1. INTRODUCTION
In open pit mining activities, one of the main problems that often faced by mining company is to determine profitable cutoff grade. The cutoff grade value indicates the ratio of mineral to the waste material content that will be processed by a mining company. A mine block material will be considered as waste and not be mined/processed further if its grade is lower than the specified cutoff grade set by the company since the cutoff grade value determines the quality, quantity and profit earned by the mining company (Dadi and Sttarvand, 2016). Hence, recently many research have been conducted to determine optimal cutoff grade through various approaches and methods.
The basic model of cutoff grade determination was first introduced by Henning (1963) through the breakeven approach and Lane (1964) through the heuristic approach (also known as Lane’s algorithm). Beside using breakeven and Lane’s approaches, there were also other approaches used in the determination of cutoff grade over the past years. Some of these approaches were mixed integer linear programming (Moosavi et al., 2014;Gholinejad and Moosavi, 2016), nonlinear programming (Yasrebi et al., 2015), dynamic and stochastic programming (Li and Yang, 2012;Thompson and Barr, 2014) and mathematical approaches through analytical solutions (Gama, 2013;Muttaqin and Rosyidi, 2017). Among these approaches, the mathematical approach through analytic solutions has several advantages over other approaches such as exact solution, easy to use, and faster optimal solution searching time.
Nowadays, it is important to give attention to environmental and social issues. So far, most researchers considered waste materials as costs for the mining companies. The waste materials can give huge effect on the environment and human health. The open pit mining acitivity usually generates huge amount of waste and it is a big environmental problem for the mining company (Gupta and Paul, 2015). Therefore, for the mining company, removing/rehabilitating these waste materials is very important (Kuranchie et al., 2015). In other side, Bellefant et al. (2013) found that the remaining waste materials still have some economic value for the mining companies. Beside managing waste materials, the mining company is also responsible for reclamation of the mine site as soon as the mining activity ends. The reclamation is very important to ensure that the mine site is no longer harmful and can be used again for future human life (Testa and Pompy, 2007).
In addition to the determining the value of cutoff grade, another problem often faced by mining companies is how to prioritize and select the mining project/location based on certain objectives or criteria. So far, there were no research publications focusing on the integration between the cutoff grade optimization model and the mining project/location selection model for the open pit mining activity. Even, there were very limited literatures exist in the mining project/location selection model, for example: Mukherjee (1994) and Mukherjee and Bera (1995). Mukherjee (1994) and Mukerjee and Bera (1995) only modelled the mining project/location selection separately regardless of the outcome of the cutoff grade optimization model. Whereas integration in that decisionmaking area is quite crucial to do because the decision on the selection of the open mining project/location is strongly influenced by economic indicators, such as the value of Net Present Value (NPV) and Return on Investment (ROI).
Mining activities can provide a very large economic impact to a country, either directly or indirectly, whilst require huge investments (Bahşiş and Kasap, 2016). Therefore a feasibility assessment of a mining project using NPV and ROI indicators is extremely important. Maximum NPV and ROI values for each location can be achieved through cutoff grade optimization. This is why the decision of mining project/location selection cannot be separated from the decision of the optimal value for the cutoff grade.
Based on above problems, it is important to develop cutoff grade optimization model and mining project/location selection model. The cutoff grade optimization model developed in this research is solved using analytical approach which considering mining cost, waste removal/rehabilitation cost, processing cost, reclamation cost, selling stage cost, fixed cost and income/revenue from selling the valuable waste materials. The main purpose of this research is to develop a sequential model that can be used by mining company decision makers in determining simultaneously the optimal cutoff grade value and selecting the best mining project/location which gives maximum NPV and ROI. To illustrate the application of the model, a numerical example is given. The output of this research is expected to be used directly by decision makers in open pit mining companies to face the global competition, increasing the efficiency demand of energy and resource and to provide responsibility on environmental and social aspects.
2. LITERATURE STUDY
2.1 Previous Research of The Cutoff Grade Determination Model
