A new global optimum solution model based on multi-value dynamic programming for the cut-off grade optimization in a surface mining complex

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 discrete multi-value dynamic programming (MDP) has been developed to calculate the global optimum COG of a large-scale 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 global 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. This model can generate results in polynomial-time from the complex of mining, milling, and smelting and refining system corresponding to various limiting conditions. The prime objective considered in the model is to optimize the COG of a metalliferous deposit. A working open-pit copper mining complex from India has been used to validate the model. 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 gradually hits zero after the project’s life cycle, confirming the typical pattern of other mining firms.


Introduction
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 (Krausmann et al., 2018;Van Vuuren et al., 1999).
Metal mining has consistently been shown to be a significant source for meeting metal demands in infrastructure, health, information systems, and other services that can support economic growth (Jowitt et al., 2020). Thus, cutoff grade ( ) 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% (Henckens & Worrell, 2020;Mudd, Mineral economics is not a new field; this can be followed back 200 years to the research of classical economists like as Smith (Smith, 1776), Ricardo (Ricardo, 1821), and Jevons (Jevons, 1866), among others. Smith (Smith, 1776) 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 (Ricardo, 1821) 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 Fathollahzadeh et al., 2021;Rezakhah & Newman, 2020). 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.

Literature Review
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 lowest grade of ore that can be processed commercially. Therefore, this criterion creates a boundary inside a particular mineral deposit between ore and waste, either accepted for milling or rejected for dumping (Taylor, 1972(Taylor, , 1986.
In several 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 subcategories: 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 (Henning, 1963) was the first who brought the idea of optimization into the mining business and a year later the seminal work of Lane (Lane, 1964) was introduced. Lane determined optimal based on limited capacities, supporting Henning's (Henning, 1963) prediction that ideal tend 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 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 (Lane, 1964(Lane, , 1988 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 (Lane, 1964(Lane, , 1988) method has been widely criticized for determination that may be more suitable in certain circumstances, making it possible to disregard the optimal estimates for (Asad, 2005). Lane's technique has stayed empirical, necessitating the development and implementation of an analytical optimization technique (Dagdelen & Kawahata, 2008.
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 (Azimi et al., 2012;Dagdelen & Kawahata, 2008;Ganguli et al., 2011;Yasrebi et al., 2015). The basics of Lane's (Lane, 1964(Lane, , 1988 technique have been expanded in most optimization approaches. Several studies employed time-value-of-money-based optimization approaches. The optimization algorithms that involve maximization lead to better outcomes than those that maximizes the profit function. Lane (Lane, 1988) 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 corresponding 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 (Dowd, 1976) demonstrated the technique that might be used for stochastic data with the objective function of maximizing the present value of future net profits. Several conventional and meta-heuristic techniques are being applied extensively to optimize pit scheduling. Later, Wang (Q. Wang, 1995) and Tolwinski (Tolwinski, 1998) studied dynamic programming to study block sequencing. Due to the periodic extraction of finite resources, the efficiency of dynamic programming approaches has shown to be helpful in scheduling mining operations for simultaneous optimization of several policies (B. King, 2000). Linear programming, for example, might be used to maximize cash flow locally, whereas dynamic programming could be used to improve on a global level.
Baidowi et al. (Baidowi et al., 2021) established a mathematical model for optimization in the open-pit metalliferous deposits based on overall profit maximisation, incorporating reclamation income and expenditures.
The authors proposed the consideration of environmental variables in the determination of . Few authors emphasized the importance and integration of blending optimization throughout the mining value chain (Chanda & Dagdelen, 1995;Haonan et al., 2021;Jamshidi & Osanloo, 2018).

