Simultaneous stochastic optimisation of mining complexes: Integrating progressive reclamation and waste management with contextual bandits

ABSTRACT

Managing environmental performance of waste dump facilities in mining complexes is an integral part of long-term production planning. Sustainable long-term production scheduling solutions are desired to mitigate risk and return the environment to a productive post-mining state. A simultaneous stochastic optimisation framework for long-term production scheduling in mining complexes is developed that integrates waste management and progressive reclamation. The waste dump placement schedule is jointly optimised with the extraction sequence, destination policy, and stockpiling decisions in a single stochastic mathematical programming framework. This includes the timing of progressive reclamation activities in parallel with production to enhance waste dump rehabilitation. Uncertainty related to the production of acid rock drainage is quantified by simulating geochemical properties of waste and managing the blending of uncertain waste properties within the optimisation framework. With respect to the framework for simultaneous stochastic optimisation, contextual bandits are explored to improve the metaheuristic solution approach and solve the corresponding large-scale optimisation model. The framework is tested in a multi-mine copper-gold mining complex leading to improved environmental performance. Risk of acid rock drainage is decreased by 52.5% in the waste dump facilities. Reclamation planning activities for meeting environmental requirements are scheduled prior to closure. The solution approach more effectively improves the objective function with contextual bandits leading to a 24% improvement in the study presented.

KEYWORDS:

• Waste management
• reclamation
• reinforcement learning
• stochastic programming
• contextual bandits
• long-term mine planning

1. Introduction

Waste management and reclamation are essential operational aspects to be considered when planning and optimising the long-term production schedule of a mining complex [1]. An integrative planning approach is desired to satisfy the diverse set of environmental constraints associated with operating a mining complex, while providing an economically viable long-term production schedule. For successful reclamation and rehabilitation, a progressive approach is taken to accelerate the reclamation of mining waste dump facilities, eliminate liabilities and improve environmental performance [2,3]. Furthermore, strategic waste management practices that mitigate acid rock drainage (ARD) and prevent metal leaching are considered to meet water-quality standards and reduce remediation costs [4]. Environmental management is vital for mitigating long-term complications that develop during and after production ceases [5]. Directly considering these components when optimising long-term production schedules reduces environmental risk and is expected to enhance the financial and environmental performance of a mining complex over its lifetime.

Depending on the processes required to recover valuable products that are sold to customers and the market, mining complexes may contain several mines, stockpiles, preconcentration, processing, and waste dump facilities [6]. Research related to the simultaneous stochastic optimisation of mining complexes aims to globally optimise all critical operational aspects in a single stochastic mathematical programming formulation [7–11]. This includes optimising the extraction sequence, destination policy, capital investments, stockpiling, ore blending, processing, transportation and operating alternative decisions, while managing uncertainty related to quantity and quantity of the material in the ground [7–10,12–18]. Decisions are optimised to maximise net present value, by enhancing the value of the products sold, and minimising technical risk related to the uncertain material supply. However, the management, treatment and reclamation of mining waste dump facilities has received limited attention to date within these frameworks.

Waste management and rehabilitation scheduling decisions are typically considered following the optimisation of the long-term production schedule [19–22], thus, advantageous synergies between the extraction sequence, destination policy and waste production schedule may not be found. Fu et al. [23] formulate a mixed integer programming model that jointly optimises an open pit mine production schedule and waste dump placement schedule. Additionally, the encapsulation of potentially acid generating (PAG) waste is included to mitigate ARD, an important waste management consideration. There are several limitations to this integrative approach that are addressed in this work. First, the method ignores material uncertainty and variability of the material properties and uses a deterministic (average type) model of the material attributes. Second, PAG and non-acid generating (NAG) waste materials are classified based on the geochemical properties within a selective mining unit (a mining block), which ignores the potential to blend waste materials of different qualities to mitigate ARD. Lastly, the progressive reclamation of waste dump facilities is not considered within the optimisation formulation. Lin et al. [24] develop a model for optimising waste dump schedules simultaneously with production. Levinson and Dimitrakopoulos [25] optimise the long-term production schedule with waste management considerations in an open-pit gold mining complex under uncertainty to manage the production of PAG waste material. However, this approach does not provide an executable waste dump production schedule for material placement. Ben-Awuah and Askari-Nasab [26] consider scheduling mining cuts and waste dumps jointly; however, this method ignores material uncertainty, aggregates mining blocks to reduce the number of decisions variables and does not produce a waste dump production schedule.

ARD and associated metal leaching have large impacts on water resources and are a serious concern for mining complexes producing wastes with sulphide bearing minerals [27]. Following exposure to oxidising conditions, sulphide waste materials can produce sulphuric acid. Sulphuric acid can be neutralised by contacting certain minerals in the waste material such as carbonates. A recent field evaluation that monitors the geochemical blending of waste materials demonstrates an opportunity for permanent prevention of ARD under certain conditions [28]. More common practices for mitigating ARD aim to eliminate or reduce the rate of oxidation with water covers, treatment methods that add chemicals to acidic water, and various other techniques [29,30]. These conventional approaches can cause long-term risk. Treatment-based approaches extend long-term recurring costs that effect the economics of a mining complex. Water covers and constructed impoundments are high risk due to the potential of failures and the extended duration which the materials must remain covered [4,31–33]. Instead, geochemical blending techniques mix waste with different geochemical characteristics to overcome these limitations by preventing the formation of ARD at the source – the waste dump facilities – without the need for water covers. Blending effectively manages the net producing ratio (NPR) or the ratio of neutralisation potential (NP) to acid potential (AP) of the mixed waste product at the final destination [34,35].

Integrating geochemical blending of waste materials in the optimisation of mining complexes increases the complexity of placing waste material. Geochemical properties of waste are required to blend materials and manage the risk of ARD. Therefore, geostatistical simulation techniques are applied to quantify the quality and quantity of the relevant attributes within the mineral deposits in a mining complex [36,37]. After extraction, the geochemical attributes are tracked across the mining complex to blend waste materials of different qualities and generate a waste product that performs as if it is non-acid generating. A waste dump production schedule or sequence of material placement is determined by optimising the mining complex to generate safe products for placement in waste dump facilities, which makes these decisions manageable in large-scale industry applications. Simultaneously optimising the long-term production schedule with the waste dump placement schedule is critical for these reasons. This work develops a long-term stochastic optimisation formulation that manages waste material blending to prevent ARD. Waste materials are mixed within waste dump cells to create a safe waste product by managing the NPR ratio. Considering the blending of waste materials to mitigate ARD can directly change the response of other production scheduling decisions as the schedule adapts to find waste with the correct geochemical properties, while still aiming to maximise net present value.

Other critical environmental management considerations related to planning and scheduling of an operating mining complex are reclamation and closure requirements outlined by the local government and stakeholders of a region [38]. Waste dump facilities cover large areas of a mining complex and must be returned to a natural and productive state to minimise negative environmental effects. This includes re-establishing tree cover, native species and ensuring landforms are returned to suitable environmental conditions for the region [39–41]. Progressive reclamation provides opportunities to monitor the results of reclamation during production and reduces environmental risk as the requirements for closure are meet [38]. Additionally, progressive reclamation allows for the direct placement of topsoil/overburden material in reclamation areas. Direct placement improves soil development at waste dump facilities leading to more productive reclamation of native species [42–44]. Integrating these critical planning components into the optimisation of the long-term production schedule provides opportunities to eliminate environmental risk by reclaiming segments of the waste dump facilities during production. This is crucial to consider in the simultaneous stochastic optimisation of mining complexes.