There were many studies related to the determination of optimal cutoff grade values through profit maximization in open pit mining activities. Several approaches/methods has been developed in the optimal cutoff grade optimization model include breakeven, heuristic, linear programming, nonlinear programming, dynamic and stochastic programming. Among these methods, the heuristic approach introduced by Lane (1964) is the most popular and widely used by researchers.
The cutoff grade optimization models based on Lane’s algorithm were developed by Ataei and Osanloo (2003), Gholamnejad (2008), Osanloo et al. (2008), Abdollahisharif et al. (2012), Asad and Dimitrakopoulos (2013), Azimi et al. (2013), DadiCetin and Dowd (2013), Qinghua et al. (2014) and Narrei and Osanloo (2015). The models were developed using Lane’s algorithm or extended Lane’s algorithm which have several main advantages such as clearer optimal solution procedures, but it also takes longer time to find the optimal solution. The cutoff grade determination models by using mixedinteger linear programming approach were developed by Moosavi et al. (2014) and Gholinejad and Moosavi (2016). The main advantage of the cutoff grade determining models developed through the mixedinteger linear programming approach is the flexibility of using multiple decision variables in the model that can consist of integer, boolean, or other variables. However, the mixed integer linear programming method has disadvantage in term of time efficiency to find the optimal solution. The cutoff grade optimization models by using nonlinear programming approach was developed by Yasrebi et al. (2015). The cutoff grade optimization models by using dynamic approach and stochastic programming were developed by Li and Yang (2012) and Thompson and Barr (2014). The dynamic and stochastic models were developed based on the uncertainties that are always present in the real system. But on the other hand these models have higher complexity level in term of the model development and finding the optimal solution. While the cutoff grade optimization models by using analytical approach were developed by Johnson et al. (2011), Gama (2013) and Muttaqin and Rosyidi (2017). Unlike other approaches, the models developed using analytical approach have an advantage in finding the exact solution.
Among above cutoff grade optimization models, the models developed by Ataei and Osanloo (2003), Li and Yang (2012), Cetin and Dowd (2013) and Qinghua et al. (2014) can be used for multiple mine deposits. In addition, some researchers also considered the uncertainty of the sales price of the ore mining (Johnson et al., 2011;Azimi et al., 2013;Thompson and Barr, 2014) and uncertainty of mine deposit distributions (Li and Yang, 2012;Asad and Dimitrakopoulos, 2013;Azimi et al., 2013). From the revenue and cost component aspects, almost all of the cutoff grade determination models considered sales revenue, mining cost, processing/concentrating cost, marketing cost and fixed cost components. However, some researchers also considered the waste removal or waste rehabilitation cost (Gholamnejad, 2008;Osanloo et al., 2008;Abdollahisharif et al., 2012;Gama, 2013;Muttaqin and Rosyidi, 2017). Moreover, the cutoff grade optimization model based on Lane’s algorithm developed by Narrei and Osanloo (2015) also accommodated the revenue from the valuable waste materials.
2.2 Previous Research of The Mining Project/Location Selection Model
Nowadays, there are very limited studies that discuss about the selection of projects/locations in the field of mining studies, especially in the open pit mining. So far, the models that widely used by researchers related to the mining project/location selection were developed through multicriteria approach. For example, AkbarpourShirazi et al. (2014) developed the In Pit CrushingConveying (IPCC) process location determination model on an open pit using the Analytical Hierarchial Process (AHP) method. In addition, using the same method of AHP, Hermann et al. (2015) developed a model for selecting landfilling methods in the mining field.
Based on the results of the literature review, the model of mining project/location selection in the field of mining studies only developed by Mukherjee (1994) and Mukherjee and Bera (1995). The model developed by Mukherjee (1994) was implemented to a case study of the coal mining industry in India. The multiple criteria integer linear programming was used in those models. Mukherjee and Bera (1995) then further developed a project selection model in the field of mining studies using goal programming approach including four objective functions (capital investment, production cost, profit and manpower goal) and one constraint (annual production).
Due to the limitations of the existing literature on the project/location selection model in the open pit mining, this study also uses several literatures to discuss the model of project selection in other fields. Those literatures are principally important in understanding the purpose, limitations and various advantages of each method/approach to solve various project selection problems in various fields. Mavrotas et al. (2006) developed a general purpose project selection model using a combination of multicriteria and integer (binary) programming approach. Hassanzadeh et al. (2014) also developed a general purpose project selection model with two objective functions, namely benefit maximization and risk level minimization of the project. The MixedInteger Linear Programming approach was used in the model with a case study of project selection in telecommunication companies.