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Dehghani and Bogdanovic (Dehghani & Bogdanovic, 2018) used a bat algorithm to estimate metal price swings, which produced better outcomes than the traditional estimation approach for forecasting copper price. This resulted in a price trend. However, it is questionable when there is a significant and rapid shift in demand and supply or a technological upgrade or technology replacement. Surface miner, for example, is a technological substitute that reduces overall equipment expenditure by around half, eliminating the requirement of drilling, blasting, and crushing. A genetic algorithm was employed by Franco-sepulveda et al. (Franco-sepulveda et al., 2019) to maximize the of a mining operation. They looked at the price of the metal from a historical date to the present day and found that the sale value was 50% of the time lower than the projected value. Su et al. (Su et al., 2020) stated that the emergence and collapse of several price bubbles are connected to supposition, macroeconomic instability, a supply-demand imbalance, and economic turmoil. Tahar et al. (Tahar et al., 2021) agreed that commodities prices shocks might have differential effects. Most of the commodities' economies are highly volatile and difficult to anticipate. Considering an incorrect pricing forecast leads to sub-optimal selections and implications.
Russell (Russell, 1970) pioneered the concept of maximizing a project's . The most commonly used economic metric in evaluating financial projects is (Creemers, 2018;Peymankar et al., 2021). It is worth noting that is acutely susceptible to revenue variations (Hustrulid et al., 2013;Oraee et al., 2011;Saleh & Marais, 2006;Wiesemann et al., 2010). Ahmadi (Ahmadi, 2018) 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 (Asad & Topal, 2011;Biswas et al., 2020). Various researchers (Ahmadi & Shahabi, 2018;Bascetin & Nieto, 2007;Qingfei Wang et al., 2010) 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 in mind. The target of the strategic mine plan is to maximize while adhering to all limitations (Jélvez et al., 2020;Samavati et al., 2018). If one doesn't optimize or check the periodically, there might be a time when the mine can face a cash crisis and needs to be shut off abruptly due to the possibility of negative in the long run. So, the mine planners should plan considering all the price dynamics, , grade-tonnage ratio, reserve, and market scenario. At the end of the exploration stage, the ore body model is first finalized, and then the planning of mine development and project scheduling are carried out accordingly. A planning job requires a multitude of functions. In general, the mine planning comprises the steps as shown in the sequential flowchart of the whole process given below in Fig. 2.

Optimization model framework
The proposed multi-value dynamic programming ( ) framework has been established with the knowledge that the optimum for a surface mining complex is dynamic. A high-level programming language (Python) has been utilized for the simulation of the model. The entire set of mathematical programming is based on algebraic equations, calculus of variation, and stepwise computation that meet the requirements of a sequentially advancing algorithm to determine the global optimum . Real working data from the surface mining complex of Hindustan Copper Limited (HCL) has been used to validate the algorithm's efficacy and applicability.

Functional model of multi-varying conditions
The optimum in an open-pit metalliferous deposit is primarily influenced by three controlling capacity restrictions of several activities, namely mining, milling, smelting, and refining. While estimating the global optimum , eight alternative solutions may be explored based on the varied limiting factors and production capacity of the three phases. (Table 1). Table 1 Eight dynamic models of potential combinations based on the multi-conditional scenario.

Refinery capacity
Restrictions or constraints ( -Limited and -Unlimited)
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 assumptions
It is presumed that the whole reserve will be mined according to the grade dissemination, but only that fraction of the mined-out material that exceeds or equals the should be sent for concentration. Furthermore, the metal price, operational expenses, capital costs, rate of interest, tax percentage, and discount rate have all been assumed to be uniform during the life of mine. However, the volatility of the commodity price indices and technical developments might reshape the entire scenario of expenses related to excavation, processing and future revenue.
To make the work comprehensible, it has been assumed that, at least on average, all ore mined within any twelve months will also be processed during that period.

