A sequentially advancing algorithm based on multi-value dynamic programming for the cut-off grade optimization in open-pit metalliferous deposits

In techno-economic concern, cut-off grade ( COG ) optimization is the key for efficient mineral liquidation from the huge metalliferous surface mining sector. In this paper, a sequentially advancing algorithm based on exact multi-value dynamic programming (MDP) has been developed to determine the optimum COG of an open-pit metalliferous deposit. The proposed COG optimization algorithm aims to overcome the limitations of straightforward classical techniques in determining the optimum COG . This discrete COG-MDP model is the first of its kind and has the novelty of dealing with the simulation of eight dynamic possibilities to achieve the maximal Net Present Value (NPV). A high-level programming language (Python) has been used to develop the computer model to deal with the complexity of handling a minimum of 500 series of dynamic variables with a precision value of 0.01% in grade bins. This model can generate results in polynomial-time from the complex mine, mill, and smelter and refinery system corresponding to various limiting conditions. The prime objective considered in the model is to optimize the COG of a metalliferous deposit. The model validation has been done using a real-life case study of an open-pit copper mine in India (Malanjkhand Copper Mine, HCL), considering the fixed yearly output of the mining, milling, and smelting and refining. In this study, the optimum COG for the Malanjkhand copper deposit has been found to be (0.33%, 0.23%, 0.52%, 0.26%, 0.27%, 0.22%, 0.24%) with a maximum NPV of ₹ (12204, 14653, 16948, 14609, 21454, 26717, 38821) million corresponding to various scenarios. The findings also show that the present value of net cash-flow grows in the early years, peaks at a specified mid-life time, and then drops as the reserve is depleted. The present value gradually hits zero after the project’s life cycle, confirming the t ypical pattern of other mining firms. advancing algorithm ● Cut-off grade ( COG ) ● Open-pit mining ● NPV maximization.