Similar research was also conducted by Eshlaghy and Razi (2015) who developed a project selection model using Gray Relational Analysis (GRA) method and the genetic algorithm. ShakhsiNiaei et al. (2015) developed a multi purpose optimization model of project selection using a combination of multicriteria approach, genetic algorithm and differential evolution algorithm. Tavana et al. (2015) also developed a multi purpose project selection model by combining multicriteria and integer programming approaches. Data Envelopment Analysis (DEA) and Technique for Order Performance of Similarity to Ideal Solution (TOPSIS) were used in the model to solve the problem. Moreover, Kalashnikov et al. (2017) also developed a project selection model with two objective functions, namely minimizing the difficulty level of project implementation and maximizing the benefits of the project. Previous research summary of the project/ location selection model is shown in Table 1.
3. PROBLEM DESCRIPTION
There are two models developed in this research. The first model is used to determine the optimal cutoff grade, while the second model is used to select one or more mining projects/locations which maximize the NPV and ROI. The calculations of NPV and ROI are based on respective optimal cutoff grade for each mining project /location. The objective function of the first model is to maximize the total profit, whereas it depends on the revenues and cost components of the system.
In this study, the cutoff grade optimization model considers two kinds of revenues, namely sales revenue and valuable waste materials revenue. Sales revenue is calculated from average mined grade of ore (T), recovery rate (U), and sales price (V). Valuable waste materials revenue is calculated from income from revaluing waste materials (I), valuable material/waste ratio (A), and waste/ore stripping ratio (R). The cost components considered in the first model are mining cost (M), processing/ concentrating cost (B), waste removal/rehabilitation cost (E), reclamation cost (C), marketing/selling cost (S), and fixed cost per ton production (F/P). In this cutoff grade optimization model, the cutoff grade (T_{c}) is the only decision variable. The objective function of the cutoff grade determination model in this study is shown in equation (1) with notations given in Table 2.
Before determining the optimal cutoff grade value, it is important for the decision maker to give attention to two criteria, namely the minimum allowable cutoff grade (T_{cm}) and the maximum allowable stripping ratio (R_{max}). A mining project/location can generate profits if the average of ore (T) is greater than the minimum allowable cutoff grade (T_{cm}) and the stripping ratio (R) is lower than the maximum allowable stripping ratio (R_{max}). The minimum allowable cutoff grade and maximum allowable stripping ratio are shown in equation (2) and (3) respectively.
To construct the optimal cutoff grade formula, it is assumed that the relationship between the cutoff grade (T_{c}) with the ore tonnage (Q), stripping ratio (R) and the average grade of ore (T) are linear (Gama, 2013), as shown in equation (4) – (6).
The above equations are then substituted into the total profit function in equation (1) to get a new total profit formula as shown in equation (7).
By setting the first derivation of equation (7) to T_{c} and solving the equation using Wolfram Mathematica 7.0 software, the optimal cutoff grade formula is obtained by equation (8).
For the second model, it is assumed that several projects/locations alternatives are available and due to some constraints such as investment costs and annual operational costs, the mining company must determine which projects/locations must be selected to be mined and explored. In this case, the mining company must select one or more profitable projects/locations based on the NPV and ROI of each mining project/location. Hence the decision variable in this mining project/location selection model, denotes by X_{j}, is expressed by a binary number and multi objective optimization method is needed since the model has two objective functions.
Since the NPV and ROI values have different units, it is necessary to transform these two objectives into one dimensionless unit (Marler and Arora, 2004). The result of multiplication of each indicator value (after being transformed) with each weight indicator will become an aggregate value of NPVROI denoted by U. Hence, to gain maximum benefits (through NPV and ROI), the mining company must find the maximum value of U through a combination of selected projects/locations.
The objective function for the mining project/location selection model is shown in equation (9), while the constraints of the mining project/location selection model are shown in equation (14), (15), (16) and (17).
where,
subject to:
The notations are listed below,

