes v ∈ V that can hold inventory, let yvkt represent the amount of product k available at the beginning of period t , and let CSvt be the maximum 11
stock level per period in tons (if the node cannot hold inventory, then CSvt = 0 ). If k is a contaminant, then the maximum amount that can be released per period is Ekt (otherwise,
Ekt = ∞ ). We assume that the firm is a price‐taker and that all the production of product k in period t can be sold at the market price pkt (clearly, pkt = 0 for non‐commercial products).
The link of the downstream processes with the upstream production phase is established
through the auxiliary variables ProdCostst and xskt . The former denotes the total production cost, while the latter represents the tons of product k produced in sector/expansion s in period t . The mathematical formulation of the downstream processes is the following:
⎛
T
max
∑∑ ∑
t =1 k∈K u∈V ∪ S
βt ⎜ ∑
∑
pk 'tTCvkk ' fuvkt −
⎝ v∈F k '∈COM
⎞ T cvkt fuvkt ⎟ − ∑∑ βt ProdCostst v∈V ∪ F ⎠ t =1 s∈S
∑
(1)
subject to
∑ f = ∑ f
xskt =
∑ ∑ TC
u∈V ∪ S k∈K
f
vkk ' uvkt
+ yvk 't −1
∑ ∑f ∑y
u∈V ∪ S k∈K
∑ ∑ ∑ TC
u∈V ∪ F
svkt
vuk ' t
+ yvk 't
∀s ∈ S , k ∈ K , t ∈ Τ
(2)
∀v ∈ V , k ' ∈ K , t ∈ Τ
(3)
uvkt
≤ CPvt
∀v ∈ V ∪ F , t ∈ Τ
(4)
vkt
≤ CSvt
∀v ∈ V , t ∈ Τ
(5)
≤ Ek 't
∀k ' ∈ CONT , t ∈ Τ
(6)
k∈K
u∈V ∪ S v∈F k∈K
v∈V ∪ F
f
vkk ' uvkt
fuvkt , yvkt ≥ 0
∀u, v ∈ V , k ∈ K , t ∈ Τ. (7)
The objective of the model is to maximize the net present value of the mines over the next T periods. The first term in the objective function (1) corresponds to the discounted benefits obtained from selling the final products. The second and third terms represent the discounted processing and production costs respectively (here the parameter β t < 1 is the discount factor). Constraints (2) feed the production from each sector/expansion into the network of downstream processes. Constraints (3) ensure the flow conservation and inventory balance at each node. Constraints (4) and (5) impose maximum processing capacities and stock levels respectively. Finally, constraints (6) limit the release of pollutants at the final stage, and constraints (7) are the usual non‐negativity conditions on flows and inventory levels. 4.2 Underground Extraction To model the production phase for underground mines, we assume that the mining method is block caving, as explained in section 3.2.1. Let the index i ∈ I ( s) denote a drawpoint or column 12
in sector s ∈ SU and let the index n ∈ N (i ) denote the blocks in that column which are numbered 0 to N (i ) − 1 from bottom to top. The main decision in the production phase is when to remove each block. For that, we introduce the binary variable zint , that equals one if the extraction of block i in column n is initiated in period t , and the continuous variable eint , which represents the fraction of the block that is removed in each period.
As part of the model's input, we need the following sets and parameters. Let TON ink be the
amount (tons) of product k contained in block (i, n) . Let EMIN (i ) denote the minimum set of blocks of column i that must be removed once production at that column is initiated. Let
IU = ∪ s∈SU I ( s ) be the set of drawpoints/columns in all the underground mines considered. Let Δ denote the maximum number of periods a drawpoint can remain open. Let α in be the height of block (i, n) and let the variable hit denote the height of column i in period t , which is given by the blocks that have been removed so far. Let δ s denote the maximum height differential allowed between two neighboring columns in sector s , and let IJ be the set of column pairs that are close enough to be considered neighbors.
