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SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA
Preprint 12-142 OPTIMIZING THE NUMBER OF OPEN PIT MINE PUSH BACKS M. Rezaee, Rezaee, Univ. of Kentucky, Lexington, KY M. Osanloo, Osanloo, Amirkabir Univ. of Tech., Tehran, Iran R. Q. Honaker, Honaker, Univ. of Kentucky, Lexington, KY algorithms, the optimum number of open pit mine push backs is determined by analytical and MADM methods. The results are then evaluated in an iron open pit mine.
ABSTRACT Push backs used in the production planning of open pit mines play an integral role in generating annual cash flow. Although several algorithms for push back designs have been proposed, none have studied the optimum number of push backs. In this paper, different algorithms have been implemented for the push back design of a cross section of a copper deposit and the results discussed. Moreover, the optimum number of push backs has been determined using analytical and multiple attribute decision making (MADM) methods based on minimizing the risk associated with grade uncertainty, and minimizing stripping ratio, while maximizing net present value (NPV). According to the findings, the optimum number of push backs has been proposed to be between three and six, as lower quantities of push backs are more likely to achieve the calculated net present value, and have smoother stripping ratios for different pushbacks. This approach was examined in an operating mine and the optimum number of push backs determined to be four.
DISCUSSION OF SOME PUSH BACK DESIGN ALGORITHMS In this study, Ramazan-Dagdelen, Gershon, Whittle, and Gholamnejad-Osanloo algorithms are applied to push back design of a copper deposit cross section. The 2D block model configuration containing the average grade and standard deviation of each block is depicted in Figures 1 and 2, respectively. The UPL, determined by LG algorithm, is also shown in Figure 1.
INTRODUCTION Open pit mining production planning is conducted based on push back design. In other words, to reduce the planning problem size, defer waste stripping, and maximize the net present value (NPV), the ultimate pit limit (UPL) is broken into some smaller pits, called nested pits or incremental pits. To make the nested pits more practical and manageable, they are considered as mining constraints, such as required operating bench width, slope stability, ramp and hauling accesses. These practical pits are known as push backs. Since long term production planning is performed based on push back design, optimum push back design has a significant role in determining annual cash flow.
Figure 1. Cross-section view of the copper deposit showing showing the average block grade (%).
In order to maximize NPV, the best ore should be extracted in the first push backs. The definition of the best ore has evolved with time. Whittle used the definition of the best ore as the one with the highest grade in his algorithm [1, 2]. Ramazan and Dagdelen defined the best ore as “the material that has the highest grade and the least amount of waste stripping” , while Gholamnejad and Osanloo defined it as “Material that has high grade, a low amount of uncertainty, and needs a low waste stripping for extraction” [4,5].
Figure 2. Cross-section view of the copper deposit showing showing the 2 standard deviations (% ).
Several algorithms have been suggested for push back design . The methods applied in these algorithms are categorized as heuristic or simulation, linear programming, Lagrangian parameterization, or trial and error . Most of these algorithms have been discussed in Table 1 (see Appendix). With computerized methods for mine sequencing, push backs are designed based on trial and error, and the optimum number of open pit mine push backs has not yet been studied. Optimization can save time and result in more efficient sequencing, which leads to optimal production scheduling and higher annual cash flow.