This manuscript presents an innovative stochastic mathematical programming formulation and optimisation framework that directly incorporates waste management and progressive reclamation considerations. Unlike previous stochastic programming formulations, the model integrates waste management through the blending and scheduling of waste materials to mitigate ARD. In addition, the production schedule for placing topsoil/overburden material is included to accelerate progressive reclamation at the waste dump facilities to minimise long-term liabilities and satisfy regulatory requirements. Directly considering these aspects within the simultaneous stochastic optimisation framework improves the overarching long-term plan but also increases the size of the stochastic mathematical program.

Intelligent solvers are required to optimise large-scale optimisation models. Contextual bandits are explored in this work for selecting heuristics within a metaheuristic solution approach to optimise the proposed stochastic programming formulation. Models that consider the flow of materials after extraction lead to difficult to solve non-linear optimisation models. This often arises due to the non-linearities in the objectives and constraints caused by blending and stockpiling materials. Bley et al. [45] explain several major challenges caused by non-linear formulations in relation to mine production scheduling with stockpiles. Zhang and Dimitrakopoulos [46] address non-linearities in recovery processes and forward contracts. Metaheuristics are advantageous for optimising large-scale stochastic programming models in reasonable timeframes while handling non-linear constraints and have been widely applied in long-term production scheduling [7–10,12,18,47–54]. Goodfellow and Dimitrakopoulos [9] adapt the acceptance criterion with a cumulative distributive function for each decision neighbourhood to improve the simulation annealing optimisation framework. Instead of modifying the acceptance criterion, a different optimisation strategy is considered in this work that utilises learnings from a contextual bandit algorithm [55]. The algorithm selects an action that it expects will lead to larger improvements in the objective function of a stochastic mathematical program, while exploring less visited actions with an $\mathit{\epsilon }$-greedy policy. During optimisation, the learner or agent adapts the actions selected given the context of the mining complex environment. Therefore, as the agent learns, the actions selected during the optimisation are targeting changes to the production schedule that are expected to provide the largest improvements to the long-term production schedule.

In the next sections, the stochastic programming formulation is introduced followed by the optimisation framework that utilises contextual bandits. Subsequently, the proposed formulation and solution approach are tested with a case study in a large-scale mining complex. Lastly, conclusions and future work follow.

2. Simultaneous stochastic optimisation of mining complexes with progressive reclamation and waste management

2.1. Stochastic mathematical programming optimization formulation

A two-stage non-linear stochastic programming model is outlined in this section [56]. The proposed formulation provides a simultaneous stochastic optimisation model for jointly optimising the extraction sequence, destination policy, stockpiling, waste dump production schedule, waste blending and progressive reclamation. The waste dump production schedule considers the placement of rock and overburden/topsoil material, while blending waste of different qualities to mitigate the risk of ARD. The objective function of the stochastic mathematical program aims to maximise net present value and minimise deviations from production targets. Figure 1 provides examples of the waste dump considerations in the proposed model. A set of equiprobable stochastic realisations of the key material attributes and the geochemical characteristics of interest are used to account for uncertainty and local variability of ore and waste materials [37]. The stochastic mathematical programming formulation is described next.

Figure 1. Blending strategy to manage NPR at waste dump facilities and precedence requirements in a simplified 2-D example of a waste dump model.

2.1.1. Notations and definitions

This section defines the notations and their definitions for the stochastic mathematical programming model. Tables 1 and 2 include the sets and parameters. Table 3 includes the decision variables.

Table 1. Sets.

Notations	Definitions
$\mathit{T}$	Set of long-term production periods to be scheduled, indexed by $\mathit{t}$.
$\mathit{S}$	Set of stochastically simulated block models of the pertinent material attributes, indexed by $\mathit{s}.$
$\mathit{A}$	Set of simulated material attributes in each block in the simulated block models, indexed by $\mathit{a}$.
$\mathcal{M}$	Set of mines.
$\mathcal{S}$	Set of stockpiles.
$\mathcal{P}$	Set of processing facilities.
$\mathcal{W}$	Set of waste dump facilities.
$\mathcal{D}$	Set of all mining complex destinations $\mathcal{D}=\mathcal{S}\cup \mathcal{P}\cup \mathcal{W}$.
$C_w^{\mathrm{O}\mathrm{B}}$	Set of waste dump cell locations at waste dump $\mathit{w}\in \mathcal{W}$ that accept topsoil/overburden material.
$C_w^R$	Set of waste dump cell locations at waste dump $\mathit{w}\in \mathcal{W}$ that accept rock material.
$C_w$	Set of all waste dump cell locations at waste dump $\mathit{w}\in \mathcal{W}$ where $C_w=C_w^{\text{OB}}{\cup }_{w^R}^{\mathit{C}}$ indexed by $\mathit{c}$.
$G^{\mathrm{OB}}$	Set of material groups that are topsoil/overburden.
$G^R$	Set of material groups that are rock.
$\mathit{G}$	Set of all material groups such that $\mathcal{G}={\mathcal{G}}^{\mathrm{OB}}\cup {\mathcal{G}}^R$, indexed by $\mathit{g}$.
$\mathcal{I}\left(\mathit{i}\right)$	Set of locations that accept material from location $\mathit{i}\in \mathcal{M}\cup \mathcal{D}$.
$\mathcal{O}\left(\mathit{i}\right)$	Set of locations that receive material from location $\mathit{i}\in \mathcal{M}\cup \mathcal{D}$.
$B_m$	Set of blocks in the block model for each mine $\mathit{m}\in \mathcal{M}$, indexed by $\mathit{b}$.
$\mathbb{O}\mathit{(}\mathit{b}\mathit{)}$	Set of overlying blocks for all blocks $\mathit{b}\in B_m$.
$\mathrm{P}{\mathrm{D}}_{\mathit{w},\mathit{c}}$	Set of predecessor cells to cell $\mathit{c}\in C_w$ that must be filled prior to placing material at cell $\mathit{c}$.

Table 2. Parameters.