Introduction and critical review of previous work done
Mineral resources are essential for contemporary society's socioeconomic development. Even though these commodities have been widely cross-referenced as "finite," their metal output has continuously risen throughout the ages. The twenty-first century has seen exponential resource extraction, corresponding to global economic expansion; consequently, long-term sustainable development is crucial. Global metal utilization is expanding at a rate of roughly 3.2% per year, boosting trade and commerce [1,2]. Metal mining has consistently shown to be a significant source for meeting metal demands in infrastructure, health, information systems, and other services that can support economic growth [3]. Thus, plays a significant role in achieving the objective of attaining techno-economic sustainability, which strives to maximize recovery utilization. The field of knowledge, exploration, findings, innovations, and technological progress required to meet a predetermined has an impact on the ultimate achievable resource capacities. The worldwide average for copper is approximately 0.49% [4][5][6].
Mineral economics is not a new field; this can be followed back 200 years to the research of classical economists like as Smith [7], Ricardo [8], and Jevons [9], among others. Smith [7] noted that the demand for mineral resources appears to be unlimited due to customers' preferences and desire for manufactured products, which need a large number of fossil fuels and mineral resources to be exploited. Ricardo [8] highlighted that all mineral commodities are exposed to significant fluctuations, either temporary or permanent in nature, and of significant economic impact.
Surface mine planning is a dynamic and complicated system with a host of variables and multi-stages ( Fig. 1), making it a complex problem that is hard to solve and demands a continuous financial risk assessment. Open-pit scheduling intends to find the best schedule of calendar programming across the mineral value cycle [10][11][12] . This is because the value of a mining project is usually affected by various underlying economic and physical variables, such as commodity prices, grades, expenses, timetables, volumes, and environmental concerns, which are unknown with confidence at the project's inception. Mining is a capital-intensive industry in which economic concerns dominate all activities. This is necessary for its long-term viability and growth. The of a mineral deposit is determined based on the techno-economic viability of production and profit obtained from the time a mine is opened until it is closed. is the grade below which the ore cannot be processed profitably. Therefore, this criteria creates a boundary inside a particular mineral deposit between ore and waste, where ore must be delivered to a mill for processing, and waste is either left undisturbed or sent to waste disposal [13,14].
In a number of contexts, mineral resources are classified as either ore or waste. The use of is a standard route to explore the ore-waste ratio and the total mineral content. All the optimization techniques can be divided broadly into traditional and modern methods. Traditional methods may further be divided into four major sub-categories: mathematical programming, integer programming, linear programming, and nonlinear programming. On the other hand, modern techniques may be divided broadly into two major categories: the evolutionary algorithm and the Monte Carlo simulation.
Out of all the techniques, the maximum number of work may be attributed to the analytical approaches, but none can yield the optimal results. Henning [15] was the first who brought the idea of optimization into the mining business and a year later the seminal work of Lane [16] was introduced. Lane determined optimal based on limited capacities, supporting Henning's [15] prediction that ideal tend to drop with time. Henning's method keeps certain operational parameters fixed throughout the project's life cycle and does not account for the potential of capacity expansion. Following Lane's work, the amount of literature on the topic has expanded exponentially.
Through, globally the breakeven and Lane's [16,17] algorithm are being used commonly to maximize profit. However, maintaining a steady breakeven across the mine life would have serious consequences, potentially exposing the mining operation to considerably sub-optimal operating results. Lane's method also solved the breakeven model's flaws. It considered the production restrictions of various phases and the relevance of the grade-tonnage characteristics.
Unfortunately, Lane's [16,17] method has been widely criticized for determination that may be more suitable in certain circumstances, making it possible to disregard the optimal estimates for [18]. Lane's technique has stayed empirical, necessitating the development and implementation of a rational optimization technique [19,20]. The application of mathematical programming to open-pit mine planning traces back to the 1960 . Acknowledging the limitations of Lane's approach, a few attempts have been made to overcome them through an alternative analytical framework of optimization [20][21][22][23]. The basics of Lane's [16,17] technique have been expanded in most optimization approaches. Several studies employed time-value-of-money-based optimization approaches. The optimization algorithms that involve NPV maximization lead to better outcomes than those that maximizes the profit function.
Lane [17] effectively used the deterministic dynamic programming concepts to the optimization of policies. However, he advocated to find an optimal operating strategy and the associated maximum present value in a specific condition, the sum of the maximum present value surface is irrelevant. Every point along the optimal strategy path must be optimal, and all related current values must be maxima. Assuming that future prices would vary, he proposed a dynamic programming method that can assure that the best is picked at all times. However, it can only be approximated by forecasting, which is fraught with risk. Dowd [24]  developed a stochastic and dynamic programming model, extending Lane's algorithm. They attempted to address the uncertainty by incorporating an assumed value of 2% per period for the case of a gold deposit, which increased annually irrespective of any uncertain futuristic changes related to demand and technology; this, in reality, is not always the case. Saliba and Dimitrakopoulos [37] applied the simultaneous stochastic operation considering the market uncertainty. The price fluctuations of commodity and the problem of optimization of the mining complex under the uncertainty of the market and supply jointly was the strength of the framework. However, it was limited in optimizing the mining capacity and processing rates with an extension of the destination policy to consider multiple variables in decisions.
Su et al. [38] stated that the emergence and collapse of several price bubbles is connected to supposition, macroeconomic instability, a supply-demand imbalance, and economic turmoil. Tahar et al. [39] agreed that commodities prices shocks might have differential effects. The most of commodities' economies are highly volatile and difficult to anticipate. Considering an erroneous pricing forecast leads to sub-optimal selections and implications. Dehghani and Bogdanovic [33] further However, a few authors have also advocated that the theory of optimization supports the ultimate objective of a mining operation through maximization of [43][44][45]. Ahmadi and Bazzazi [46] proposed that choosing the optimal maximizes the and the total profit of the project. The optimization of the considering the maximum achievable over the life of the mine is one of the critical issues in the mining of open pits. The criteria that control the development of policy aligns with an operation's strategic objectives to maximize the discounted value over the life of the process [47]. Russell [48] pioneered the concept of maximising a project's . The most commonly used economic metric in evaluating financial projects is [49,50]. It is worth noting that is acutely susceptible to revenue variations [51][52][53][54].
Ahmadi [55] optimized the based on Lane's theory to maximize the using MATLAB. The optimum is the grade that maximizes the chosen objective function, usually the [56]. Various researchers [57][58][59] have given several algorithms to determine optimum . They concluded that evaluating the based on maximizing the produces more accurate findings than the other methods. The that leads to the highest for the project must be selected.
Determination of optimum is an essential and crucial aspect to exploit any metalliferous deposits efficiently and economically. Metal prices are dynamic in nature and follow an asymmetrical trend. So, one cannot fix the production schedule having a random without optimizing the objective function to reach the maximum . The stage pits can be scheduled strategically and optimally once the ultimate pit has been set, keeping the maximum discounted and undiscounted block value