Ramazan-Dagdelen algorithms concluded that the traditional push back design algorithms, like Whittle, does not always maximize the NPV of the mining projects, because they do not necessarily take into account the waste stripping required to extract the highest grade ore. Therefore, they developed their algorithm based on minimizing the stripping ratio criteria . This algorithm utilizes dynamic programming, which is difficult to apply. The result of this algorithm is illustrated in Figure 3, with the stripping ratios rounded to one decimal placed. The result shows that blocks with the same stripping ratio can be categorized in different nested pits, but the stripping ratios of the blocks in this cross section are descending. Therefore, the minimizing stripping ratio algorithm is not applicable to push back design of this cross section. For this mine, earlier exploitation of the upper blocks with higher stripping ratios is obligatory. It could be generally said that,
In this study, based on the latest definition of the best ore, the optimum number of push backs is “the one that has the minimum risk associated with grade uncertainty, the highest NPV, and the lowest cumulative stripping ratio of push backs (CSRP), while satisfying mining constraints such as minimum push back width, haul roads, and geotechnical parameters”. After evaluating some push back design
SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA minimizing stripping ratio algorithm are not applicable for reserves that are very deep with a large amount of overburden, because upper blocks have higher stripping ratio than lower blocks. Furthermore, this algorithm just considers the stripping ratio of blocks for exploitation sequence, and it does not take into account the block grades. Therefore, for some ore reserves like iron or even copper deposits which have been leached and have higher grade zones in lower levels, this algorithm cannot lead to achieving high NPV. In other words, this algorithm generally suggests that upper blocks with lower stripping ratio should be extracted earlier, but for these deposits extracting the lower blocks even with higher stripping ratio results in getting higher NPV. However, using this algorithm in conjunction with the algorithms, which consider the ore grades in push back design, results in obtaining a higher NPV for certain open pit mine projects .
blocks. This unnecessary waste stripping results in decreasing the NPV. The reason for this problem is that the positional grade of those waste blocks is higher just because they are located in upper levels and their downward cone contains more blocks. For instances, block 14 should be extracted before 12, and 13; the blocks 16, 17, and 21 can be extracted before block number 15; 30 before 27 and 28; 47 before 44, 45, and 46; 122,123,124,125, and 127 before 115 up to 121. Besides, earlier exploitation of some waste blocks with even lower positional grade could lead the earlier access of valuable ore blocks, if the slope stability allows the exploitation, e.g., block 19 could be extracted before 18. After determining the blocks sequences, push backs were designed for the copper cross section. For the push back design, the approach of the Wang and Sevim algorithm, which uses a different set of nested pits with a nearly constant size increment, was used. Different push backs lives years from 3 to seven were considered for the design. To compare the results for a final decision, the cumulative discounted push back risk indicator (DPRI), the cumulative stripping ratio of push backs (CSRP), and the NPV were determined. The values for the first push back were also calculated, because they are more critical than the values for other push backs. Since the blocks sequence has already been determined by the Gershon algorithm, the overall NPV of the designs is the same and is equal to $249281. In this paper, to determine the NPV, a 15% discounted ratio is considered in the calculation. It was also assumed that 5 blocks are extracted in each year. To determine blocks economic values (BEV), the copper price of $7,000/ton was considered. In addition, based on data obtained from the Sarcheshmeh copper mine in Iran, a mining cost of $13/ton, a processing cost of $600/ton, and an 80% recovery were applied in block economic value (BEV) calculation.
Figure 3. Stripping ratios of the copper Cross-section. The Gershon algorithm was also applied to the push back design. This heuristic algorithm is based on ranked positional weight, and first determines the block sequences based on the order of index of desirability, which is also called the positional weight or the positional grade. This index is the sum of the ore qualities within the cone generated from a block within the ultimate pit. Since the value of blocks decreases by depth, a discount factor should be applied to block values at each level [17, 18]. The positional grades and block exploitation sequences are shown in Figures 4, and 5, respectively. It should be noted that the discounted factor was assumed to equal 1.
In order to calculate the discounted push back risk indicator which is the index of risk associated with grade uncertainty, a risk indicator should first be defined. This indicator is set to estimate the variance of each block (?2). Blocks with a higher variance have a higher risk. The sum of the undiscounted uncertainties associated with the push backs is considered as “push back risk indicator” (PRI) and can be calculated as follows: nk
PRI k = ∑ i =1
2 σ i
+ ∑∑ cov(i, j )
i =1 j =1
Where PRI k is undiscounted risk indicator for k th push back and k = 1, 2,…, K. K is total number of push backs within the optimum pit limit, Cov (i,j ) is the covariance between block i and block j within the k th th push back, and nk is the total number of blocks within the k push back. Because the risk in the early years of the production period is more critical, a discount rate should be applied to each push back risk indicator. The ‘discounted push back risk indicator’ (DPRI) for a series of push backs can then be calculated as: Figure 4. Positional grades of the copper deposit cross section calculated by the Gershon method.