Notations	Definitions
$w_{\mathit{b},\mathit{s}}$	Mass of block $\mathit{b}\in B_m$ in scenario $\mathit{s}\in \mathit{S}$.
$g_{\mathit{a},\mathit{b},\mathit{s}}$	Grade or quality of attribute $\mathit{a}\in \mathit{A}$ in block $\mathit{b}\in B_m$ and scenario $\mathit{s}\in \mathit{S}$.
$\mathrm{s}{\mathrm{f}}_{\mathit{b},\mathit{s}}$	Swell factor of block $\mathit{b}\in B_m$ in scenario $\mathit{s}\in \mathit{S}$ which represents the adjustment factor for converting bank volume to the loose material volume.
$\mathrm{s}{\mathrm{v}}_{\mathit{b},\mathit{s}}$	Specific volume (inverse of density) of the material in block $\mathit{b}\in B_m$ in scenario $\mathit{s}\in \mathit{S}$.
${\theta }_{\mathit{b},\mathit{g},\mathit{s}}$	Block membership parameters set to one if block $\mathit{b}\in B_m$ is in group $\mathit{g}\in \mathit{G}$ in scenario $\mathit{s}\in \mathit{S}$, zero otherwise.
$\mathit{d}$	Economic discount rate.
$\mathrm{rd}$	Risk discount rate for deferring risk to later production periods.
$p_{\mathit{a},\mathit{i},\mathit{t}}$	Selling price of a product with attribute $\mathit{a}\in \mathit{A}$ at location $\mathit{i}\in \mathcal{P}$ in period $\mathit{t}\in \mathit{T},p_{\mathit{a},\mathit{i},\mathit{t}}=p_{\mathit{a},\mathit{i},1}/{\left(1+\mathit{d}\right)}^t$.
$r_{\mathit{a},\mathit{i},\mathit{t}}$	Recovery of product produced with attribute $\mathit{a}\in \mathit{A}$ at location $\mathit{i}\in \mathcal{D}$ in period $\mathit{t}\in \mathit{T}$.
$c_{\mathit{i},\mathit{t}}$	Unit cost to extract, handle, or process material at location $\mathit{i}\in \mathcal{M}\cup \mathcal{D},c_{\mathit{i},\mathit{t}}=c_{\mathit{i},1}/{\left(1+\mathit{d}\right)}^t$.
$\mathrm{r}{\mathrm{c}}_{\mathit{w},\mathit{t}}$	Reclamation cost per unit area at waste dump $\mathit{w}\in \mathit{W}$ in period $\mathit{t}\in \mathit{T}$.
${\psi }_c$	Topsoil/overburden thickness requirement for cell $\mathit{c}\in C_w^{\mathrm{O}\mathrm{B}}$.
${\mathrm{p}\mathrm{c}}_{i,t}^{+/-}$	Penalty cost for positive (+) or negative (-) deviations from the short-term capacity at location $\mathit{i}\in \mathcal{M}\cup \mathcal{D}$ in period $\mathit{t}\in \mathit{T}$.
${\mathrm{p}\mathrm{c}}_{a,w,t}^{+/-}$	Penalty cost for positive (+) or negative (-) deviation from the upper or lower grade threshold of attribute $\mathit{a}\in \mathit{A}$ at waste dump location $\mathit{w}\in \mathcal{W}$ in period $\mathit{t}\in \mathit{T}$.
${\mathrm{p}\mathrm{c}}_{w,t}^{\mathrm{N}\mathrm{P}\mathrm{R}}$	Penalty cost for a unit deviation from the NPR threshold at waste dump location $\mathit{w}\in \mathcal{W}$.
${\mathrm{p}\mathrm{c}}_t^{\mathrm{R}\mathrm{e}\mathrm{c}}$	Penalty cost for a unit deviation from the reclamation target for the area reclaimed in production period $\mathit{t}\in \mathit{T}.$
$V_{\mathit{w},\mathit{c}}$	Volume capacity for dumping material at waste dump $\mathit{w}\in \mathcal{W}$ in cell $\mathit{c}\in C_w$.
$L_{\mathit{i},\mathit{t}}$	Lower bound on the capacity at location $\mathit{i}\in \mathcal{M}\cup \mathcal{D}$.
$U_{\mathit{i},\mathit{t}}$	Upper bound on the capacity at location $\mathit{i}\in \mathcal{M}\cup \mathcal{D}$.
$U_{\mathrm{T}\mathrm{a}\mathrm{rget}}^a$	Upper grade/quality threshold of metal leaching attribute $\mathit{a}\in \mathit{A}$ in the waste facility.
$L_{\mathrm{T}\mathrm{a}\mathrm{rget}}^a$	Upper grade/quality threshold of metal leaching attribute $\mathit{a}\in \mathit{A}$ in the waste facility.
$L_{\mathrm{T}\mathrm{a}\mathrm{rget}}^{\mathrm{N}\mathrm{P}\mathrm{R}}$	Lower limit for NPR. Waste dump facility cells with a NPR ratio below this threshold are considered at risk of ARD. On the contrary, cells with an NPR ratio greater than this threshold are considered to be safe given the neutralisation properties of the material within a waste dump cell.
NP	Indicating attribute of neutralisation potential at the waste dump.
AP	Indicating attribute for acid production at the waste dump.
$R_t$	Annual reclamation target for waste dump facilities in period $\mathit{t}\in \mathit{T}$.

Table 3. Decision variables.

Notations	Definitions
$x_{\mathit{b},\mathit{t}}$	Binary variable set to one if block $\mathit{b}\in B_m$ is extracted in period $\mathit{t}\in \mathit{T}$, zero otherwise.
$y_{\mathit{i},\mathit{j},\mathit{t}}$	Continuous variable with range [0, 1] determines the fraction of material sent from stockpile location $\mathit{i}\in \mathcal{S}$ to destination $\mathit{j}\in \mathcal{O}\left(\mathit{i}\right)$ in period $\mathit{t}\in \mathit{T}$.
$z_{\mathit{g},\mathit{j},\mathit{t}}$	Binary variable set to one if material in group $\mathit{g}\in \mathit{G}$ is sent to destination $\mathit{j}\in \mathcal{D}$ in period $\mathit{t}\in \mathit{T}$, zero otherwise.
${\lambda }_{\mathit{g},\mathit{t},\mathit{s}}$	Continuous variable denoting the mass of material extracted from group $\mathit{g}\in \mathit{G}$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}.$
${\gamma }_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}$	Continuous variable denoting the mass of attribute $\mathit{a}\in \mathit{A}$ extracted from group $\mathit{g}\in \mathit{G}$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}$.
${\overline{\mathrm{mass}}}_{\mathit{i},\mathit{t},\mathit{s}}$	Continuous variable denoting the mass of material passing through location $\mathit{i}\in \mathcal{M}\cup \mathcal{D}$ in period $\mathit{t}\in \mathit{T}$ in scenario $\mathit{s}\in \mathit{S}$.
${\overline{\mathrm{mass}}}_{\mathit{a},\mathit{i},\mathit{t},\mathit{s}}$	Continuous variable denoting the recovered quantity of attribute $\mathit{a}\in \mathit{A}$ at location $\mathit{i}\in \mathcal{D}$ in period $\mathit{t}\in \mathit{T}$ in scenario $\mathit{s}\in \mathit{S}$.
${\overline{\mathrm{mass}}}_{\mathit{a},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}$	Continuous variable denoting the mass of attribute $\mathit{a}\in \mathit{A}$ at waste dump facility $\mathit{w}\in \mathcal{W}$ cell $\mathit{c}\in C_w$ in period $\mathit{t}\in \mathit{T}$ in scenario $\mathit{s}\in \mathit{S}$.
${\kappa }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}$	Continuous variable in range [0,1] representing the percentage fill at waste dump $\mathit{w}\in \mathcal{W}$ cell $\mathit{c}\in C_w$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}$.
${\alpha }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}$	Binary variable denoting whether waste dump $\mathit{w}\in \mathcal{W}$ cell $\mathit{c}\in C_w$ is available for dumping material in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}$.
${\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}$	Continuous variable denoting the mass of material from group $\mathit{g}$ sent to waste dump $\mathit{w}\in \mathit{W}$to cell $\mathit{c}$ given $\left(\mathit{g},\mathit{c}\right)\in G^R\times C_w^R\cup G^{\mathrm{OB}}\times C_w^{\mathrm{O}\mathrm{B}}$ in period $\mathit{t}\in \mathit{T}$, and scenario $\mathit{s}\in \mathit{S}$.
${\bar{v}}_{\mathit{g},\mathit{w},\mathit{t},\mathit{s}}$	Continuous variable denoting the volume of material from group $\mathit{g}\in \mathit{G}$ sent to waste dump $\mathit{w}\in \mathcal{W}$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}.$
$g_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}$	Continuous variable denoting the grade, quality, or concentration of attribute $\mathit{a}\in \mathit{A}$ within the material extracted from the mines in group $\mathit{g}\in \mathit{G},$period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}.$
${\delta }_{i,t,s}^{+/-}$	Recourse variable measuring positive (+) and negative (-) deviation from production targets at location $\mathit{i}\in \mathcal{M}\cup \mathcal{D}$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}$.
${\delta }_{a,w,c,t,s}^{+/-}$	Recourse variable denoting positive (+) and negative (-) deviations from material quality constraints for attribute $\mathit{a}\in \mathit{A}$ at waste dump $\mathit{w}\in \mathcal{W}$ at cell $\mathit{c}\in C_w$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}.$
${\delta }_{w,c,t,s}^{\mathrm{N}\mathrm{P}\mathrm{R}}$	Recourse variable denoting deviations from the NPR threshold in the blended product in waste dump $\mathit{w}\in \mathcal{W}$ cell $\mathit{c}\in C_w$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}.$
${\delta }_{t,s}^{\mathrm{R}\mathrm{e}\mathrm{c}}$	Recourse variable denoting deviations from the reclamation targets for each production period $\mathit{t}\in \mathit{T}.$
$\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}$	Continuous variable denoting the specific volume of material extracted from group $\mathit{g}\in \mathit{G}$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}$.
$\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}$	Continuous variable denoting the swell factor of material extracted from group $\mathit{g}\in \mathit{G}$ in period $\mathit{t}\in \mathit{T}$ and scenario $\mathit{s}\in \mathit{S}$