The extraction precedence relationship among columns is given by the set SECU ,
where (i, j ) ∈ SECU if the extraction of column i must begin before it takes places at column j . Initiating the production at column i incurs a fixed cost ai and removing block (i, n) takes a minimum of γ in days. There is also a variable cost bst per ton of rock extracted from sector s in period t . Let Dt denote the duration of period t measured in days, and for sector s , the maximum (minimum) incorporated area and extracted tons allowed in that period are given by MAX_AREAst ( MIN_AREAst ) and MAX_EXTst ( MIN_EXTst ) respectively. Finally, let
WIN ( s ) denote the time window in which sector s can be in production. The following constraints complete the model for underground mines.
xskt =
ProdCostst =
∑z t∈Τ t
int
∑e g =1
ing
∑ ∑ TON
i∈I ( s ) n∈N ( i )
∑ az
i∈I ( s )
i i 0t
≤1
e
ink int
+ bst ∑ xskt k∈K
∀s ∈ SU , k ∈ K , t ∈ Τ
(8)
∀s ∈ SU , t ∈ Τ
(9)
∀i ∈ IU , n ∈ N (i ) t
≤ ∑ zing g =1
13
(10)
∀i ∈ IU , n ∈ N (i ), t ∈ Τ (11)
T
T
∑ ∑
t =1 n∈EMIN ( i )
eint ≥ ∑ EMIN (i ) zi 0t t =1 t
t
∑z g =1
in +1 g
t
∑z g =1
zi 0t +
j0g
∑e
∑
n∈N ( i )
ing
≤ ∑ ei 0 g
∀(i, j ) ∈ SECU , t ∈ Τ
≤1
∀i ∈ IU , n ∈ N (i ), t ∈ Τ (15)
g =1
γ in eint ≤ DPt hit = hit −1 +
∑α
n∈N ( i )
in
hit − h jt ≤ δ s MIN_AREAst ≤
∑
i∈I ( s )
eint
(14)
∀i ∈ IU , t ∈ Τ
(16)
∀i ∈ IU , t ∈ Τ
(17)
∀i, j ∈ IJ , s ∈ SU , t ∈ Τ (18)
AREAi zi 0t ≤ MAX_AREAst
MIN_EXTst ≤ ∑ xskt ≤ MAX_EXTst k∈K
∑z
(12)
∀i ∈ IU , n ∈ N (i ), t ∈ Τ (13)
g =1 t
T
g =t+Δ
≤ ∑ eing
∀i ∈ IU
∀s ∈ SU , t ∈ Τ
(19)
∀s ∈ SU , t ∈ Τ
(20)
=0
∀s ∈ SU , t ∈ Τ \ WIN ( s ) (21)
zint ∈ {0,1}, eint , hit ≥ 0
∀i ∈ IU , n ∈ N (i ), t ∈ Τ (22)
i∈I ( s )
i 0t
Constraints (8) define the production per sector, product and period. Constraints (9) define the production cost for underground mines which has a fixed cost component for each column that starts production and a variable cost per ton of material removed. Constraints (10) ensure that each block is removed at most once. Constraints (11) establish the logical link between variables eint and zint so that the extraction of a block cannot occur in a period prior to when it is initiated. Constraints (12) ensure that the minimum set of blocks is removed if a column enters production. Constraints (13) require the extraction of block n to finish before the extraction of block n + 1 can start at each drawpoint. Constraints (14) restrict the order in which the drawpoints can enter production.3 Constraints (15) impose the maximum duration of a drawpoint. Constraints (16) limit the extraction rate of each column. Constraints (17) define the height of the column in each period and constraints (18) guarantee that neighboring columns have similar heights so that the rock breaks in a smooth fashion. Constraints (19) and (20) impose bounds on the area incorporated and the rock extracted per period respectively. Constraints (21) ensure that a sector is not extracted outside its time window, and constraints (22) restrict the domain of the decision variables. 3
Note that the left hand side of constraints (11), (13) and (14) is formulated as a cumulative sum over time, which helps to tighten the LP relaxation.