DPRI = ∑ k =1
PRI k t
(1 + ik ) k
where t k is the year at which the exploitation of the k th push back is finished, and ik is the discount rate for k th push back. By knowing the yearly discount rate and also each push back’s life, ik is easily determined [4, 5]. In calculations the values of covariance have been neglected. A summary of the values for the different designs is shown in Table 2. Mine designers could select one of these options based on the parameters’ values, or combine different push backs of different options and do more analysis. Multiple attribute decision making (MADM) methods could be also helpful in making the final decision.
Figure 5. Block exploitation sequences determined by the Gershon algorithm. As the results indicate, one of the disadvantages of this algorithm is that some waste blocks are extracted earlier than some valuable ore
SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA while considering 45° degree for slop stability. The NPV was $258,142 for 27 year life of this mine.
Table 2. Summary of the Values for the Designed Push Backs in Gershon Algorithm Number of 1st push back Life of Number blocks/nested each of nested pits (except nested Stripping DPRI NPV ($) pits last nested ratio pit (year) pit)
Figure 6. Nested pits designed by the Whittle algorithm. Table 3. Summary of the Values for the Nested Pits Designed by the Whittle Algorithm. Nested Ore No. of Blocks Pit Blocks
The block economic value (BEV) which is applied in different algorithms is defined in Equation 3: (3) where TOi is the total amount of ore in the block I , is the average estimated grade of i th block, R is the percent of total recovery by processing the ore; P is the price per unit of the final product, PC is the cost of processing the block as ore rather than treating it as waste; TRi is the total amount of rock (ore and waste) in the i th block, and MC is the cost of mining a ton of material (the material can be ore or waste. in case of waste, the BEV will become negative). The Whittle algorithm has been widely used in push back design. Whittle realized that mining and processing cost would probably be affected equally by inflation, but metal price is more subjected to the uncertainty, and therefore has a greater impact on the blocks’ values. To minimize the risk due to changes in economic conditions; and also minimizing the massive quantity of data processing by changing all the economic factors in calculating BEVs and determining nested pits, Whittle reduced the number of factors in the BEV, and reduced the number of economic variables by dividing equation 3 by MC ,
As Figure 6 indicates, some of them nested pits are too small and are not applicable for the push back design. Furthermore, the first nested pit in the Whittle algorithm is much larger than the others. This problem is called “gap problem”, which is the large difference between the numbers of blocks of two adjacent push backs. One of the sequencing goals is minimizing the scheduling problem size, and if there is a gap problem in push back design, the size problem still exists and affects the role of push backs in effective yearly production planning.
(4) Or, (5) In Equation 5, V i represents the value generated by i th block per unit cost of mining. There are only two factors in the Whittle equation. The first one is P/MC , or , which is the main variable. 1/ is called “Metal cost of mining” and is the amount of product which must be sold to pay for the mining a ton of waste. The second one is PC/MC , , or “cost ratio”, which changes only if new mining or processing methods are introduced.
Lastly, the push back design algorithm incorporated with grade uncertainty was used in this study [4,5]. Dimitrakopoulos categorized the uncertainties of mining projects in three groups: • •
For push back design, is changed step by step and LG algorithm is implemented on the new block economics values. Whenever ultimate pit limit changes, a new nested pit is generated. To design push backs of the copper cross section based on Whittle algorithm, has been changed from 539 (calculated at $7,000/ton) down to 30, for which no pit is generated. Fifteen nested pits were generated with? values of 539, 343, 214, 179, 107, 84, 67, 51, 45, 40, 39, 36, 34, 32, and 31. These nested pits are illustrated in Figure 6. Table 3 reflects the values calculated for the nested pits designed. These nested pits are considered for the mine constraint such as minimum push back width required, slop stability, and ramp for the final push back design.
Uncertainty of the ore body model and related in-situ grade variability and material type distribution. Uncertainty of technical mining specifications such as slope constraints, excavation capacities, etc. Uncertainty of economic issues including capital and operating costs, and commodity prices.