2.1.2. Objective function

The objective function is presented in Equation (1). Part I and II of the objective function maximise the revenues obtained from processing valuable attributes to sell to the market and minimise the costs to produce them. Part III of the objective function minimises the reclamation costs per a unit area of reclaimed land at the waste facilities. This can include the placement, re-sloping, water management and planting costs for reclamation. Part IV minimises deviations from production targets related to the capacities of each location in the mining complex. Part V and VI minimise deviations from the blend of materials within the waste dump cell to prevent the production of ARD by managing the NPR ratio and metal leaching requirements. Part VII encourages progressive reclamation to place materials in locations that are ready for reclamation. Maximising the NPV by including the discounted revenues and costs associated with mining, processing and stockpiling material within the objective function has been considered in previous stochastic programming formulations while managing risk related to capacities [7,9]. However, the contribution of this optimisation model is that reclamation costs are directly considered in the objective function and the balancing of NPR and metal leaching material attributes are included to prevent ARD. Lastly, progressive reclamation is encouraged to mitigate liabilities and improve environmental performance of the mining complex.

(1) $\begin{array}{rl}& \max \frac{1}{S}\left[\underset{\mathrm{PartI}}{\underbrace{\sum _{\mathit{s}\in \mathit{S}}\sum _{\mathit{t}\in \mathit{T}}\sum _{\mathit{i}\in \mathcal{P}}\sum _{\mathit{a}\in \mathit{A}}{p}_{\mathit{a},\mathit{i},\mathit{t}}{\overline{\mathrm{mass}}}_{\mathit{a},\mathit{i},\mathit{t},\mathit{s}}}}\right.-\underset{\mathrm{PartII}}{\underbrace{\sum _{\mathit{s}\in \mathit{S}}\sum _{\mathit{t}\in \mathit{T}}\sum _{\mathit{i}\in \mathcal{M}\cup \mathcal{D}}{c}_{\mathit{i},\mathit{t}}{\overline{\mathrm{mass}}}_{\mathit{i},\mathit{t},\mathit{s}}}}\\ & -\underset{\mathrm{PartIII}}{\underbrace{\sum _{\mathit{s}\in \mathit{S}}\sum _{\mathit{t}\in \mathit{T}}\sum _{\mathit{w}\in \mathcal{W}}\sum _{c\in C_w^{\mathrm{O}\mathrm{B}}}\sum _{\mathit{g}\in {\mathit{G}}^{\mathrm{OB}}}(\mathrm{r}{\mathrm{c}}_{\mathit{w},\mathit{t}}/{\psi }_c)\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}}}\\ & -\underset{\mathrm{PartIV}}{\underbrace{\sum _{\mathit{s}\in \mathit{S}}\sum _{\mathit{t}\in \mathit{T}}\sum _{\mathit{i}\in \mathcal{M}\cup \mathcal{D}}\left({\mathrm{p}\mathrm{c}}_{i,t}^{-}{\delta }_{i,t,s}^{-}+{\mathrm{p}\mathrm{c}}_{i,t}^{+}{\delta }_{i,t,s}^{+}\right)}}-\underset{\mathrm{PartV}}{\underbrace{\sum _{\mathit{s}\in \mathit{S}}\sum _{\mathit{t}\in \mathit{T}}\sum _{\mathit{w}\in \mathcal{W}}\sum _{c\in C_w}{\mathrm{p}\mathrm{c}}_{w,t}^{\mathrm{N}\mathrm{P}\mathrm{R}}{\delta }_{w,c,t,s}^{\mathrm{N}\mathrm{P}\mathrm{R}}}}\\ & \underset{\mathrm{PartVI}}{\underbrace{\sum _{\mathit{s}\in \mathit{S}}\sum _{\mathit{t}\in \mathit{T}}\sum _{\mathit{w}\in \mathcal{W}}\sum _{c\in C_w^R}\sum _{\mathit{a}\in \mathit{A}}\left({\mathrm{p}\mathrm{c}}_{a,w,t}^{-}{\delta }_{a,w,c,t,s}^{-}+{\mathrm{p}\mathrm{c}}_{a,w,t}^{+}{\delta }_{a,w,c,t,s}^{+}\right)}}\\ & \left.\underset{\mathrm{PartVII}}{\underbrace{\sum _{\mathit{s}\in \mathit{S}}\sum _{\mathit{t}\in \mathit{T}}{\mathrm{p}\mathrm{c}}_t^{\mathrm{R}\mathrm{e}\mathrm{c}}{\delta }_{t,s}^{\mathrm{R}\mathrm{e}\mathrm{c}}}}\right]\end{array}$(1)

The objective function is maximised subject to the following constraints.

2.1.3. Constraints

Block precedence and reserve constraints ensure that geotechnical slope constraints are enforced and that a single mining block can only be mined once:

(2) $\begin{array}{c}{x}_{\mathit{b},\mathit{t}}\le \sum _{\mathit{k}=1}^t{x}_{\mathit{b}{\mathit{ }}^{\mathrm{\prime }},\mathit{k}}\mathrm{\forall b}\in B_m,\mathit{b}{ }^{\mathrm{\prime }}\in \mathbb{O}\left(\mathit{b}\right),\mathit{t}\in \mathit{T}.\end{array}$(2)

(3) $\sum _{\mathit{t}\in \mathit{T}}{x}_{\mathit{b},\mathit{t}}\le 1\mathrm{\forall b}\in B_m,\mathit{m}\in \mathcal{M}$(3)

Mass and attribute tracking constraints calculate material quantities and qualities throughout the mining complex.

Constraints (4)–(5) compute the mass and attribute quantity extracted from the mines in each period.