14
In the model, we need constraints (11) and (13) together with the binary requirement
on zint to ensure that the blocks of a column are extracted one by one from the bottom to the top. However, if the blocks at the bottom have the highest grades, then the binary requirement can be relaxed since the sole fact that this is a maximization problem will force the extraction to occur sequentially in the desired order. Hence, in that case the LP relaxation provides a near‐ integer solution. This is an important observation that is relevant to our solution method. Therefore, we state it formally in the following lemma. Lemma 1. If the blocks n ∈ N (i ) in column i are identical except for the grade of the commercial minerals, which are decreasing, then the optimal solution to the LP relaxation satisfies zint ∈ {0,1} for all blocks except the first and last one extracted in period t . 4.3 Open Pit Extraction In contrast with underground mines, the mining method for open pit excavations allows for a more aggregated model since there is no need of getting into the details at the block level. Here the index i ∈ I ( s ) represents a drawpoint or bench in expansion s ∈ SO , which are numbered 1 to I ( s ) from top to bottom. The decision variables are: zit ∈ {0,1} , which equals one if the extraction of bench i is initiated in period t ; and eit ≥ 0 , which is a continuous variable that represents the amount (tons) extracted per period. Let I O = ∪ s∈SO I ( s ) be the set of drawpoints/benches in all the open pit mines under consideration. Let S (m) be the set of all expansions in the open pit mine m ∈ M O . Let SECO denote the set of all pairs (i, j ) such that the extraction at bench i must be initiated before it takes place at bench j . Note that these could be benches from different expansions. All the other parameters are the same as in the underground case. The following constraints complete the production plan optimization model for open pit mines.
xskt = ProdCostst =
∑z t∈Τ t
it
∑ TON
i∈I ( s )
∑ az
i∈I ( s )
i it
≤1 t
∑ eig ≤ ∑ zig g =1
g =1
15
e
ik it
+ bst ∑ xskt k∈K
∀s ∈ SO , k ∈ K , t ∈ Τ
(23)
∀s ∈ SO , t ∈ Τ
(24)
∀i ∈ I O
(25)
∀i ∈ I O , t ∈ Τ
(26)
T
T
∑e ≥ ∑ z it
t =1
t
∑z g =1
i +1 g
∀i ∈ I S
(27)
∀i ∈ I O , t ∈ Τ
(28)
∀(i, j ) ∈ SECO , t ∈ Τ
(29)
∀s ∈ SO , t ∈ Τ
(30)
≤ MAX_EXTmt
∀m ∈ M O , t ∈ Τ
(31)
=0
∀s ∈ SO , t ∈ Τ \ WIN ( s ) (32)
t =1 t
it
≤ ∑ eig g =1
t
t
g =1
g =1
∑ z jg ≤ ∑ eig ∑γ
i∈I ( s )
MIN_EXTmt ≤
i
eit ≤ DPt
∑ ∑x
s∈S ( m ) k∈K
skt
∑z
i∈I ( s )
it
zit ∈ {0,1}, eit ≥ 0
∀i ∈ I O , t ∈ Τ.
(33)
Constraints (23) and (24) define the production and costs variables respectively, similar to constraints (8) and (9) in the underground case. Constraints (25) ensure that each bench is removed at most once. Constraints (26) establish the logical link between variables eit and zit so that the extraction of a bench cannot occur in a period prior to when it is initiated. Constraints (27) ensure that bench is not partially extracted. Constraints (28) require the extraction to take place from top to bottom within an expansion. Constraints (29) impose the extraction sequence among benches to comply with geomechanical requirements. Constraints (30) limit the extraction rate per expansion and constraints (31) impose bounds on the total extraction per period. Finally, constraints (32) impose an extraction time window and constraints (33) restrict the domain of the decision variables. 4.4 Model Discussion and Strengthened Formulation We begin this section stating the model's complexity. The following proposition justifies the heuristic methods described in the next section. Proposition 2: The long‐term production planning problem for underground and open pit mines given by Equations (1)‐(33) is strongly NP‐hard. Given Proposition 2, we turn to the LP relaxation of the model to generate near optimal solutions. In the case of underground mines, from Lemma 1, if ore grades are decreasing, then most of the zint variables in the LP solution have integer values. There is a natural rounding heuristic that follows from this observation which is described in the next section. Underground mines are usually designed in such a way that the lower blocks of a column have higher grades. 16
This provides support for using our heuristic which is confirmed by its good performance in practice (see section 6.1). For open pit mines, there is no equivalent to Lemma 1. In fact, several benches of overburden material must be removed before the ore deposit is reached, and even then, the best ore grades are usually deeper in the ground. In terms of our model, this means that the solution of the LP relaxation is highly fractional since it is optimal to reach the lower benches as soon as possible. Indeed, most of the zit variables have small fractional values very close to zero. This makes it hard to construct feasible integer solutions based on the LP relaxation. In particular, branch and bound or rounding heuristics become very inefficient. To overcome this obstacle, we strengthen the formulation for open pit mines by adding a network structure to the production phase. This is explained next. For each expansion, we generate a directed graph where the nodes represent a bench‐ period grid, as shown in the left hand side of Figure 5. At a given node, there is an outgoing arc to all the benches that can be reached in the next period, according to the conditions imposed by constraints (28) and (30)‐(32). An artificial staring bench and period is added to the upper level of the grid, and similarly an artificial ending bench and period is added to the lower part (see the right hand side of Figure 5). Hence, timing the extraction of the expansion is equivalent to moving one unit of flow from the upper left corner to the lower right of the grid.