Gholamnejad and Osanloo  recognized that the optimal scenario for push backs is sensitive to the uncertainties concerned with the inputs to the optimization model. They realized that the uncertainty ofan ore body model and grade variability is the major issue for not achieving the anticipated production planning in the early stages of mining projects. Consequently, they developed the method of “grade parameterization with variance” to push back design. In the BEV equation, instead of average grade they used grade of blocks which can be obtained from average grade and its standard deviation. The block economic value in this algorithm is calculated as below:
To determine the NPV, production planning was performed on these nested pits based on the worst case production planning. In other words, exploiting each nested pit cannot be started unless the exploitation of the previous one has been finished. Also, blocks in each nested pit are extracted from the least to the highest economic value
SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA where n is a constant number, which its variation results in generating the different pits, and i standard deviation of i th block grade. For estimating the range of n when th e normal curve is used, n belongs to:
there are still some blocks that are worth being extracted and they generate a nested pit, since they have a high grade and small standard deviation. Although the NPV of the first push back of this algorithm is smaller, since it has fewer blocks, the DPRI(1.08%) of the first push back of this algorithm is much lower than that with the Whittle algorithm (11.75%). Consequently, the planning based on this push back design could be more reliable, because of less risk associated with grade uncertainty in the first years of operation after start up.
(7) In other cases where there is no information about the probability distribution of block grade, according to the Chebyshev’s theorem, n should be less than 4.5 since mean and variance values are available.
The sequence for extracting the blocks in this algorithm would occur in this order: high grade zones with low amount of uncertainty, high grade zones with medium amount of uncertainty, high grade zones with high amount of uncertainty, medium to low grade zones with low amount of uncertainty, and finally medium to low grade zones with a high amount of uncertainty. This ordering is related to the change of the value of n in the equation 6, and brings into higher achievable NPV [4,5].
The first step of this approach is calculating the block values using the lowest value of n (zero), and determining UPL by applying LG algorithm. Blocks outside of the UPL can be excluded from further calculation. Then n is further increased step by step and UPLs are determined. Nested pits are generated by comparing the UPLs for different n values. By increasing the n value, average grade of ore blocks decreases, and consequently the block economic values are reduced. The higher standard deviation of the blocks, the greater decreasing in block economic values. Therefore, smaller nested pits contain more valuable, and less uncertain, ore blocks .
ANALYTICAL METHOD After designing different numbers of nested pits by “grade parameterization with variance”, discounted push back risk indicator, cumulative stripping ratio and net present values have been determined for each number of nested pits, because the defined optimum number of push backs in this study is related to these values. The correlation of DPRI, NPV, and CSRP with the number of nested pits, are illustrated in Figures 8, 9, and 10, respectively. Since having the ultimate pit limit, or two nested pits are not usually suitable for push back design, analyses were performed for nested pits equal to or larger than 3.
By analyzing the “grade parameterization with variance” method, it has been observed in this study that this algorithm is flexible to generate different number of push backs if different Δ n values are applied. Different numbers of nested pits generated by this algorithm were subjected to the optimization of the number of open pit mine push backs. After determining UPL for the copper cross section in n=0, BEV values and their related UPLs for different n values (from 0 to 4.5 in increasing increments of 0.1 in each step) were determined. The 45 UPLs were compared by applying different Δ n values. To evaluate the effect of price inflation, optimization was performed for two different copper prices. In the first trial, $1,100/ton was used,with a $5/ton mining cost, and a $20/ton for processing cost . A price of $7,000/ton was then applied, while keeping mining cost, processing costs, and other variables the same as the data obtained from the Sarcheshme Copper mine in Iran. For the $7,000/tonprice, different numbers of nested pits from 1 to 23 (1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, and 23 nested pits) were generated. Some ?n values resulted in the same number of nested pits, while sizes of the nested pits were different. For example, all the values in range of 2 to 3.9 for ? n generate 3 nested pits. Among these scenarios, one can select the final option regarding desirable NPV, risk, and stripping ratio. For example, one can choose the nested pits with the least amount of DPRI for further analyses and optimization. One of the nested pit series designed by this algorithm is shown in Figure 7. 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 17, and 24 nested pits were generated at the copper price of $1,100/ton.
Figure 8. Correlation of DPRI with the number of nested pits.