Optimization model framework
The proposed model has been developed considering that the optimum for a metalliferous deposit is dynamic in nature.
The programming language used for the model has been designed using Python. The complete sequence of mathematical and computational techniques is based on simple algebraic equations, calculus of variation, and iterative algorithm steps fulfilling the criteria of a sequentially advancing algorithm for the optimization and maximization of subject to the condition that the capacities of mining, concentrating, and refining are either limited or unlimited. The model has been validated using real-life data from the study of the Malanjkhand Copper Project of Hindustan Copper Limited (HCL) to demonstrate the effectiveness of the model's algorithm.

Operational model
In metalliferous open-pit mine planning and design, the optimum is mainly affected by three governing capacity constraints of various operations, i.e., mine, mill, and smelting and refinery. Eight different possibilities can be considered while calculating the optimum based on the different limiting conditions the production capacities of the three stages (Table 1). Mine capacity is limited 3.
Mine and mill capacities are limited 6.
Mine and refinery capacities are limited 7.
Mill and refinery capacities are limited 8.
All (mine, mill and refinery) capacities are limited

Model prerequisites
Prerequisites for the application of the optimization model include the development of the ultimate pit limit or pit extent; calculation of mineable ore reserves in terms of mineral grade and tonnage distribution within the pit limit; the mining, processing and refining stage capacities; the operating costs at these stages; and the technical and other economic parameters including the metal price.

Model assumptions
It has been presumed that the total deposit, as specified by the grade distribution, will be excavated, but only that portion of the excavated material that exceeds or equals the should be sent for concentration. The production, concentration, and refining capacities have been assumed to be stable during the project life. Furthermore, it has been assumed that the metal price, operational costs, capital costs, rate of interest, tax rate, and discount rate remain stable over the project life though the metal price index volatility and technical developments can change the entire scenario of mining, milling, and refining costs, as well as, the future revenue. To simplify the problem, it has been assumed that, at least on average, all ore mined within any twelve months will also be processed during that period.