Figure 5: Extraction graph of the production phase for open pit mines. To formulate the extraction graph for an expansion s ∈ SO , we first need to introduce some additional notation. Let t = 0 and t = T + 1 denote the artificial starting and ending periods respectively. Similarly, let Or ( s ) and De( s ) denote the artificial starting and ending benches respectively. Let AN (i, t ) and SU (i, t ) denote the antecessors and successors of node (i, t ) in the extraction graph respectively. Let wijt be a binary decision variable that equals one if the unit of flow (i.e., the extraction) goes from bench i to bench j in period t . In other 17
words, if wijt = 1 , then it implies that the extraction of benches i and j must begin in periods t − 1 and t respectively, and given the precedence constraints (29) all the benches in between must be fully extracted in period t . Note that if there is an arc from node (Or ( s ), T ) to ( De( s), T + 1) , then the model could still decide not extract the expansion at all (see Figure 5). The mathematical formulation of the extraction graph is the following:
∑
wOr ( s ) j1 = 1
∀ s ∈ SO
(34)
w jDe ( s )T +1 = 1
∀ s ∈ SO
(35)
∀i ∈ I O , t ∈ Τ
(36)
∀i ∈ I O , t ∈ Τ
(37)
∀i, j ∈ I O , t ∈ Τ.
(38)
j∈SU ( Or ( s ),0)
∑
j∈AN ( De ( s ),T +1)
∑
j∈AN ( i ,t )
w jit =
∑
j∈SU ( i ,t )
i −1
zit = ∑
wij (t +1)
∑
h = 0 j∈SU ( h ,t −1) j ≥i
wijt ∈{0,1}
whjt
Constraints (34) and (35) impose that a unit of flow starting from the upper left node of the extraction graph in period t = 0 must finish at the lower right in period t = T + 1 . Constraints (36) establish the flow conservation at each node of the graph. Constraints (37) link the flow variables wijt with the decision variable zit from the original formulation, and constraints (38) impose the binary condition. By construction, constraints (34)‐(38) are valid for the original formulation of the open pit production phase. It is well known that in the absence of other constraints, the LP relaxation of the network formulation given by constraints (34)‐(38) yields an integer solution. When the precedence constraints and the downstream processes are considered, some of the wijt variables are fractional. However, the LP relaxation is tighter than the original formulation, and more importantly, the fractional values are closer to one. The latter provides a good starting point for the rounding heuristic that we describe in the next section. 5. SOLUTION APPROACH Our approach to solve the model described in section 4 consists in solving the LP relaxation and then using a heuristic rounding procedure in order to find a feasible integer solution. In sections 5.1 and 5.2 we provide the details for the underground and open pit cases respectively. Note that in the relaxed problem we can assume with no loss of optimality that the constraints (11) hold as an equality so that the zint and eint variables take the same values. Therefore, we omit
18
the latter (idem for zit and eit ). The procedures described here below could apply to other minerals besides copper. 5.1 Underground Mining Rounding Heuristic In the case of underground mines, the LP relaxation does not present many fractional values. This can be explained primarily by two reasons: (i) the model formulation itself is very tight; and (ii) the copper grade in a column is typically decreasing, then larger benefits are obtained when the inferior blocks are extracted first (see Lemma 1). To obtain an integer solution, the rounding heuristic proceeds iteratively from the inferior blocks to the higher ones fixing those that already have an integer value. During this process the tons of ore extracted from each column is fixed to the value obtained in the relaxed solution. Figure 6 shows the passages of the heuristic for a particular column.