Model framework based on multi-value 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 ( -) variance of a mineral deposit is the input state of the framework with an individual domain ( , ), as correlated in Fig. 3. For every range of ( , ), the -variance is distributed inside the range of , and , . Therefore, the functional relation is given by ∶  . Here, the (₵) for each set of ranges from , must be chosen, and the stages of transition from to must be determined for every sequence of values. The values from individual stages are stored, and the optimization for the following state begins. is the lowest grade at which a metalliferous deposit may be extracted profitably (or processed). Material with a grade higher than is termed ore, whereas a material with a grade lower than is considered waste. The average grade (ḡ ( , ) ) is determined from the ( -) variance of a mineral deposit and the concentration of metallic value present ( ) in it by following the weighted average formula expressed in equation (1).
Where is the total material ( + ) to be mined out.
The stripping ratio ( , ) is a fundamental parameter to be considered while calculating the of any mining complex. Mining professionals play a crucial part in determining Sα,β figures related to their mine advancement, total ore-waste ratio, and whether to maintain the value constant or increase it in a trend that affects their .
Equation (4) Mining companies make money by mining mineral resources and refining them into marketable products. During operation, the mining complex incurred both variable and fixed expenses. The revenue is determined by the tonnes of ore treated, ḡ ( , ) , the percent recoveries over-processing, and the product's current market value. Variable expenses are further divided into those related to mining and processing the ore and those related to digging and disposing of the waste. Drilling and blasting, transportation, disposal, restoration, and other expenditures are examples of variable expenses. General and administrative expenses and any additional costs that are not dependent on output are examples of fixed costs. The may be calculated by discounting capital expenses and subtracting them from the discounted cash flow. Now sequentially utilizing all of the variables mentioned above, the , of a project is determined by equation (5): The original ruling criteria for all stages are ( , ) , ( , ) , ( , ) for achieving the optimized returns ( , ), these input decision variables ( ( , ) , ( , ) , ( , ) ) are regulated by the various limiting capacity functions.
Depending upon the (₵) and the stage transition ( ≤ ≤ ), the tonnage of outputs at distinct stages, i.e., mining, milling, and smelting and refining, are estimated consequently by following any of the logical statement devised through equation (6) to (8). Subject to: If is the overall metallurgical recovery, then = 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, optimization has been observed as a nonlinear dynamic system and can be solved using Dynamic Programming (DP) to obtain a globally optimum solution. 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 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. After that, 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 from all the values having their 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 maximum with their corresponding optimal mine life and production; among all the optimum values, the one with the highest is the global optimum .
The following assumptions, as given below, have been made for the sequence of working.
1. The activity time frame is deterministic.
2. The per-period resource requirements for activities and project resource availability are set and known.
3. The cash flow associated with each activity is not susceptible to change in the 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 aims to maximize the project's and is expressed as mathematical programming; the following problem statements are developed in the form of equations to solve the dynamic optimality scenario. .
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 (Taylor, 1986) and the modified Taylor's rule by Long (Long, 2009).    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.
) ⨉ ( , ) ⨉ ( 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 the capacity of mining and refining are the controlling factors, 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 the capacity of the milling and refining are the controlling factors, then the decision state is, 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 capacities of mining, milling and smelting and refining are the controlling factors, then the decision   Ultimately the grade at which it contributes the maximal value decides the global 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.
Where ḡ (α, β) is the average mill-head grade. -: When milling and refining capacities are the ruling units, then the decision state is -: When refining and mining capacities are the ruling units, then the decision state is accordingly either. The value of ( , ) and ( , ) are calculated using equations (36) and (37).
Following the stage transition, the decision variables ( ( , ) , ( , ) , ( , ) ) are stored as the final or ultimate decision variables ( ( , ) , ( , ) , ( , ) ) and will now be the optimum solutions for each ( , ) state of variables. These decision variables will now be utilized to calculate the project's ( / ), The selection of the optimal is followed by a multi-staged sequential decision problem (Li & Yang, 2012;Biswas et al., 2020) with the output from each state feeding as an input for the following stages. Each stage has multiple different states correlated with it, as illustrated in Fig. 5. Bellman's principle of Optimality (Bellman, 1972)  There are returns after every stage, and the return as cash flow ( , ) of the ' ' stages has been calculated as follows. The selling price = ( , ) , operating expenses of mining, milling, and smelting and refining = ( ( , ) , ( , ) , ( , ) ), fixed costs of mining, milling, and smelting and refining = ( ( , ) , ( , ) , ( , ) ), any The objective function is to maximize the for the total mineral deposit throughout the entire life of mine ( ).
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).