(4) $\begin{array}{c}{\lambda }_{\mathit{g},\mathit{t},\mathit{s}}=\sum _{\mathit{m}\in \mathcal{M}}\sum _{b\in B_m}{\theta }_{\mathit{b},\mathit{g},\mathit{s}}{w}_{\mathit{b},\mathit{s}}{x}_{\mathit{b},\mathit{t}}\mathrm{\forall g}\in \mathit{G},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(4)

(5) $\begin{array}{c}{\gamma }_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}=\sum _{\mathit{m}\in \mathcal{M}}\sum _{b\in B_m}{\theta }_{\mathit{b},\mathit{g},\mathit{s}}{w}_{\mathit{b},\mathit{s}}{g}_{\mathit{a},\mathit{b},\mathit{s}}{x}_{\mathit{b},\mathit{t}}\mathrm{\forall a}\in \mathit{A},\mathit{g}\in \mathit{G},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(5)

Constraints (6) track the total amount of material mined at each mine in the mining complex:

(6) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathit{m},\mathit{t},\mathit{s}}=\sum _{b\in B_m}{w}_{\mathit{b},\mathit{s}}{x}_{\mathit{b},\mathit{t}}\mathrm{\forall m}\in \mathcal{M},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(6)

Constraints (7)–(8) track the mass and the recovered quantity of each attribute at the various destinations in mining complex. Constraint (9) ensures that only a fraction of materials within a stockpile can be transported to a subsequent downstream destination in a single period and constraint (10) ensures that for all non-stockpiling destinations $\mathit{i}\in \mathcal{D}\mathrm{\setminus }\mathcal{S}$ that all materials are transferred to the subsequent location $\mathit{j}$. The constraints now follow:

(7) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathit{j},\left(\mathit{t}+1\right),\mathit{s}}=\sum _{\mathit{g}\in \mathit{G}}{\lambda }_{\mathit{g},\mathit{t},\mathit{s}}{z}_{\mathit{g},\mathit{j},\left(\mathit{t}+1\right)}+\sum _{\mathit{i}\in \mathit{J}\left(j\right)\cap \mathcal{S}}{\overline{\mathrm{mass}}}_{\mathit{i},\mathit{t},\mathit{s}}{y}_{\mathit{i},\mathit{j},\left(\mathit{t}+1\right)}\\ +{\overline{\mathrm{mass}}}_{\mathit{i},\mathit{t},\mathit{s}}\left(1-\sum _{\mathit{k}\in \mathcal{O}\left(j\right)}{y}_{\mathit{i},\mathit{k},\mathit{t}}\right)\mathrm{\forall j}\in \mathcal{D},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(7)

(8) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathit{a},\mathit{j},\left(\mathit{t}+1\right),\mathit{s}}=\sum _{\mathit{g}\in \mathit{G}}{r}_{\mathit{a},\mathit{j},\left(\mathit{t}+1\right)}{\gamma }_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}{z}_{\mathit{g},\mathit{j},\left(\mathit{t}+1\right)}+\sum _{\mathit{i}\in \mathcal{I}\left(j\right)\cap \mathcal{S}}{r}_{\mathit{a},\mathit{j},\left(\mathit{t}+1\right)}{\overline{\mathrm{mass}}}_{\mathit{a},\mathit{i},\mathit{t},\mathit{s}}{y}_{\mathit{i},\mathit{j},\left(\mathit{t}+1\right)}\\ +{\overline{\mathrm{mass}}}_{\mathit{a},\mathit{j},\mathit{t},\mathit{s}}\left(1-\sum _{\mathit{k}\in \mathcal{O}\left(j\right)}{y}_{\mathit{i},\mathit{k},\mathit{t}}\right)\mathrm{\forall a}\in \mathit{A},\mathit{j}\in \mathcal{D},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(8)

(9) $\begin{array}{c}\sum _{\mathit{i}\in \mathcal{O}\left(i\right)}{y}_{\mathit{i},\mathit{j},\mathit{t}}\le 1\mathrm{\forall i}\in \mathcal{S},\mathit{t}\in \mathit{T}.\end{array}$(9)

(10) $\begin{array}{c}\sum _{\mathit{i}\in \mathcal{O}\left(i\right)}{y}_{\mathit{i},\mathit{j},\mathit{t}}=1\mathrm{\forall i}\in \mathcal{D}\mathrm{\setminus }\mathcal{S},\mathit{t}\in \mathit{T}.\end{array}$(10)

Constraints (11)–(13) calculate the swell factor, specific volume and volume of material sent from each group to the waste dump facility. The swell factor and specific volume of a group are calculated based on the input simulated block parameters and the extraction sequence selected such that:

(11) $\begin{array}{c}\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}{\lambda }_{\mathit{g},\mathit{t},\mathit{s}}=\sum _{\mathit{m}\in \mathcal{M}}\sum _{b\in B_m}\mathrm{s}{\mathrm{f}}_{\mathit{b},\mathit{s}}{\theta }_{\mathit{b},\mathit{g},\mathit{s}}{w}_{\mathit{b},\mathit{s}}{x}_{\mathit{b},\mathit{t}}\mathrm{\forall g}\in \mathit{G},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(11)

(12) $\begin{array}{c}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}{\lambda }_{\mathit{g},\mathit{t},\mathit{s}}=\sum _{\mathit{m}\in \mathcal{M}}\sum _{b\in B_m}\mathrm{s}{\mathrm{v}}_{\mathit{b},\mathit{s}}{\theta }_{\mathit{b},\mathit{g},\mathit{s}}{w}_{\mathit{b},\mathit{s}}{x}_{\mathit{b},\mathit{t}}\mathrm{\forall g}\in \mathit{G},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(12)

(13) $\begin{array}{c}{\bar{v}}_{\mathit{g},\mathit{w},\mathit{t},\mathit{s}}=\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}{z}_{\mathit{g},\mathit{w},\mathit{t}}\sum _{\mathit{m}\in \mathcal{M}}\sum _{b\in B_m}{\theta }_{\mathit{b},\mathit{g},\mathit{s}}{w}_{\mathit{b},\mathit{s}}{x}_{\mathit{b},\mathit{t}}\mathrm{\forall g}\in \mathit{G},\mathit{w}\in \mathcal{W},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(13)

Based on the material sent to the waste dump facilities the materials are distributed to different cells to blend materials of different qualities, as shown in Figure 1. Constraints (14)–(15) ensure that the volume of material extracted and sent to the waste dump facility are distributed to the waste dump cells that accept materials of specific types (rock and overburden):

(14) $\begin{array}{c}{\bar{v}}_{\mathit{g},\mathit{w},\mathit{t},\mathit{s}}-\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}\sum _{c\in C_w^R}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}=0\mathrm{\forall g}\in G^R,\mathit{w}\in \mathcal{W},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(14)

(15) $\begin{array}{c}{\bar{v}}_{\mathit{g},\mathit{w},\mathit{t},\mathit{s}}-\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}\sum _{c\in C_w^{\mathrm{O}\mathrm{B}}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}=0\mathrm{\forall g}\in G^{\mathrm{OB}},\mathit{w}\in \mathcal{W},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(15)

Constraints (16)–(17) ensure that the volume of material placed in each cell does not violate the available volume of material within each waste dump facility cell:

(16) $\begin{array}{c}\sum _{\mathit{g}\in {\mathit{G}}^{\mathrm{OB}}}\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\le V_{\mathit{w},\mathit{c}}{\alpha }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\mathrm{\forall w}\in \mathcal{W},\mathit{c}\in C_w^{\mathrm{O}\mathrm{B}},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(16)

(17) $\begin{array}{c}\sum _{g\in G^R}\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\le V_{\mathit{w},\mathit{c}}{\alpha }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\mathrm{\forall w}\in \mathcal{W},\mathit{c}\in C_w^R,\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(17)

Constraints (18)–(19) update the fill level of a waste dump cell:

(18) $\begin{array}{c}\sum _{\mathit{g}\in {\mathit{G}}^{\mathrm{OB}}}\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}=V_{\mathit{w},\mathit{c}}{\kappa }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\mathrm{\forall w}\in \mathcal{W},\mathit{c}\in C_w^{\mathrm{O}\mathrm{B}},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(18)