i: Index for alternatives i = {1, 2, …, n} with number of alternative, n = 2^{m})

j: Index for mining project/location, j = {1, 2, …, m} with m denotes the number of mining project /location

X_{ij} : Decision binary variable for each project /location selection j (0 or 1)

N_{ij} : NPV of alternative i for project/location j ($)

Y_{ij} : ROI of alternative i for each project/location j (%)

L_{ij} : Investment cost of alternative i for each project/location j ($)

W_{ij} : Annual operational cost of alternative i for each project/location j ($)

θ_{max} : Available investment funds ($)

φ_{max} : Available annual operational funds ($)
In the project/location selection model, there are at least four constraints needed. The first constraint in equation (14) expresses the total investment cost that must not exceed the maximum investment funds provided by the company. The second constraint in equation (15) is needed to ensure the total annual operational cost of all project/locations will not exceed the maximum available annual operational funds.
4. SOLUTION PROCEDURES
It can be emphasized that the mining project/location selection model is a nonlinear formulation. Hence, it cannot be solved directly by solvers. That is why this 3staged solution procedure is proposed. The solution procedure of the model can be divided into three main stages: first, determine the optimal cutoff grade; second, determine the value of NPV and ROI; and third, select the mining project/location.

Step 1: For each mining project/location (j), determine the optimal cutoff grade (T_{c}) that maximize the total profit (Z).

Step 1a: Obtain the main input of the cutoff grade determination model (mining grade distribution and economic parameters)

Step 1b: Check the mining feasibility using minimum allowable cutoff grade (T_{cm}) and maximum allowable stripping ratio (R_{max}) indicators using equation (2) and equation (3)

Step 1c: Calculate regression coefficient for variable Q, R and T using equation (4), (5) and (6)

Step 1d: Determine the optimal cutoff grade using equation (8), then calculate the total profit using equation (7)

Step 2: For each mining project/location (j), determine the value of NPV and ROI .

Step 2a: Find the limiting mining stage using data of capacity limits for each mining stage, estimate years of completion

Step 2b: By adding the investment cost, calculate the NPV and ROI using equation (18) and (19)

Step 3: Select the combination value of mining project/location selection decision (X_{ij}) that maximize the value of U.

Step 3a: Identify constraints by setting the available investment funds (θ_{max}) and available annual operational funds (φ_{max}). The annual operational funds (γ) for every mining project/location can be determined using equation (20)