Figure 6: The rounding heuristic for underground mining. 5.2 Open Pit Mining Buffer‐based Heuristic The heuristic to solve the open pit problem is also performed in an iterative fashion. As before, the starting point is the solution of the LP relaxation. Then, a subset of extraction variables is fixed, and the LP relaxation is solved again. This procedure continues until there are no fractional variables remaining. The overall structure of the heuristic is the following: Step 0: Initialization. Solve the LP relaxation of the open pit problem and let zitLP ( q ) denote the solution for the extraction variables where q represents a counter. Initialize q = 1 and initialize Λ (m) to contain all the expansions in the open pit mine m ∈ M O . Step 1: Extraction simulation. For each expansion s ∈ ∪ m∈M O Λ (m) , apply Algorithm 1 to simulate its extraction. Start with the inner expansions of the open pit and continue outwards. Algorithm 1 is described below in pseudo‐code. Note that IND( A) is an indicator function that equals one if the argument A is true and is zero otherwise. The sets Buf1 ( s ) and Buf 2 ( s ) are used to store the last bench that is reached in each period of the simulated extraction of expansion s . 19
let t = 1 let i = Or(s) let j = Or(s)+1 let Buf1(s) = Buf2(s) = φ while (t≤T and jt {z LP } jg jg
if (f1 = f2 = f3 =1) (q) let z SIM = 1 jt update downstream flows let j = j+1 else (q) let z SIM = 0 jt let Buf1 ( s ) = Buf1 ( s ) ∪ { j} and Buf 2 ( s ) = Buf 2 ( s ) ∪ {( j , t )} let i = j - 1 let t = t+1 end end Algorithm 1: Extraction simulation for an expansion s ∈ ∪ m∈M O Λ (m) . Step 2: Expansion selection. For each mine m ∈ M O , select the expansion that minimizes the squared difference between the extraction simulation (from the previous step) and the solution of the q ‐th LP relaxation. In formal terms, let sq∗ (m) = arg min s∈Λ ( m )
∑
i∈ I ( s ), t∈Τ
( zitSIM ( q ) − zitLP ( q ) )2 .
Step 3: Tighten LP formulation. Add the following two constraints to the open pit problem:
zit = zitSIM ( q ) zit + zit +1 = 1
∀i ∈ I ( sq* (m)) \ Buf1 ( sq* (m)), m ∈ M O , t ∈ Τ (39)
∀(i, t ) ∈ Buf 2 ( sq* (m)), m ∈ M O LP ( q +1) it
Solve the q + 1 ‐th LP relaxation and let z
(40)
denote the solution for the extraction
variables. For each m ∈ M O , redefine Λ(m) = Λ(m) \{sq* (m)} . If ∪ m∈M O Λ (m) is empty, then STOP. Otherwise, increment the counter q = q + 1 and go to Step 1. The premise of the heuristic, as in most rounding procedures, is that the fractional solution of the LP relaxation is a good starting point to build near‐optimal integer solutions. After initializing the variables (Step 0), the outcome of the q ‐th LP relaxation is used in Step 1 to build an integer solution for each expansion, which is denoted zitSIM ( q ) . The superscript 20
SIM stands for "simulation" of the extraction phase. This is performed in Algorithm 1, which traverses the extraction network from the upper left corner down to the lower right exit node (see Figure 5). At a given node ( j , t ) , the algorithm must decide whether to move down (i.e.,
extract bench j + 1 ) or to the right (i.e., transition to period t + 1 ). The latter occurs by default unless three conditions are met, which are given by the flags f1 , f 2 , and f3 respectively.