(19) $\begin{array}{c}\sum _{g\in G^R}\mathrm{s}{\mathrm{f}}_{\mathit{g},\mathit{t},\mathit{s}}\mathrm{s}{\mathrm{v}}_{\mathit{g},\mathit{t},\mathit{s}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}=V_{\mathit{w},\mathit{c}}{\kappa }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\mathrm{\forall w}\in \mathcal{W},\mathit{c}\in C_w^R,\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(19)

Constraints (20) enforce precedence of the waste dump facility cells directly. The precedence constraints consider vertical and horizontal precedence relationships directly (a simplified example is shown in Figure 1):

(20) $\begin{array}{c}{\alpha }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\le {\kappa }_{\mathit{w},\mathit{c}{\mathit{ }}^{\mathrm{\prime }},\mathit{t},\mathit{s}}\mathrm{\forall w}\in \mathcal{W},\mathit{c}\in C_w,\mathit{c}{ }^{\mathrm{\prime }}\in \mathrm{P}{\mathrm{D}}_{\mathit{w},\mathit{c}},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(20)

Constraints (21)–(23) compute the grade, quality or concentration of an attribute within a group which is used to determine the blended concentration of an attribute in the waste dump facility cells for overburden and rock material placement, respectively:

(21) $\begin{array}{c}{g}_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}={\gamma }_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}/{\lambda }_{\mathit{g},\mathit{t},\mathit{s}}.\end{array}$(21)

(22) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathit{a},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}=\sum _{\mathit{g}\in {\mathit{G}}^{\mathrm{OB}}}{g}_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\mathrm{\forall a}\in \mathit{A},\mathit{w}\in \mathcal{W},\mathit{c}\in C_w^{\mathrm{O}\mathrm{B}},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(22)

(23) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathit{a},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}=\sum _{g\in G^R}{g}_{\mathit{a},\mathit{g},\mathit{t},\mathit{s}}{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\mathrm{\forall a}\in \mathit{A},\mathit{w}\in \mathcal{W},\mathit{c}\in C_w^R,\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(23)

Constraints (24) manage grade blending to meet the NPR threshold target within each waste dump cell with a linear function $\mathit{f}$ of the indicating geochemical attributes NP and AP. By enforcing, the following constraints it is possible to mitigate ARD in perpetuity by ensuring the ongoing prevention with a blended waste product that is non-acid generating:

(24) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathrm{NP},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}-{\delta }_{w,c,t,s}^{\mathrm{N}\mathrm{P}\mathrm{R}}\ge L_{\mathrm{T}\mathrm{a}\mathrm{rget}}^{\mathrm{N}\mathrm{P}\mathrm{R}}\mathit{f}\left({\overline{\mathrm{mass}}}_{\mathrm{AP},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\right)\mathrm{\forall w}\in \mathcal{W},\mathit{c}\in C_w,\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}\end{array}$(24)

Constraints (25)–(26) manage the risk of the concentration of geochemical attributes related to metal leaching exceeding the upper and lower concentration limits at the cells in a waste dump facility:

(25) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathrm{a},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}-{\delta }_{a,w,c,t,s}^{+}\le U_{\mathrm{T}\mathrm{a}\mathrm{rget}}^a\sum _{g\in G^R}\sum _{\mathit{\tau }=1}^t{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\\ \mathrm{\forall a}\in \mathit{A},\mathit{w}\in \mathcal{W},\mathit{c}\in C_w^{\mathrm{R}},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(25)

(26) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathrm{a},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}+{\delta }_{a,w,c,t,s}^{-}\ge L_{\mathrm{T}\mathrm{a}\mathrm{rget}}^a\sum _{g\in G^R}\sum _{\mathit{\tau }=1}^t{\bar{m}}_{\mathit{g},\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\\ \mathrm{\forall a}\in \mathit{A},\mathit{w}\in \mathcal{W},\mathit{c}\in C_w^{\mathrm{R}},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(26)

Constraints (27) aim manage the risk of meeting progressive reclamation targets for completed areas in each waste dump facilities:

(27) $\begin{array}{c}\sum _{c\in c_w^{\mathrm{O}\mathrm{B}}}{\kappa }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}+{\delta }_{t,s}^{\mathrm{R}\mathrm{e}\mathrm{c}}\ge R_t\mathrm{\forall w}\in \mathcal{W},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(27)

Constraints (28)–(29) manage the risk of deviating from the capacity constraints at each location in the mining complex:

(28) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathit{i},\mathit{t},\mathit{s}}-{\delta }_{i,t,s}^{+}\le U_{\mathit{i},\mathit{t}}\mathrm{\forall i}\in \mathcal{M}\cup \text{script}\mathit{D}\text{nolimitsD},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(28)

(29) $\begin{array}{c}{\overline{\mathrm{mass}}}_{\mathit{i},\mathit{t},\mathit{s}}+{\delta }_{i,t,s}^{-}\ge L_{\mathit{i},\mathit{t}}\mathrm{\forall i}\in \mathcal{M}\cup \text{script}\mathit{D}\text{nolimitsD},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(29)

Constraints (30) only allow a group of materials to be sent to a single destination in each production period:

(30) $\begin{array}{c}\begin{array}{c}\sum _{\mathit{j}\in \mathcal{D}}{\kappa }_{\mathit{g},\mathit{j},\mathit{t}}=1\mathrm{\forall g}\in \mathit{G},\mathit{t}\in T^{\mathrm{ST}}\cup T^{\mathrm{LT}}.\end{array}\end{array}$(30)

Lastly, non-negativity and binary constraints (31)–(32) are enforced, indices omitted for clarity.

(31) $\overline{m},\mathit{y},\mathit{\gamma },\mathit{\delta },\mathit{\kappa },\mathit{\lambda }\ge 0$(31)

(32) $\begin{array}{c}\mathit{x},\mathit{z},\mathit{\alpha }\in \left\{0,1\right\}\end{array}$(32)

The following section will outline the optimisation approach for the proposed stochastic programming formulation. The optimisation approach manages non-linearities in the objective function and constraints by decoupling the optimisation of the extraction sequence and downstream decision variables, which is explained subsequently.

2.1.4. Simulated annealing, contextual bandits and stochastic programming

The two-stage stochastic programming formulation is solved with a hybrid framework that utilises a metaheuristic with embedded contextual bandits. The metaheuristic applied is a simulated annealing-based approach that allows the model to handle non-linear objective function and constraints since the flow of materials are dependent on the product of several key decision variables. Simulated annealing explores the neighbouring solutions by modifying the extraction sequence decision variables $\left(x_{\mathit{b},\mathit{t}}\right)$ [9,57]. Similar to past work, the extraction sequence decisions are modified within the simulated annealing framework with a set of perturbations that ensure feasibility of the resulting extraction sequence [9,12,58]. Then, the remaining downstream decision variables are optimised with the branch-and-cut algorithm implemented in CPLEX v20.1 given the temporarily fixed extraction sequence decisions to obtain the objective function value for a given iteration of the simulated annealing algorithm. This ensures that each extraction sequence evaluated within the simulated annealing algorithm has optimal downstream decisions. The algorithm occasionally accepts solutions that decrease the objective function to escape local optima based on an acceptance criterion. The probability of accepting a deteriorating solution is adapted throughout the optimisation by using a cooling schedule and annealing temperature parameter. The implementation of simulated annealing in the optimisation of mining complexes has been applied in several previous works [7,9].