Step 3b: Find the combination value of X_{ij} that maximizes U value and satisfies constraints in equation (14) until equation (17)
5. NUMERICAL EXAMPLE
In this study, a numerical example is provided to illustrate the implementation of both model of cutoff grade and project/location selection. Suppose an open pit mining company wants to explore five iron ore mining projects/locations (P1P5). In order to generate maximum benefits, the company must determine which mining project/location will be selected based on NPV and ROI criteria.
Based on preliminary studies by the company, mine grade distribution data on each of the mining project /location are shown in Table 3. In addition, economic parameters and other input variables are also known as shown in Table 4. To solve this problem, the mining company must first calculate the optimal cutoff grade value, and the NPV and ROI values of each mining project/location based on the optimal value of cutoff grade. Afterward, based on the value of NPV and ROI that have been obtained, the mining company can determine which mining project/location will be selected.
5.1 Determination of Optimal Cutoff Grade Value and NPVROI Calculations for Each Mining Project/Location
The mining distribution data in Table 3 and the economic parameter data in Table 4 are the main input to obtain the optimal cutoff grade for each mining project/location. The first step to obtain the optimal cutoff grade is by calculating the profitability of each mining project/location through the minimum allowable cutoff grade and maximum allowable stripping ratio using equation (2) and equation (3). Then, the regression coefficients in equation (4), equation (5) and equation (6) must be determined first using statistical software based on the raw data. Finally, using equation (7) and equation (8), the optimal cutoff grade and total profit for each mining project/location can be obtained.
For example, for the 2^{nd} project/location (P2), by using equation (2), the minimum allowable cutoff grade is 31.20%. Since the average grade of iron ore for the minimum stripping ratio (R = 0) is above the minimum allowable cutoff grade (33.24% > 31.20%), then the mining activity is profitable. Based on the regression coefficients presented in Table 5, it can be concluded that the regression coefficients for three variables (Q, R and T) are fit enough to be used as input for the cutoff grade determination model.
Using the coefficient values of the three variables (Q, R and T), mining grade distribution data and economic parameter data, the optimal cutoff grade and total profit for the P2 project/location can be calculated. As a result, the optimal cutoff grade for the P2 project/location is 25.54% with the total profit of $151,710,259.
Using the same steps, the optimal cutoff grade and total profit for each project/location can be calculated. The results of optimal cutoff grade and total profit for each mining project/location are presented in Table 6.
5.2 NPVROI Calculations for Each Mining Project/Location
Because NPV has highly related with the concept of time value of money, the time completion for each mining project/location must be estimated first before determining the NPV and ROI value by comparing the tonnage of mined material and tonnage of iron ore to the capacity limits for each of the mining, processing and marketing stages (shown in Table 7). The limiting stage is used as the basis to estimate the completion time of the mining project, because mining, processing and marketing stages are in the same sequential mining production system. Once the estimation is completed, then the NPV and ROI values can be determined.
The mining stage and processing stage use the amount of mined material (Q_{m}), while the marketing stage use the amount of ore material (Q_{p}) as the input to be processed. For example for the P2 mining project/location, using equation (21) and equation (22), the amount of mined material (Q_{m}) is 34,953,488 tons and the amount of iron ore material (Q_{p}) is 9,083,678 tons.
Using the capacity limits of mining, processing, marketing stages in Table 7 and calculation of tonnage of mined materials and tonnage of iron ore, it is found that the P2 project/location is estimated to be completed within 12 years.
After the estimations are completed, then by using the data of investment and total profit for each project, the NPV and ROI values can be calculated using equation (18) and equation (19).
Assuming that the profit is distributed equally each year, then for the P2 project/location with the total profit of $151,710,259, the project will generate a profit of $12,642,522 each year. The result of NPV calculation for the P2 project/location is shown in Table 8.
Based on the calculations shown in Table 8, the NPV for the P2 project/location is $28,312,510. Therefore, by using the $50,000,000 investment cost, then the ROI for the P2 project/location is 56.63%.
Using the same procedure, the completion time, NPV and ROI for each project/location can be determined. The calculation of the NPV and ROI for the five mining projects/locations are shown in Table 9.
5.3 Mining Project/Location Selection Based on NPV and ROI
The first step to determine the selected mining project/location is to collect the NPV and ROI values of each mining project/location. Then, the models constraints based on equations (14) to equation (17) are applied. In this numerical example, the mining company has an investment fund limit (θ_{max}) of $1,000,000,000 and an annual operational fund limit (φ_{max}) of $1,100,000,000 per year. To calculate the annual operational cost of each mining location/project, equation (20) is used.
The next step is to find the combination value of X_{ij} that maximizes U value and satisfy constraints in equation (14) until equation (17) which contains a decision whether each j project/location in each alternative i will be selected (X_{ij} = 1) or not (X_{ij} = 0). In this numerical example, since there are j = 5 mining projects/locations, there will be as many as i = 2^{j} = 2^{5} = 32 alternative decisions. In each alternative, the total NPV and total ROI are calculated using equation (12) and equation (13). The total value of each NPV and ROI indicator are transformed using equation (10) and equation (11). After multiplied by the respective weight indicator, it will generate the value of U which represents the NPVROI value of each alternative. In this numerical example, we determined that the NPV indicator and the ROI indicator have 0.6 and 0.4 weight respectively.