Figure 9. Correlation of NPV with the number of nested pits. The results show that there is no significant change in the NPV after ten nested pits, and that DPRI increases, but cumulative stripping ratio increases more. Therefore, as an initial estimation, it could be stated that the maximum number of push backs is ten and there is no benefit in designing more than ten nested pits. Since ten nested pits did not exist with a copper price of $7,000/ton, eleven nested pits can be considered as the maximum number. Further analyses were performed by calculating the relative percent change in the NPV, DPRI, and CSRP as compared with the values at ten or eleven nested pits. In both of the price ranges, it was observed that by decreasing nested pits from 10 to 6, the NPV decreases by around 5%, while DPRI and CSRP decreased by approximately 12% and more than
Figure 7. Five push backs designed for Δ n =1.3 by applying grade parameterization method at price of $1,100/ton. As figure 7 shows, the first nested pit of this method is much smaller than that with the Whittle algorithm, and the gap problem has been decreased. The reason is that in this algorithm the values of the blocks do not decrease at the same rate, unlike the Whittle algorithm. By increasing the value of n, the economic values of the blocks with smaller standard deviation decrease less than the values of the blocks with the bigger standard deviation. As a result, by decreasing the BEVs
SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA 30%, respectively. Hence, for the maximum number of push backs, it can be said that the maximum number of nested pits for this open pit mine is six, and now the nested pits, in the range of 3 to 6, can be applied for push back design. These analyses are independent of the mine size or tonnage, and significant different price differences have the same results. In conclusion, the optimum number of open pit mine push backs is suggested to be in the range of 3 to 6. Among these four options, a designer can select the final option with regards to risk desirability, cumulative stripping ratio of push backs, and desirable net present value. The final option is considered for mine constraints to design the push backs. The stripping ratio graphs for some of the nested pit designs are presented in Figures 11 to 13. These figures show that decreasing the number of push backs leads to smoother stripping ratios for the push backs, allowing for better production planning. Decreasing the number of push backs also increases the push back sizes, and larger equipment could be used for exploitation, resulting in higher annual production.
TOPSIS METHOD Multiple attribute decision making methods were applied for ranking of the nested pits. Hwang and Yoon  developed the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method based on the concept that the chosen alternatives should have the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution. Descriptions of the fundamentals of TOPSIS method and related applications can be readily found in . For implementation of TOPSIS, three main attributes (DPRI, NPV, and CSRP) and also possible alternatives (different number of nested pits) are used. Each appraisal attribute cannot be assumed to have equal importance because each appraisal attribute has various meanings. There are many methods that can be employed to determine weights, such as the eigenvector method, weighted least square method, entropy method, AHP, as well as linear programming techniques for multidimensional analysis preference (LINMAP). The method that is chosen depends on the nature of the problem. In this study, the Entropy method was chosen for the attribute weighting. Subsequently, the TOPSIS method is implemented for ranking alternatives. The positive-ideal solution must have the least amount of DPRI, the least amount of CSRP, and the highest amount of NPV, and vice versa for negative-ideal solution. The ranking of the different numbers of nested pits, based on three mentioned attributes,is shown in Table 4. Based on alternatives ranking, three nested pits is the best number and 24 is the worst number. Also, Alternatives 10, 9, and 8 (4, 5, and 6 nested pits, respectively) are the second, third, and fourth ranking in Table 4. Also, the large jump between the ranking of 6 nested pits and next one, and also the jump between the 10 or 11 and the next ones, verify the steps of the analytical method.
Figure 10. Correlation of cumulative stripping ratio with the number of nested pits.
Table 4. Ranking of cross-section view different numbers of nested pits by TOPSIS method. Results at the price of $1100/ton
Figure 11. Stripping ratios for the 23 nested pit design.