In this work, contextual bandits are integrated into the simulated annealing algorithm as a learning framework that aims to determine which perturbations (production schedule modifications) provide the largest improvements in the resulting objective function in given some context of the modification. The actions and contexts that are used in this work are denoted by the sets $\mathit{K}$ and $\mathit{C}$, respectively. Actions or perturbations that lead to larger improvements in the resulting production schedule of a mining complex in a given context are visited more frequently. This provides an effective way to improve the objective function by utilising past learning to select more impactful perturbations during the optimisation. Given the improvement and context of the optimisation the resulting estimates of each action are updated for subsequent action selection. These estimates are referred to as estimates of the action-value function $\mathit{Q}\left(\mathit{c},\mathit{k}\right)$ which provides the expected value of taking an action $\mathit{k}\in \mathit{K}$ in a given context $\mathit{c}\in \mathit{C}$. Within the simulated annealing framework, each action is selected with an $\mathit{\epsilon }$-greedy policy which allows for exploration of different actions ($\mathit{\epsilon }\in \left[0,1\right]$ denotes the degree of exploration). The contextual bandit approach for selecting modifications to the production schedule is outlined in Algorithm 1 [55].

Given that the optimisation of mining complexes is a difficult learning task, the agent or learner must learn to track the best actions to take over time. This is because certain actions that perform better early in the optimisation process (modifications to a larger number of decision variables) may not be the top performing modifications once the solution has been improved. For instance, smaller fine-tuning actions may become more beneficial at this time. Therefore, the goal is to learn a policy that maps a context to an action which is expected to improve the objective function by the greatest amount. The learned policy is constantly adapting to new information as the optimisation algorithm proceeds. The algorithm proposed works as follows. First, the state-value function $\mathit{Q}\left(\mathit{c},\mathit{k}\right)$ estimates are initialised to an optimistic value $\mathit{O}$. This encourages early exploration of all actions. Similarly, the visit count of each context-action pair $\mathit{N}\left(\mathit{c},\mathit{k}\right)$ is initialised to zero. During the optimisation process, each time a mining block is selected to modify the extraction sequence decision variables, the context is retrieved from the mining complex. Then, an action is taken with $\mathit{\epsilon }$-greedy policy where the maximum action for the given context is taken with probability $1-\mathit{\epsilon }$. Otherwise, an action is selected uniformly at random. The action is applied to modify the extraction sequence decisions variables and the objective function value is updated after modifying these decisions. The difference between the current objective and the previous objective is calculated and used to update the action-value function. Lastly, the visit count for that action in this context is updated, which is used as a tie-breaking rule. The proposed formulation and solution approach are tested at a large copper-gold mining complex in the following section.

3. Application in a large copper-gold mining complex

3.1. Overview of the copper-gold mining complex

The stochastic optimisation formulation and solution approach are applied at a large-scale copper-gold mining complex to optimise the long-term production schedule. The mining complex under study has been operating for 10 years and has approximately 5 years of operational life remaining prior to closure where reclamation must be completed. The mining complex is presented in Figure 2. There are three open-pit mines, a run-of-mine stockpile, a processing facility, and three waste dump facilities that require mitigation of ARD and progressive reclamation. The open-pit mines supply materials to the processor within the mining complex to produce copper and gold products for delivery to customers and the market. Waste materials are separated into two material classes rock and overburden/topsoil. The geochemical characteristics of the materials in the deposit include simulations of the sulphur and total inorganic carbon content (assay samples are retrieved with an element scan with ICP-MS and aqua regia digestion and coulometry, respectively), which are used to calculate the NPR ratio in waste dump cells. In addition, the overburden/topsoil quantities are simulated to represent the uncertainty and variability in the quantity of material available for reclamation purposes [59].

Figure 2. Mining complex with three open-pit mines, three waste dump facilities, stockpile, process plant that generates two valuable products.

The three open-pit mines that supply material within the mining complex have a joint mining capacity of 42 M tonnes per year and the process plant has a maximum capacity of 13 M tonnes with a recovery of 90% and 80.4% for copper and gold, respectively. Mining blocks are not aggregated and a set of 1.84 million blocks are considered for extraction with dimensions 10 m × 10 m × 16 m. Fifteen stochastically simulated orebody models with the pertinent attributes and geochemical properties are used as input to the optimisation formulation to directly manage risk [12,60]. The stochastic simulations of the orebody are simulated directly at block support and consider the correlations between attributes using an efficient method for quantifying the uncertainty and local variability of the material attributes in a large deposit [37]. The slope angle considered for mining block precedence is 47 degrees based on geotechnical studies completed by the operators of the mining complex.

The mining complex production schedule and waste dump placement schedule are optimised for the remaining five production years in this mining complex to minimise financial losses incurred to satisfy closure requirements. A waste dump model of the areas available for material placement in the waste dump facilities is constructed, similar to a block model, where the cell locations represent the volumes available for dumping materials in different areas. The ultimate goal of the long-term production schedule is to maximise net present value, while mitigating ARD in the waste dump facilities to prevent additional long-term monitoring costs. Additionally, a progressive reclamation strategy is implemented to start rehabilitating completed waste dump areas and return them to their original environmental state during production. This helps reduce liabilities and satisfy regulatory requirements. In addition, the progressive reclamation approach provides opportunities to improve reclamation practices if certain sections are unsuccessful. To begin reclamation in the waste dump facilities, underlying waste dump cells that are below reclamation cells must be completed to ensure the full utilisation of areas available for waste disposal. A progressive reclamation goal of 50 hectares (ha) is targeted during each production year to reduce liabilities and satisfy regulatory requirements prior to closure.

Waste dump cells are 100 × 100 × 15 m³. An assumption made here is that the waste dump cell size allows for the model to adequately blend waste materials of different quality to prevent ARD. Considering the development of waste dumps forms in thin slices of rock, as trucks dump off the waste dump facility lifts, it is expected that adequate blending would occur with cell sizes of these dimensions. A NPR ratio, which measures the neutralisation potential of mining wastes, is measured in each waste dump cell by accounting for the blended geochemical properties [34]. NPR ratios greater than 2 are targeted in the waste dump cells to prevent the production of ARD. Geotechnical and operational constraints are considered when building the precedence relationships between waste-dump cells, which includes satisfying a 37.5-degree angle of repose and ensuring access is present from available roads. This is completed in a preprocessing step to define the precedence relationship in Equation (20)(20) $\begin{array}{c}{\alpha }_{\mathit{w},\mathit{c},\mathit{t},\mathit{s}}\le {\kappa }_{\mathit{w},\mathit{c}{\mathit{ }}^{\mathrm{\prime }},\mathit{t},\mathit{s}}\mathrm{\forall w}\in \mathcal{W},\mathit{c}\in C_w,\mathit{c}{ }^{\mathrm{\prime }}\in \mathrm{P}{\mathrm{D}}_{\mathit{w},\mathit{c}},\mathit{t}\in \mathit{T},\mathit{s}\in \mathit{S}.\end{array}$(20) .

3.2. Simultaneous stochastic optimization with waste management and progressive reclamation

In the remainder of this section, the results from applying the proposed simultaneous stochastic optimisation framework for long-term production are discussed and compared to a base case production schedule that does not consider integrated waste management and reclamation. The p-10, p-50, and p-90 represent a 10%, 50% and 90% chance of obtaining quantities below the values in the risk profiles that are presented throughout this section. The lower (LB) and upper (UB) bounds are drawn in the following figures when a constraint is applied. To begin with, the impacts of integrating waste management and progressive reclamation will be discussed to highlight the additional production scheduling considerations addressed in this case study. Then, the remaining production schedule decisions are discussed.