Flag f1 checks whether node j is accessible from the node reached in the previous period,
i.e., node (i, t − 1) . Flag f 2 checks the precedence constraints (29) and the availability of downstream capacity to process the material from bench j in period t . The last flag f 3 checks whether in the q ‐th LP solution the amount extracted from bench j up to period t is greater than the maximum of the fractional values in the remaining periods. The latter measures the desirability of extracting bench j in period t . This rule contrasts with the usual rounding criterion that simply picks the largest fractional value and rounds it up to one. Figure 7 shows four examples that illustrate the two criterions for a given bench j . Each example shows a different solution of the LP relaxation. In examples (a), (b) and (c), the rounding criterion based on f 3 yields the same outcome as the criterion that rounds up the largest fractional value. However, in example (d), f3 would allow the extraction to start in the second period, whereas the criterion that considers the largest fractional value would delay the extraction until the fourth period just as in example (b). In other words, the criterion based on f3 attempts to start extracting the bench earlier and will wait only if the fractional values of the LP solution are considerably skewed. f3 holds (a)
0.7
0.1
f3 holds 0.1
0.1
bench j
0.1
periods (c)
0.4
0.2
0.2
0.1
0.1
0.7
(b)
0.4
(d)
periods 0.2
bench j
f3 holds
0.2
0.2
0.2
f3 holds
Figure 7: Examples of the rounding criterion for open pit mining.
In Step 2 of the heuristic, the expansions to be fixed are selected based on how much the
extraction simulation zitSIM ( q ) differs from the LP solution zitLP ( q ) . Finally, in Step 3, additional constraints are added to the open pit problem to fix the extraction variables for the expansions selected in Step 2. Constraint (39) sets the variable zit equal to 0‐1 according to zitSIM ( q ) . For a bench that is extracted in two consecutive periods, constraint (40) is imposed, which allows the LP relaxation to decide what fraction of the bench to extract in each period. These benches act 21
as buffers between periods and give the LP relaxation more flexibility to satisfy all the constraints. Note that when there are few expansions left to be fixed, usually it is possible to solve the MIP formulation to optimality in reasonable time. Therefore, Algorithm 1 can be seen as a way of reducing the MIP until it can be solved by a commercial solver. 6. IMPLEMENTATION AND IMPACT AT CODELCO All the data need by the model was extracted directly from Codelco's databases. The equipment, machinery and facilities with a lifespan shorter than the planning horizon were prorated and included as variable costs. Some standard pre‐processing routines similar to Kuchta et al. (2003) were performed in order to reduce the number of binary variables. Representative instance sizes after pre‐processing are shown in Table 1.
Model N° Constraints N° Variables 0‐1 Variables Underground mine 446,521 535,639 196,386 Open pit mine 245,391 898,742 160,386 Table 1: Example of instance sizes after pre‐processing. The model was implemented using the modeling language GAMS and was solved using the parallel barrier interior point algorithm of CPLEX on a computer running Microsoft Windows. Given the large timeframe spanned by the project, it has benefited from several upgrades in the commercial software and the hardware performance. In sections 6.1 and 6.2 we describe large‐ scale case studies performed at Codelco El Teniente (underground) and Codelco North Division (open pit) respectively. In section 6.3 we summarize the current uses of the model and its impact on Codelco's operations. 6.1 Case Study at El Teniente4 El Teniente is the largest underground mine in the world. It is located in the Andes Mountains at 2,200 meters above the sea level and 80 kilometers south of Santiago, Chile's capital. Its origins date back to pre‐colonial times but officially its exploitation started in 1904. It currently has nearly 2,400 kilometers of underground tunnels. Our case study considered all the sectors that were in operation during the 2002 planning cycle. The horizon was divided in six periods, the first two of 1 and 4 years respectively, and the last four of 5 years each, spanning 25 years overall. The annual discount rate was 10% and all sunk costs were excluded. A fixed price of 85¢ per pound of fine copper was used throughout the horizon.5 Table 2 shows the results of the legacy planning approach explained in section 3.3 and the solution based on the optimization model. The comparison considered the same investment 4 5
A preliminary version of this case study without the financial impact was presented in Epstein et al. (2003). As of February 2010 the price was $3.4/lb.
22
projects in both cases. Hence, the differences come from the selection of reserves. The mining plan obtained with the model‐based approach increased El Teniente's NPV by 5.1%.