Model formulation using sequentially advancing algorithm based on multi-value driven dynamic programming
To determine the optimum for an open pit metalliferous deposits, one must express the problem as a mathematical programming model. The mathematical models for dealing with multi-varying or production scheduling are nonlinear discrete optimization models. The computational complexity of such issues necessitates the development of an efficient solution technique. The creation of the mathematical models and the solution technique have been described below.
The grade tonnage distribution of a metalliferous deposit having an independent domain ( , ) for input values is shown in Here, it is required to select the (₵) for each set of ranges from , and it is required to determine the stages of transition from to for each set of values. The values in each stage are saved and the optimization for the next state proceeds. The grade equal to or above the is treated as ore, and the rest is treated as waste and that is sent to the waste dump.
The average grade of the metal deposit ḡ ( , ) is determined from the grade-tonnage distribution and the concentration of metal contained ( ) in it. It is evaluated using the weighted average formulae given by equation (1).
The tonnage of ore ( ) and tonnage of waste ( ) from the grade-tonnage distribution will be determined using the equations 2(a), 2(b), 2(c) and (3).
Where, is the total material ( + ) to be mined out.
Stripping ratio ( , ) is a critical parameter, which plays a significant role in evaluating the of the deposits. Mine planners play a vital role in ascertaining Sα,β values corresponding to their pit progress whether to keep the value uniform or in an increasing trend affecting their ' . The value of , is determined using equation (4).
Mining companies make money by extracting and processing raw materials into a marketable product. The product of tonnes of ore processed, average mill head grade, % recovery over concentrating and refining, and the product's selling price are used  (5): For obtaining the optimized returns ( , ) → ( , ) , ( , ) , ( , ) are the initial decision variables for all the stages.
Decision variables are governed by the limiting capacities (constraints). Based on the cut-off grade (₵) and the stage transition ( ≤ ≤ ), the amount of production in different stages, i.e., mine, mill and refinery are determined subsequently by any one of the following conditions given in equations (6) through (8): Subject to: If is the overall metallurgical recovery, then Where: , = present value cash flow , = cash flow = discount rate = period (year) indicator ( , ) = maximum mining capacity (tonnes / yr) , = maximum mill capacity to process ore (tonnes / yr) ( , ) = maximum refinery capacity of metal production (tonnes / yr) = actual mine production rate (ore and waste) (tonnes / yr) = actual ore processing rate in the mill (tonnes / yr) = actual metal production rate in the refinery (tonnes/ yr) ( , ) = mining recovery ( , ) = mill recovery ( , ) = refinery recovery Generally, COG optimization has been observed as a nonlinear dynamic system and can be solved using Dynamic Programming (DP) to obtain a global optimum. The basic concept behind DP is to break down the optimization issue into a series of sub-problems, each representing the optimization of a single control action at a particular time. The value function, which translates the current state of the system to the NPV of future income, connects these sub-problems. The best control action at each time can be obtained by solving a sub-problem at that moment after the value function has been computed. These sub-problems are, in general, significantly easier than the original optimization issue; each sub-global problem's optimum may be quantified effectively.

Module 1: All Unlimited Conditions
When all the three stages are unlimited, it brings the most exciting and novel part of the optimization combination. This is being the first mathematical approach, which deals with all the unlimited capacities. Here all the three components , mine production, and mine-life, have been optimized simultaneously at the same time to determine the optimal . Here, for each of the − the corresponding ore-waste tonnage, stripping ratio, average grade, and cumulative tonnage has been calculated.
Each value will be optimized starting with the total tonnage as the mine production from year 1 to 100. This − mine life and mine production optimization will give the year-wise production rate for the same value for years 1 to 100.
Thereafter, the is calculated from year 1 to 100 for the same value. This whole process will be repeated for all the values of the grade-tonnage distribution curve. After this there will be a final set of data that came from all the values having their maximum with their corresponding optimal mine life and production; among all the optimum values, the one having the highest is the most optimal .
The following assumptions, as given below, have been made for the sequence of working.
2. Activity per-period resource requirements and project per-period resource availabilities are fixed and known.
3. The cash flow associated with each activity is not susceptible to change in course, hence no stochastic values have been used for the determination of .
4. There is no interruption in the performance of the activity once it has begun (no pre-emption).

5.
No cancellations are permitted. Each activity must be finished to check its optimality performance.
6. Each activity is completed following the sequentially advancing algorithm. That paradigm is associated with a predictable activity length, fixed resource requirements, and a set cash flow.
7. When an activity is completed, net non-negative cash flows will occur.
8. The stockpiling option has not been considered.
Under the prior assumptions, the optimization issue to maximize the of the project is written as a mathematical programming problem as follows: , Besides the optimization technique mentioned above for the determination of ( , ) , the empirical-formulae based have also been determined for the comparison and efficacy of the model utilizing the existing thumb rule given by Taylor [14] and the modified Taylor's rule by Long [60].