In Figures 3 and 4, the risk profiles for the NPR ratio at 51 randomly selected waste dump cells in the base case and integrated production schedule are shown. In the base case production schedule, there is significant risk of producing ARD as many waste dump cells have a blended NPR ratio that is less than the lower limit of 2. For example, the p-50 in waste dump cells 5–7 show that there is a large risk of producing acid in those areas of the waste dump due to the geochemical properties of waste placed. The integrated production schedule that considers waste management directly in the formulation is able to resolve these issues by blending materials to mitigate ARD. Although the method significantly reduces the risk of ARD and the frequency of occurrence, there are still deviations from the target NPR ratio, see cell 13 and 22 in Figure 4. Alternative methods such as sourcing additional materials or placing limestone for neutralising materials may also need to be considered to completely prevent ARD in the waste dump facilities. The other cells in the waste dump facilities show similar improvements when considering the integrated approach and reduce the number of cells with unsafe NPR ratios by 52.5%. This significantly reduces risk of ARD in the integrated production schedule with the proposed simultaneous stochastic optimisation framework. The waste placement schedule is impacted by blending materials with different geochemical properties, which leads to distinct changes in the waste dump placement schedules in the base case versus the integrated production schedule shown in Figure 5.

Figure 3. Risk profiles for cell NPR ratios in the waste dump cells in the base-case production schedule.

Figure 4. Risk profiles for cell NPR ratios in the waste dump cells in the integrated production schedule.

Figure 5. Base case (left) and integrated (right) waste dump placement schedule for waste dump facility (WD) 1 and 2.

Progressive reclamation and rehabilitation of waste dump facilities is desired to accelerate the closure of the mining complex and return these areas to a productive environmental state. Overburden/topsoil material covers were to be directly placed at the waste dump facilities to enhance plant growth and improve rehabilitation. However, the waste dump facility design must be considered, and full utilisation of the available waste dump cells is required before starting the reclamation process to efficiently use the existing waste dump footprints. Figure 6 shows that the progressive reclamation process in the mining complex is unable to meet targets in years 1–2 in both production schedules. Progressive reclamation was ignored in previous long-term optimisation studies leading to many areas of the waste dump being used and a lack of focus on completing areas to begin reclamation. The proposed optimisation model targets areas of the waste dump to place waste material that provide opportunities to begin reclamation and satisfy the target 50 ha per a year. As the waste dump cells are filled to capacity, reclamation can begin in the areas that reach the final design and targets can be more easily meet in both production schedules in years 3–5. The advantage of integrating progressive reclamation into the optimisation of mining complexes demonstrated in this application is that it provides engineering support for advancing the waste dump facilities to fulfil reclamation activities. These costs are accounted for directly in the integrated model and are not considered in the base-case causing changes to the resulting production schedule. In addition, Figure 7 demonstrates that greater then 60% of reclamation activities on waste dumps 1 and 2 can be completed in parallel with production, if scheduled appropriately.

Figure 6. Risk profiles for the integrated (blue) and base-case (green) progressive reclamation progress at the waste dump facilities.

Figure 7. Progressive reclamation progress in the integrated production schedule (left) waste dump 1 and (right) waste dump 2 from a top view.

As a result of introducing these additional production scheduling constraints into the stochastic programming formulation, there are impacts to the optimised production scheduling decisions. In Figure 8, the risk profiles for the processing facility throughput, mining production rate, copper head grade and copper production at the processing facility are shown. There are larger changes to optimised mining rate that occur in years 4 and 5. In addition, the processing facility throughput varies in each of the production schedules presented. A higher throughput with lower head grade is found at the processing plant with the integrated production schedule that considers waste management and reclamation. However, with the change in production scheduling decisions, the forecasted cashflows and metal production remain similar between the two production schedules. The cumulative discounted cashflows shown in Figure 9 demonstrate that the mining complex can break-even over the 5 production years while, preventing ARD and preparing for closure with additional waste management and reclamation costs incurred. Similar production cashflows are obtained, however, the integrated schedule improves waste management (Figure 4) and accounts directly for progressive reclamation completion and associated costs (Figure 6).

Figure 8. Risk profiles for the integrated and base-case for the processing plant annual throughput, mining production rate, copper head grade, and copper production.

Figure 9. Risk profiles for the integrated and base-case discounted and cumulative discounted cashflows.

4. Discussion

The proposed optimisation framework simultaneously optimises several major components in a mining complex considering uncertainty. This leads to critical differences in the resulting production schedules and also the mining extraction sequences in each mine. Cross sections for Mines 1–3 are shown in Figures 10 and 11, which highlight major differences between the extraction sequences produced when comparing the base case to the integrated schedule. The small arrows identify some of the visual differences in the schedule for Mine 1 and Mine 2. These changes in the extraction sequence impact the subsequent components in the mining complex and allow for improved reclamation and waste management by maximising the value of the products sold and accounting for the impact of critical attributes across the entire mining complex.

Figure 10. Comparison of the differences between the base case and integrated extraction sequence for Mine 1, an east-west cross-section.

Figure 11. Comparison of the differences between the base case and integrated extraction sequence for Mine 2 and 3, an east-west cross section.

Optimising long-term production schedules in industry scale mining complexes is a challenging optimisation problem especially when considering several additional components for integrating waste management and progressive reclamation. The mining complex optimised in this work required 2.4 M continuous decision variables and 9.2 M binary decision variables. The optimised solution was reached in 80 hours. Contextual bandits were applied to optimise large-scale mining complexes more effectively, as demonstrated in this work. The results of the optimisation using contextual bandits with an $\mathit{\epsilon }$-greedy policy was compared to those obtained by employing a random approach for selecting perturbations within the simulated annealing framework. Figure 12 shows that a 24% higher objective function value is reached after 10,000 iterations and that the contextual bandit learnings are shown to improve the heuristic selection approach after only 2,000 iterations of the simulated annealing algorithm. Context related to the areas to be mined allows for improved efficiency in the optimisation approach by understanding which areas are more likely to improve the production schedule by delaying them or advancing them in the extraction sequence.

Figure 12. Contextual bandit performance compared to random perturbation selection.

5. Conclusions and future work

A simultaneous stochastic optimisation framework for optimising large-scale mining complexes is proposed that integrates waste management and progressive reclamation considerations into the stochastic programming model. The framework directly manages supply uncertainty. A simulated annealing metaheuristic framework is applied to solve the optimisation model with contextual bandits, which is tested at an open-pit copper-gold mining complex. The case study compares the simultaneous stochastic optimisation with integrated waste management and progressive reclamation to a base case production schedule that does not consider these additional components in the optimisation. The comparisons show that similar net present value can be achieved while, accounting for waste management and mitigating ARD by blending of materials with different geochemical properties. The integrated approach reduces the number of cells at risk of ARD by 52.5%. Progressive reclamation considerations allow for 60% of reclamation activities to be completed on waste dumps 1 and 2 during production. In addition, a reclamation and waste dump placement schedule are obtained that improve environmental performance. Lastly, contextual bandits are shown to increase the efficiency of the solution approach by selecting heuristics that are expected to improve the long-term production schedule based on context taken directly from the optimisation of the case-study presented. Future work should investigate the integration of production cycle times particularly with regards to the load and haul fleet in open-pit mining complexes to accurately understand the impact of waste management and reclamation on fleet performance. In addition, further developments for improving metaheuristics through learning-based methods could help improve the solution approach.

Acknowledgments

This work is funded by the National Science and Engineering Research Council of Canada (NSERC) CRD Grant 500414-16, NSERC Discovery Grant 239019, the COSMO Stochastic Mine Planning Laboratory and mining industry consortium (AngloGold Ashanti, Anglo American, BHP, De Beers, IAMGOLD, Kinross, Newmont Mining, and Vale), and the Canada Research Chairs Program.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the Natural Sciences and Engineering Research Council of Canada [CRD Grant 500414-16; Discovery Grant 239019].