Application of KKT in determining the final destination of mined material in multi-processing mines

Highlights

• The highest amount of NPV and cash flows is obtained when the cut-off grades of the bio-heap leaching, vat leaching and concentration methods are respectively 0.14%, 0.17% and 0.19%.

• This study reveals that as the copper price changes from 4300$ to 8300$, the amount of NPV gets increased from 36 million dollars to 125 million dollars.

• The cash flows of novel algorithm increase the cash flows by 2% rather than using two processing methods algorithm.

• As the process recovery of the bio-heap leaching method increases up to 95%, the amount of the cut-off grades of bio-heap leaching, vat leaching and concentration are decreased up to 0.04%, 0.05% and 0.07%.

• The increase of investment costs makes the optimum cut-off grades of bio- heap leaching method differ much more than other cut-off grades of processing methods.

Abstract

Determining the final destination of mined materials is counted as one of the most important challenges in mining. Cut-off grade is a criterion specifying the destination of mined materials relying on the commodity price, treatment method recovery, investment and operating costs.

The development of mineral processing science makes these methods apply simultaneously in mines, thus, it is needed to specify minerals destination. Regarding environmental and technical considerations and operation costs, the optimum cut–off grades of processing methods are estimated. Minerals destination might be changed due to the differences of investment costs of processing methods. Optimization has been done through the Karush–Kuhn–Tucker (KKT) theorem in this study. Thus, we tried to innovate a novel algorithm in copper mine. Results indicated that applying KKT theorem can increase the mining NPV by 5% rather than traditional approach of determining cut-off grades. It is also demonstrated that processing recovery and operating and investing costs affect cut-off grades.

Introduction

Different processing methods are developing to achieve higher and cheaper productions. These methods are totally divided into pyrometallurgical and hydrometallurgical ones. Hydrometallurgical methods are exceedingly applied in base metal production for the sake of low investment and operating costs (Dreisinger, 2006) and less environmental pollution (Uqaili and Harijan, 2012). However, they are not practiced a lot in copper production because of their less production in comparison to pyrometallurgical methods (Schlesinger et al., 2011). Their chemical properties and grades are very significant in both hydrometallurgical and pyrometallurgical methods (Gupta, 2006). In fact, these methods mark an optimum application in specific grade ranges. Holistically, technical grade range should be scrutinized to specify the cut-off grade of processing methods as well as economical parameters.

The model developed by Lane describes the theory of cut-off grades optimization as well as its effects on the mine planning in an open pit mine (Lane, 1964, Lane, 1984, Lane, 1988). In 1998, the presented model has been executed in the optimized mine planning software (Whittle and Vassiliev, 1998) and after that several development of the novel Lane's model are argued (Dagdelen, 1992, Dagdelen, 1993, Wooler, 2001, Osanloo and Ataei, 2003, Nieto and Bascetin, 2006, Nieto and Zhang, 2013, Osanloo et al., 2008, He et al., 2009, Gholamnejad, 2009, Dimitrakopoulos, 2011, Abdollahisharif et al., 2012, Khodayari and Jafarnejad, 2012, Hustrulid et al., 2013, Asad and Dimitrakopoulos, 2013; Thompson and Barr, 2014, Rahimi et al., 2015). These researchers try to compensate the defects of the algorithm.