Application and development of the computer-aided model
A computer-aided -model has been established for determining the optimal using the formulae mentioned-the proposed computational framework is derived from the multi-value dynamic programming approach for estimating the global optimum . 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.

Corroboration of outputs using a working surface mining complex
The work requires an actual case paper that describes a study of domain knowledge of an open-pit copper ore mine with all or at least three phases of ore production, namely extraction, concentration, and smelter and refinery, to evaluate the models. The model has been corroborated utilizing data from India's most extensive open-pit copper resources (surface mining complex).

A brief description of the mining complex
Malanjkhand Copper Project (MCP) is a subsidiary of Hindustan Copper Limited, which was commissioned on 12 th November 1982. The mining lease was approved in favor of HCL in 1973. The latitude and longitude of this mine are 22°00 ′ 59" to 22°02'24" and 80°41'51" to 80°42'38" and forms a part of Survey of India Toposheet No. 64 /12, located in the Balaghat District of Madhya Pradesh in India. The strike length of the deposit is 2600 m. MCP, situated at a distance of 20 away from the Kanha National Park, contributes about 80 percent of HCL's overall copper supply. This is the largest copper mine in India (Fig. 7), with an average orebody width of 75 m. It has a total lease area of 479.9 ℎ .  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 Malanjkhand Copper Project is a massive metalliferous deposit amounting to 200 ( ) of copper ore 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  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 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.
Surface mining (open-pit) will be continued for exploiting the deposit. As per the revised mine plan schedule, mining is contemplated to expand the pit to a depth of 340 . A view of the working pit and the modelled Ultimate Pit design of the Malanjkhand Copper Project is shown in Fig. 9 (a) and (b). The computations are made using the grade-tonnage variance and operating variables (mine, mill, and refinery capacities). Economic values have been assessed in accordance with the Indian context (in ₹). Table 2 shows the mineral content inside the grade bins.

Model-inputs
Geo-mining details and techno-economic variables are the vital input parameters for optimizing the objective function for maximization of . The essential techno-economical parameters used for the -model is shown in Table 3. The following data has been used as input data to determine global optimum %. This data will be imported into the -optimizer tool for the calculation and software input.

Result-outputs
The data has been loaded into the program and analyzed by the -model, and the final optimized output for each component was obtained and shown in Table 4.   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.

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. Table 5 Conclusive summary sheet of the output variables of all the eight modules for the open-pit metalliferous deposit (MCP).

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 (Taylor, 1986) mine life rule, by Long's (Long, 2009) 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.
. 38821 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 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 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 (Lane, 1964). The computer can produce the result in seconds from the complex system of mine, mill, and smelter and refinery comprising of eight different possibilities. Thus, it will be a useful industry-oriented tool to plan a mining project for a given deposit considering all the possible dynamic conditions. An alternate way to Lane's algorithm and other current methodologies, based on mathematical programming, is described here as a practical and generic method for determining optimal , production rate, mine-life and the global .
The Malanjkhand copper complex, for example, faces a variety of production restrictions. Besides, these operations will generally have a well-defined operating range, which may vary in actual situations. The model ignores the volatility and variations in economic characteristics, such as commodity pricing, stockpiling, dilution, multimineral aspects, and mine-restoration. The stockpiling option has not been included as that may increase the complexities enormously and 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 optimization framework methodology can handle technically more acceptable relations, functions, and multidynamic aspects. 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.