≤ ₵ ≤
According to the mine-life, we calculate the production capacities and the following stage capacities, and then it follows the sequentially advancing optimization approach to determine the optimal value. Ultimately, the grade at which it contributes the Global maximal value decides the optimal value. The overview of the multi-value driven nested algorithmic approach is shown in Fig. 4. Module II: When mining capacity is limited and the rest two stages are unlimited, then the maximum value of from the array is considered, neglecting that production rate is less than production capacity, taking the corresponding , and deciding the final capacities of the rest two stages.
Module III: When mining capacity and smelter & refinery capacities are unlimited, and mill capacity is limited, then the maximum value of from the array is considered where the mill production is less than or equal to the mill capacity and take the corresponding , and hence deciding the final output from the rest two unlimited stages.
Module IV: When mining capacity and mill capacity are unlimited, and smelter and refinery capacity is limited, then the maximum value of from the array is considered, which satisfies the condition that corresponding smelter and refinery production is less than or equal to the smelter and refinery capacity and take the corresponding , and hence deciding the final output from the rest two unlimited stages.
When smelting & refining capacity is the governing constraint, then the decision state is, Module V: When mining capacity and mill capacity are limited, and smelter & refinery capacity is unlimited, then the maximum value of from the array is considered that satisfies the condition that corresponding mine production and mill production is less than or equal to their capacity and take the corresponding , and hence deciding the final smelter and refinery capacity.
Module VI: When mining capacity and smelter capacity are limited, and mill capacity is unlimited, then the maximum value of is considered from the array that satisfies the condition that corresponding mine production and smelter production is less than or equal to their capacity and take the corresponding , and hence deciding the mill capacity.
When mining and refining capacity are the governing constraint, then the decision state is, Module VII: When milling capacity and smelter and refinery capacity are limited, and mine production capacity is unlimited, then the maximum value of is considered from the array that satisfies the condition that corresponding mill, smelter and refinery production is less than or equal to their capacities and take the corresponding , and hence deciding the final mine capacity.
When milling and refining capacity are the governing constraint, then the decision state is, ≤ ( , ) ( , ) ∀ and, Module VIII: When all the three stages are limited, the maximum value of from the array that satisfies the condition corresponding to all three-production stages will be considered. Among the three stages, the minimum capacity stage is the determining factor to decide the final output from the rest two stages, and thus the corresponding calculation will take place.
When all the mining, milling and smelting & refining capacity is the governing constraint, then the decision state is, Ultimately the grade at which it contributes the maximal value decides the optimal value.
The given stage-wise logical conditions are implied to all the eight modules for obtaining the final ( , ) , ( , ) , ( , ) values according to the various limiting conditions from the set of mining, milling, smelting and refining.

Stage -II:
When milling and refining capacity is the governing constraint, then the decision state is

Stage -III:
When refining and mining capacities are the governing constraints, then the decision state is accordingly either. The value of ( , ) and ( , ) are calculated using equations (36) and (37).
Determination of the Optimum is a - [61], and output from each state is fed as an input to the next. Each stage has several possible states associated with it, as shown in Fig. 5.
The objective function is to maximize the for the whole deposit over the whole Mine life ( ). It is the cumulative summation of all the cash flows ( 0 , 1 , 2 … ), where 0 is the capital invested having a fixed discount rate of ' ′ up to ℎ year given by equation (44) .

Implementation and development of the computer model
A computerized model based on the dynamic programming method has been established for determining the optimal using the formulae mentioned. The programming language used to develop the computer package is ℎ 3.7, the ideal multi-paradigm programming language for developing scientific and business applications. It imparts a great extent of calculation speed and data accuracy. ℎ is garbage-collected and dynamically typed. It supports multiple programming paradigms, including structured (mainly procedural), object-oriented and functional programming. A framework, version-5, is used for this model. The Graphical User interface ( ) has been developed using ℎ 3.7 and designer that comes with 5 . In particular, there are some of the core packages like , , and that has been used exclusively for this model. The automated model offers more operational capability and a graphical user interface (GUI) based on the user interface software. The software package consists of three main components: input-data, output-result, and result-graphical.
The input-data component can be categorized into two parts: Part-1− Mineral inventory data, and Part-2− Costs and other parameters Input. The mineral inventory input data consist of grade class intervals (Grade ranges) and the tonnage distribution

Validation of outputs using the real case study
To validate the models, the work necessitates a real case study in terms of domain knowledge of an open-pit copper ore mine with all or at least three phases of ore production, namely extraction, concentration, refinery, and smelter. The model has been validated using data from one of India's largest open-pit copper deposits.