The effect of environmental and economical parameters on cut-off grades has been analyzed in various modeling (Ataei and Osanloo, 2004, Asad, 2005, Rashidinejad et al., 2008, Gholamnejad, 2008, King, 2009, Ganguli et al., 2011, Moosavi et al., 2014, Narri and Osanloo, 2015, Rahimi and Ghasemzadeh, 2015). Inflation rate is identified as one of the most influential economical parameters. Regarding this rate in cut-off grade equations, determine the market fluctuations (Asad, 2007, Azimi and Osanloo, 2011, Azimi et al., 2012, Azimi et al., 2013). Furthermore, studies indicate that environmental considerations in cut-off grades algorithm increase the total NPV of mines (Rahimi et al., 2014). Remarking environmental costs reduces reclamation costs and it is also influential on mines optimum designing (Akbari and Rahimi, 2016). Lots of numerical and mathematical optimizations have been done in cut-off grades calculations. In fact, dynamic cut-off grades of underground mines (Johnson et al., 2011), stockpiles grade (Asad and Topal, 2011) and multi metal mines (Ataei and Osanloo, 2003a, Ataei and Osanloo, 2003b) are calculated. The net present value is the optimization criterion in most studies although the output rate and IRR can be identified as the objective function. Some studies have been done to complete presented models by Lane, Davy and Whittle. In most modeling, the effect of the difference between processing methods and their constraints haven’t been evaluated.

Each one of the algorithm determining optimum cut-off grades hold some failures according to recovery and different environmental costs of treatment methods. In addition, the algorithm specifying deposits destination should consider the investment and operating costs due to significant differences of related costs of hydrometallurgical and pyrometallurgical methods. We defined a comprehensive algorithm of cut-off grade in order to solve these problems. These algorithms scrutinize various processing methods and their constraints as well as effective parameters.

The KKT theorem is applicable. Karush–Kuhn–Tucker (KKT) Theorem is the most central theorem in constrained optimization and In mathematical optimization, it is first order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.

The system of equations corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations. The KKT conditions were originally named after Harold W. Kuhn, and Albert W. Tucker, who first published the conditions in 1951 (Kuhn and Tucker, 1951). Later scholars discovered that the necessary conditions for this problem had been stated by William Karush (Karush, 1939) in his master's thesis in 1939. In this theorem, Suppose that the objective function $f:R^n\stackrel{}{\to }R$ and the constraint functions $g_i:R^n\stackrel{}{\to }R$ and $h_i:R^n\stackrel{}{\to }R$ are continuously differentiable at a point $x^{*}$. If $x^{*}$ is a local optimum and the optimization problem satisfies some regularity conditions, then there exist constants ${\mu }_i\left(i=1,\dots ,m\right)$ and ${\gamma }_i\left(j=1,\dots ,m\right)$ called KKT multipliers, such that (Relations (1), (2), (3), (4), (5))

$$\mathrm{For}\mathrm{Maximaization}f\left(x\right)=\nabla f\left(x^{*}\right)=\sum _{i=1}^m{\mu }_i\nabla g_i\left(x^{*}\right)+\sum _{j=1}^l{\mathit{\gamma }}_j\nabla h_j\left(x^{*}\right)$$

$$\mathrm{For}\mathrm{Minimazation}f\left(x\right)=-\nabla f\left(x^{*}\right)=\sum _{i=1}^m{\mu }_i\nabla g_i\left(x^{*}\right)+\sum _{j=1}^l{\mathit{\gamma }}_j\nabla h_j\left(x^{*}\right)$$

$$\mathrm{Primal}\mathrm{feasability}\{\begin{array}{c}{g}_i\left(x^{*}\right)\le 0,\mathrm{for}\mathrm{all}i=1,\dots ,m\\ h_j\left(x^{*}\right)=0,\mathrm{for}\mathrm{all}j=1,\dots ,m\end{array}$$

$$\mathrm{Dual}\mathrm{feasability}{\mu }_i\ge 0,\mathrm{for}\mathrm{all}i=1,\dots ,m$$

$$\mathrm{Complementary}\mathrm{slackness}{\mu }_i{g}_i\left(x^{*}\right)=0,\mathrm{for}\mathrm{all}i=1,\dots ,m$$

The schematic of the KKT theorem application is shown in Fig. 1.

Optimization is done to maximize NPV and it uses KKT theorem and an iteration algorithm. The optimum cut-off grades of processing methods are the first assumption of KKT theorem and they are calculated by programming. Holistically, this algorithm makes the highest rate of NPV. In addition, the effect of the commodity price, operating, investment costs and recovery of processing methods on NPV is assessed.