A2(17 NP) A3(14 NP)
Results at the price of $7000/ton
A8(6 NP) A9(5 NP)
1. NP: Nested pits, 2. Di , and Di : Distance of each alternative from positive and negative ideal solution, 3. Ci: the relative closeness to the ideal solution
VALIDATION This approach has been tested for Chadormaloo iron open pit mine in Iran. After determining the UPL with LG algorithm, different numbers of push backs have been designed for this mine. The blocks dimensions 25 m x 25 m x 15m, and the iron price, mining cost, and processing cost are $497/ton, $3/ton, and $90/ton, respectively. The cutoff grade for this mine is 20% with an average ore output rate of 1.4 x 107 ton/year. A discounted rate of 15% was used for NPV calculation. The correlation between NPV and CSRP with different number of push backs are shown in Figure 14.
Figure 12. Stripping ratios for the 7 nested pit design.
Figure 14 shows that the cumulative stripping ratio increases with the number of push backs. The NPV increases steeply up to 3 push backs, while the NPV from 3 to 5 push backs increases moderately. Beyond 5 push backs, the NPV remains flat. Therefore, 3, 4, and 5
Figure 13. Stripping ratios for the 23 nested pit design.
SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA push backs are considered for more analyses. The graph of the cumulative NPV for 3, 4, and 5 push backs is shown in Figure 15.
Figure 16. Plan view of 4 push backs. Figure 14. The correlation between NPV, and CSRP with different numbers of push backs
Figure 17. Cross section view of four push backs. CONCLUSION Usually, the ultimate pit limit of an open pit mine is divided into some nested pits. Considering the minimum push back width, access to pit, and other constraints, these nested pits can be converted to push backs for use as a guide during the scheduling process. In computerized open pit production planning, push backs are designed by trial and error. This paper presents a study on optimizing thenumber of push backs. First, some of the push back design algorithms were applied to a 2-D configuration for a copper mine. The results showed that the push backs algorithm based on minimizing the stripping ratio is not applicable for all mine reserves, especially for those like copper and iron that have been leached. In other words, the higher grades zones, which cause a higher NPV, are located in lower levels with these types of deposits, and their exploitation requires a large amount of the waste stripping. The Gershon algorithm could cause unnecessary waste stripping, while some valuable ore blocks could be extracted at the same time. Moreover, designing push backs by this algorithm needs a trial and error method and a high level of design experience. The Whittle algorithm has a gap problem. Moreover, the block economic values in this algorithm decrease at the same rate to a point where there is no pit to extract, while there are still a set of valuable blocks that can be extracted. This is because this algorithm does not take into account the uncertainty of the ore body model and its related in situ grade variability, and the material type distribution. The method of “parameterization of grade with variance” also has been applied for push back design. This algorithm minimizes the gap problem, and extracts high grade zones with low amount of uncertainty in the earlier production periods. Consequently, it minimizes the risk associated with grade uncertainty and is also flexible for generating different numbers of push backs. Different numbers of push backs generated by this algorithm were subjected to optimizing the number of open pit mine push backs. The optimum number of push backs in this study was defined as the one with the lowest risk associated with grade uncertainty, the lowest amount of
Figure 15. Cumulative NPV Vs. push backs for 3, 4, and 5 push backs. Figure 15 shows that 4 push backs, as compared with 3 push backs, have a higher total NPV, and also higher NPV of the first push back, which is very critical in mining projects. Furthermore, the NPV of the first three push backs in the 4 push back design is much higher than those of the 5 push back design. Table 5 reflects the values for the 4 push back design and shows that the lives of the push backs are close together with no gap problem. Therefore, 4 push backs would be the best option for production planning. These four push backs have a total NPV of $14,708,007,549, with an overall stripping ratio of 0.83. The cross section and plan views of the 4 push backs areshown in Figures 16, and 17, respectively. Table 5. Values determined for four push backs in Chadormaloo iron open pit mine. Number of Push backs
SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA cumulative stripping ratio of push backs, and the highest net present value. It also considers mining constraints, such as minimum push back width, haul roads, and geotechnical parameters. The analytical and TOPSIS method showed that the optimum number of open pit mine push backs is in the range of 3 up to 6. Lower numbers of push backs have a less discounted push back risk indicator and higher probability of achieving the calculated net present value. Also the less number of push backs has smoother push backs’ stripping ratios, which is more effective for production planning. A lower number of push backs causes an increase in push back dimensions which leads to using larger equipment for exploitation and thus higher annual production in open pit mines.