The objective function of the mining project /location selection model in this study is to select the alternative which has the biggest U value. However, since in this model there are several constraints to be considered, not all of the alternatives are feasible. For example, if the mining company wants to select all projects/locations, then the total investment that the company should spend ($1,670,000,000) will exceed the available investment fund ($1,000,000,000). Therefore, to solve the optimization problem in this model, the Oracle Crystal Ball optimization software is used. The optimization results for the numerical example are shown in Figure 1.
Based on Figure 1, we can see that the optimization result gives a U value of 0.780 with the decision result for X_{1} = 0, X_{2} = 1, X_{3} = 1, X_{4} = 0 and X_{5} = 1. Thus, it can be concluded that to get maximum NPV and ROI, the mining company must select the P2, P3 and P5 project/ location. In this case, the results of the optimization of U value and the decision for each project/location are influenced by several factors/variables. One of them is the weight for each objective function. In this mining project/location selection model, the weight of each indicator depends on the preference of the decision maker. In addition, because the model in this study is a sequential model, sensitivity analysis is also done by changing the value of the selling price parameter. Therefore, to study the effect of weight objective function to the value of the U value and decision variables, a set of sensitivity analysis is performed by changing the weight of each objective function (NPV and ROI) from 0 to 1, and sales price value by 5% and 10% from the initial value. The results of complete sensitivity analysis are presented in Table 10 and Figure 2.
Based on the results of sensitivity analysis in Table 10 and Figure 2, it can be concluded that the weight of NPV and ROI criteria affect decisions variable and the value of U for each mining project/location. In this numerical example, if the NPV indicator is weighted 0 to 0.1, then the best decision is to select P5 mining project/ location. If the NPV is weighted 0.2, then the best decision is to select P2 and P5 mining projects/locations. Whereas if the NPV is weighted 0.3 to 1, then the optimal decision is to select the P2, P3 and P5 mining project/ location. Therefore, because the value of weighting greatly affects the optimal decision, then the decision makers should be carefully determining which indicator has a higher priority.
Based on the results of the sensitivity analysis, it is also known that the higher sales price value, the U value generated based on the optimization results has a tendency to decrease further. The reason is because the higher sales price value, the NPV and ROI generated from a mining project/location will also increase. With a large NPV and ROI value, the difference between the NPV and ROI values in a mining project/location with others will also be greater. With large difference in NPV and ROI, the U value that presents the NPV and ROI values as transformed as shown in equations (9) will be lower. Therefore it can be concluded that the decrease in U value means that if the sales price increases, then by not choosing (eliminating) one or several mining projects/locations, the potential loss for the company will increase. If the sales price increases, the company should be able to choose many mining projects/locations. If not, then the potential loss of the company will increase. This is the reason why the value of U will have a downward trend inversely proportional to the increase in the sales price.
6. CONCLUSION AND FURTHER RESEARCH
This research proposes an integrated model of cutoff grade and mining project/location selection. The model was solved sequentially by first determine the optimal cutoff grade. Then based on the cutoff grade for every mining project/location, an optimal combination of several mining projects/locations can be found. The objective function of the first model was to maximize the total profit while for the second model was to maximize simultaneously the NPV and ROI. To understand the application of the proposed model that has been developed, this study also presents a numerical example. Based on the results of model development and numerical example given in this study, it is expected that open pit mining companies can solve problems related to the determination of cutoff grade and selection of mining project/location.
The result of sensitivity analysis points out that the changes in selling price give significant effects on the model’s optimal solution. The higher the selling price, the lower the optimal cutoff grade and the higher the total profit will be. At a certain level of decrease in selling price, the optimal decision is to not select any mining project/location. The reason is because the selling price is too low, so it will not be beneficial to select any of the projects. Determination of selling prices is important in this case, since the model solution turns out to be sensitive to this parameter. Analysis result from the weight of importance of each indicator, it can be concluded that decision makers play an important role in influencing the optimization result and decisions about the mining project/location selection problem. In this case, determination of the NPV and ROI indicator weights must be done properly so that optimization can give the best results.
One of the main limitations in this study is the decision to choose a mining project/location depends only on the value of NPV and ROI on each project/location. For future research, the model in this study can be more developed by adding other economic indicators as well as noneconomic aspects such as price uncertainty, risk and difficulty level for each mining project/location.