Brief Introduction about the Mine
Hindustan Copper Limited (HCL), a public sector company under the administrative supervision of the Ministry of Mines, Government of India, was established on 9 th November 1967. It is the only copper producer that is vertically integrated, with operations of mining, beneficiation, smelting, refining, and downstream saleable products. From the run-off mine grade of 0.8 to 1.2% , the ore concentrate of grade 25-26% is prepared. Subsequently, copper concentrate is fed to the smelter of HCL located in Moubhandar (Ghatsila) to prepare copper bars and plates as per buyers' specifications (Fig. 6). reserves. MCP has rich mineable copper reserves of 143 , which is more than 70% of the country's available resources.
India's largest copper mine, Malanjkhand deposit extends vertically from the surface at 580 mean reduced level ( ) to a depth of (-)300 . Open-pit mining was planned initially to a depth of 204 (376 ), which is now extended to 240 Three distinct forms of mineralization are clearly visible. Firstly, the mineralization is primarily correlated with quartz, which is generally limited to the quartz reef as a fracture fill form and can be known as quartz ore. Secondly, the mineralization of the stringer pattern has quartz and calcite joints. This stringer ore is found primarily in granites. Lastly, the disseminated ore inside granites or micro granites is present within the interstices of inherent grains of the related rocks. The most predominant ores are chalcopyrite, chalcocite, and malachite in order of abundance. The secondary and oxidized ore minerals are confined to the upper part of the deposit. Due to hydrothermal vein formation, the Malanjkhand copper ore deposit geometry varies in the strike and dip directions. The host rock is quartz reef and granite.
Open-pit mining will be continued for exploiting the deposit. The calculation module is based on the grade-tonnage distribution and operational data (mine, mill, and refinery capacities).
Financial parameters have been considered as per the Indian scenario (in ₹). The mineral inventory is given in Table 2.

Model input
Geo-mining details and techno-economic variables are the vital input parameters for optimizing the objective function for maximization of . In addition to the grade-tonnage deposit (mineral inventory) data, the required technical and financial data are given below in Table 3. The following data has been used as input data to determine optimum %. This data will be imported into the -optimizer tool for the calculation and software input.

Results output
The data mentioned above was provided to the software and processed by the optimizer model, after which the final optimized output of every section was generated and listed in Table 4.   In this model, the input values (operational and economical) have been taken irrespective of the time. Therefore, the values and results will behave accordingly as per the user's input values. Thus, the mine planners or experts will run their module as per the current situation and get their results to optimize accordingly. If there will be a significant increase in the pricing dynamics in the coming years, the planning experts will run through the program and optimize accordingly. Therefore, there is no need to introduce the price forecasting technique in this model because there are a host of variables where the cost-escalation factor will be applicable, including the entire mine, mill, and smelter and refinery stages. Therefore, considering all the variables will make things more complex and may lead to an indefinite solution.

Graphs of the results output
The Graphical User Interface of the output data module for the above results output as mentioned in the graphical result module are shown through graphs illustrated in Figures 10, 11, 12 and 13, respectively.

Discussions of results
The results after the simulation from all the eight modules have been presented through tables and figures. The observations according to the given grade-tonnage and techno-economic variations are discussed below in Table 5.

Module -8 (A):
Taylor [14] , , ( , , ) ( )/ ( )/ ( ) When all the capacities (mining, concentrator, and smelter and refinery) are unlimited, the optimal (%) and the corresponding maximum , mine production and mine life have been calculated according to Taylor's [14] mine life rule, by Long's [60] rule and by the DP based algorithm (the presented -model). It has been found that in all the three cases, mining capacity becomes the limiting function. Therefore, the logic sequentially advances and decides the final and values, and correspondingly the limiting logical conditions on and values will contribute to the final results. It may be observed that among the three methods for the unlimited conditions, the DP based algorithm gives the maximum of ₹ 388201 million at (%) of 0.24.
A major advantage of the DP based algorithm is that it optimizes all the four variables − %, , mine production and mine life simultaneously.