Model‐based vs. Legacy Percentage Difference Revenues 2.4% Production Costs 1.2% Investments 0.0% NPV 5.1% Table 2: Legacy vs. model‐based planning approach at El Teniente. A detailed comparison of the solutions showed that both provided production schedules that were very similar in terms of total tons extracted per period and sector, but the model obtained higher grades of copper and molybdenum, which translated into a higher income, as seen in Table 2. This improvement came mostly from considering the mine as a whole and from timing the production of each sector. The legacy approach that looked at each sector separately did not maximize the throughput of the downstream processes. The model‐based solution consistently reached the capacity of the intermediate processes and it provided an appropriate ore blend from all sectors so that the concentration plant received a constant supply. Moreover, the model was solved with an explicit constraint to limit the total amount of arsenic produced (as a byproduct) each period. The legacy approach had difficulties incorporating such kind of environmental restrictions. 6.2 Case Study at the North Division Codelco North Division is located 1,650 kilometers north of Santiago and at approximately 3,000 meters above the sea level. It comprehends several open pit mines at different stages of their life‐cycle. Currently, the two most important mines are Chuquicamata (better known as "Chuqui") and Radomiro Tomic (RT). Chuquicamata has been in operation for nearly a century and its pit today covers more than 1,300 hectares and is almost one kilometer deep. In contrast, the exploitation of RT only started in 1995, but it ramped up production very quickly. While Chuquicamata is mainly composed of sulfides, RT's reserves are mostly oxides. The case study analyzed at Codelco North Division considered both mines, Chuquicamata and RT, and we used the geological, technical and economic information available in the 2004 planning period. We considered a 10‐year planning horizon divided in yearly periods with a fixed price for copper of 85¢/lb. As before, sunk costs were excluded from the evaluation, so investments were mostly the acquisition of major extraction machinery including trucks, drills and loaders.
23
Like in the case study at El Teniente, we conducted an analysis to quantify the impact of the implementation of our proposed methodology. However, a simple comparison of the net present values is not enough to assess the economic impact. Since one of the advantages of our approach is that it can be used to simultaneously plan several mines and plants, the implementation of our methodology facilitated the first joint plan for Chuquicamata and RT. Therefore, to isolate the impact of our planning methodology from the benefits due to integrating both mines we compared our proposed solution against three benchmarks: i. Legacy‐based independent plans ( l − ind ): This is the optimized baseline where we impose the flow levels determined by the legacy planning approach and then we solve the model to find the best solution under those conditions. Here, different mines are planned independently so the downstream processes are not shared. ii. Model‐based independent plans ( m − ind ): In this instance the two mines are not allowed to interact (as in l − ind ) but we no longer impose the flow levels from the legacy iii.
approach, so the production plan for each mine is dictated by the model. Legacy‐based integrated plan ( l − int ): In this instance the production plan is determined by the legacy planning process (as in l − ind ) but the material is allowed to flow between the two mines, so it provides an estimation of the value of integrated planning under the legacy approach.
We denoted our proposed solution m − int , which stands for model‐based integrated plan.
In Table 3 we compare m − int against the three benchmarks. The respective percentage increases are shown in Figure 8. First, we note that the total economic benefit from using the model‐based methodology to obtain an integrated plan corresponds to a gain of 8.2% with respect to the optimized baseline l − ind . From Figure 8, the total impact has the following breakdown: 3.2 percentage points (pp) are due to integrated planning (see l − int vs. l − ind ), 4.7pp are due to the model‐based approach (see m − ind vs. l − ind ), 0.2pp can be attributed to synergies between integrated planning and the model‐based approach (to see this, compare the parallel arrows), and finally 0.1pp is the multiplicative effect. In other words, the specific impact of the model‐based approach is in the order of 5% which is consistent with the result obtained in the case study done at El Teniente (see section 6.1). The fact that there are synergies is another interesting observation meaning that integrated planning and the model‐ based approach reinforce each other.