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SME Annual Meeting Feb. 19 - 22, 2012, Seattle, WA APPENDIX Table 1. Push Back Design Algorithms. Author (publish Method year) Learchs Grossmann (LG) (1965)  Vallet (1976) 
Some Advantages and/or Disadvantages Adv.: Generates a more optimum ultimate pit limit. Disadv.: Repetitive and time consuming, does not consider uncertainties in mine projects , has gap problem, and only considers economic parameters, especially exploitation cost. Adv.: Generates a set of nested pits in one run. Disadv.: the same as the LG algorithm, except this algorithm is not as time consuming. Adv.: Provides a nested sequence of optimal pits for mine scheduling , is mathematically approved, Disadv.: has gap problem, ore mining and processing costs vary with the project size and production rate and cannot be determined precisely without knowing the ultimate pit limit (UPL), and does not consider uncertainties. Adv.: The exploitation of higher grade zones in i nitial years, sensitivity analysis of UPL with regard to metal price, and only one variable required to solve the problem. Disadv.: Not applicable for multiple metal ores, has gap problem, does not consider the risk of grade variation, and the assuming of the same processing and exploitation rate of change is questionable. Adv.: Gives an idea for block sequence exploitation in short and long term. Disadv.: Strips some unnecessary waste blocks while there is access to ore blocks, is difficult to apply to push back design, and does not consider uncertainties. Adv.: Solves the gap problem in mine sequencing. Disadv.: does not assure the opti mum UPL and maximum NPV, and does not consider uncertainties.
Economical parameterization (Maximization of “ P = aQ – bT –cV ” with different values of parameters to 1 obtain a set of nested pits) . Modification of LG 3D Graph Tree algorithm
Reserve technical parameterization (Searches for Matheron the set of nested pits maximizing t he (1962, 1973, 1975) parameterization function of “Q- θ T- λV ” for each [12,13,14,15, 16] 1 value of θ and λ) .
Whittle (1988) [1,2]
Economical parameterization (Finds a set of nested pits Maximizing “P = aQ – bT – cV ” while a/c ratio 1 varies) 
Gershon (1989) 
Heuristic algorithm (Based on positional ranked weight and considering the quality of ore, the position of ore blocks, and the quality of the blocks under the desired ore block) [17,18]
Wang – Sevim (1995) 
Heuristic algorithm (Obtains a set of nested pits and simultaneously determines UPL, production rate, and mine life)
Ramazan – Dagdelen (1998) 
Adv.: Causes a higher NPV as compared with other methods like Whittle, especially at high interest rates. Disadv.: Difficult to apply, is not applicable for multiple metal ores, just considers blocks as ore or waste, does not take into account blocks grades, and does not consider uncertainties. Adv.: Minimizes the risk associated with grade uncertainty, minimizes the gap problem, and is applicable for mult iple metal ores. Disadv.: Does not consider technical and economic uncertainties.
Dynamic algorithm (based on the graph theory and minimizing the push backs’ stripping ratio)
Gholamnejad Grade parameterization with variance (incorporates Osanloo the grade uncertainty with push back design, and (2007) [4,5] applies LG algorithm) Notes: 1 - In the parameterization equations, a, b, and c parameters are commodity price, processing cost, and exploitation cost, respectively. V, Q and T are the amount of material contained in the project, the amount of ore, and the metal content, respectively . θ and λ have the same dimension as cut-off grade . 2 - Matheron algorithms have been extended or discussed continuously by different researchers such as Francois-Bongarcon, and Guibal [20, 21, 22], Dagdelen and Francois-Bongarcon . 3 - A disadvantage of most of the push back design algorithms is that they do not consider average grade for push backs, which may not achieve the yearly mine scheduling targets. Also, most of the mentioned algorithms do not consider the minimum required run-of-mine material for processing plants. Therefore, they may fail to reach an optimum solution. 4 - The trial-and-error approach in push back design, which is applied in computerized sequencing, has been well discussed by several authors: Couzens , Mathieson , Ilies and Perry , Whittle [1,2], Kim and Zhao, and Tolwinski [27,28].