Module -8 (C):
DP based algorithm ( -model described in the paper) , , ( , , ) It may be observed that the results obtained from all the different modules using the real life data behave dynamically as per the existing grade-tonnage, techno-economic criteria, and other limiting conditions.

Conclusion and recommendations
The significance of developing a new evaluation technique, such as the one proposed in this study, lies in providing a pragmatic analytical approach to the mining industry that allows the mine planner or analyst to obtain a more realistic estimate of an open-pit project value accounts for the given techno-economic conditions. The three primary stages utilized in optimization depend upon the limiting conditions of mining, milling, and smelting and refining. In each of these sectors, several smaller divisions may be established. By individually optimizing mine and in planning stages, a guarantee of obtaining the highest is generally lost.
This necessitates developing a suitable approach for finding the optimum of ore, and hence a multi-value dynamic programming has been developed and used to compute for a metalliferous deposit. In this model, discrete values have been utilized for Dynamic Programming ( ), a more deterministic approach than stochastic one. The developed model is based on the concept of dynamic programming and thus finds the solution much more straightforward than other optimization techniques. The fundamental framework and dynamics guide this model rather than tailoring it to time series. This increases the model's generic validation, but it takes a long time to design and parameterize. Its key strength is that it optimizes several policies simultaneously, that the results are connected and mutually reliable, and effectively validates against facts. As in this model, optimization has been carried out after finalizing the Ultimate Pit, and the pit optimization follows the objective function of maximizing profits irrespective of time. Therefore, whenever one does mining, it will create wealth, and the economics will behave as per the existing scenario. As a result, mine planners must re-run the -optimizer model depending on market conditions and modify their intermediate stages of excavation following demand and economics. In general, the mine planners can choose their optimization in three stages.
• First stage by initially excavating higher-grade ore to recover capital, operating, and running costs and reduce the payback period.
• Second stage, by recovering the depreciation and tax amounts, can mine with the average-grade or lower grade.
• Third stage, when the machines or plants are fully depreciated, they can go for mining the lower grade ore and sustain the working of mines.
This work has the novelty of dealing with eight different optimization possibilities. So far, only six potential outcomes have been dealt with as on date by Lane [16]. The computer can produce the result in seconds from the complex system of mine, mill, and smelter and refinery comprising of eight different possibilities, and thus it will be a useful industry-oriented tool to plan a mining project for a given deposit considering all the possible conditions. An alternative method to Lane's algorithm and other existing approaches, a technique based on computer programming algorithms, has been presented here as a feasible and general method for the solution of optimal and production rate determination.
Open-pit mines such as the Malanjkhand copper mine has diversified production constraints. Besides, these operations will generally have a well-defined operating range, which may vary in actual situations. The model does not consider the uncertainty in economic parameters, especially the uncertainty and fluctuations on metal prices, stockpiling or blending, mining dilution, multi-mineral deposits, and mine rehabilitation. The stockpiling option has not been included as that may increase the complexities enormously and may lead to an incomplete solution. The objective of this model is to introduce a universal approach for resolving the optimality scenario related to of open-pit metalliferous deposits.
This method is capable of handling technically more acceptable relationships and definitions. However, the software provides enough flexibility to include or exclude the parameter(s) by changing the design's code; therefore, there is ample room for the expansion and modification of the model to cover additional scenarios. The intention here is to introduce a general method for the solution with a deterministic mathematical approach.
However, there is a scope to modify the program by including the stochastic behavior of the variables (commodity price, operating and maintenance expenses, fixed cost, hauling distance, discount rate etc.) to make it more applicable but at the cost of increasing the complexities. However, based on mining conditions and economics, the material categorized as waste could become economically viable to process in the future, necessitating a shift in policy. So, the utility and complexity of stockpiling is a more extensive area to explore.