24
m − int l − ind m − ind l − int Revenues 1.4% 0.0% 1.7% 1.1% Production Costs ‐1.6% 0.0% 0.8% 0.0% Investments ‐27.3% 0.0% ‐31.8% 0.0% NPV 8.2% 0.0% 4.7% 3.2% Table 3: Economic evaluation of the model‐based plan ( m − int ) vs. benchmarks at Codelco North Division. Figures are shown as percentage changes with respect to the baseline ( l − ind ). Integrated Planning Yes
No
Production Plan
Legacy Approach
l − ind 4.7%
Model-Based Approach
m − ind
3.2% 8.2%
3.4%
l − int 4.9%
m − int
Figure 8: Impact of the model‐based integrated production plan ( m − int ) on NPV with respect to the three benchmarks: l − ind , m − ind , and l − int . In Table 4 we compare our solution m − int with the three benchmark plans. We observe
that the total mined material is almost identical in all instances, so the total costs are comparable (see Table 3). This is intuitive because the ore with higher grades is located at the bottom of the pit and to get there all the material above it needs to be extracted. The total extraction might be different if the model decides not to extract a whole expansion, but that did not happen in this exercise. The economic improvement of the model‐based over the legacy approach comes mainly from two sources. First, the model‐based approach requires less investment in machinery because it can achieve better utilizations by smoothing production whereas the legacy approach needs to make additional investments to accommodate higher production peaks.6 Second and more importantly, the model‐based approach produces more copper than the legacy approach. In particular, with respect to the baseline l − ind , the solution
m − int sends 5.5% more ore to the plants reducing the average grade by 2.6%. Unlike traditional methodologies that use a cutoff grade per mine and period, the model‐based approach simultaneously considers multiple elements to decide when and where to process each bench of the pit resulting in a better usage of the available capacity. For instance, we
6
The reduction in investments can be seen in Table 3. Percentagewise, these are important reductions. However, in absolute terms they are small compared to total revenue and costs.
25
verify that in our solutions, when the benches are close to the plant it decides to send ore of relatively lower grade than those generated from benches that are away from the plants.
m − int l − ind m − ind l − int Total Extraction (Kton) 2,882,457 2,889,825 2,874,050 2,889,825 Ore to Process (Kton) 1,468,104 1,391,384 1,456,711 1,418,244 Average Ore Grade (%) 0.660 0.678 0.663 0.674 Average Recovery Rate (%) 80.4 81.0 82.1 80.5 Copper produced (Kton) 7,792 7,643 7,926 7,693 Table 4: Mining plan summary of the model‐based vs. benchmarks at North Division. Material flows are shown in kilo‐ton, grades shown in percentage. An interesting observation comes from comparing the independent and integrated model‐ based solutions ( m − ind and m − int respectively). The former has a better recovery rate and produces more copper but has a lower NPV than the latter. The reason is that the integrated solution redirects mineral from Chuquicamata to a plant next to RT which had some slack capacity ( 0 and ein* < 1. (41) Due to the decreasing grades, the tonnage from block n that reaches the concentration plant is more profitable than an equivalent amount from block n + 1 . Except for this difference at the commercial level, the blocks are identical. Hence, shifting production from block n + 1 to block n is feasible since the downstream processes and the other constraints remain unaffected. In
other words, if (41) holds, it is possible to shift a certain amount ε > 0 from ein* +1 to ein* and the objective value would improve. This means that {ein* }n∈N ( i ) cannot be optimal, but that would be a contradiction. So condition (41) cannot hold, and ein* +1 can only be positive if ein* equals one. Therefore, ein* ∈ {0,1} for all blocks except the last one extracted (i.e., except for the highest n such that ein* > 0 ). From constraints (11) and (13) we have that ein ≤ zin ≤ ein −1 ∀n > 0 . This implies that the optimal solution of the LP relaxation must satisfy zin ∈ {0,1} for all blocks except the first and last one extracted, which concludes the proof. Proof of Proposition 2 Consider a single period, a single product, and a single downstream process with finite capacity. Let the columns in the underground mines have a single block. Set TCvkk ' = 1 , EMIN (i ) = 1 ,
Δ = T = 1 , WIN ( s) = Τ , δ s = ∞ , γ in = 0 , Ekt = ∞ , pkt = 1 and set all the cost parameters equal to zero. Set the minimum incorporated area and extraction requirements equal to zero, and set the maximum values equal to infinity. Then, the production planning model reduces to the precedence constraint knapsack problem in which the profits are equal to the weights. The latter is strongly NP‐hard (see section 13.2 of Kellerer et al. 2004). Since we have shown a polynomial reduction, the long‐term production planning problem is strongly NP‐hard as well. 30
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