Reducing the Dimensions of the Stochastic Programming Problems of Metallurgical Design Procedures

Process design procedures under uncertainty result in stochastic optimization problems whose resolution is complex due to the large uncertainty space, which hinders the application of optimization approaches, as well as the establishment of relationships between input and output variables. On the other hand, supervised machine learning (SML) offers tools with which to develop surrogate models, which are computationally inexpensive and efficient. This paper proposes a procedure based on modern design of experiments, deterministic optimization, SML tools, and global sensitivity analysis (GSA) to reduce the size of the uncertainty space for stochastic optimization problems. The proposal is illustrated with a case study based on the stochastic design of flotation plants. The results reveal that surrogate models of stochastic formulation enable the prediction of the structure, profitability parameters, and metallurgical parameters of designed flotation plants, as well as reducing the size of the uncertainty space via GSA and, consequently, establishing relationships between the input and output variables of the stochastic formulation

Editorial for Special Issue “Modeling, Design, and Optimization of Multiphase Systems in Minerals Processing, Volume II”

The manuscripts published in the 2019 Special Issue “Modeling, Design, and Optimization of Multiphase Systems in Minerals Processing” [...

Design of Flotation Circuits Using Tabu-Search Algorithms: Multispecies, Equipment Design, and Profitability Parameters

The design of a flotation circuit based on optimization techniques requires a superstructure for representing a set of alternatives, a mathematical model for modeling the alternatives, and an optimization technique for solving the problem. The optimization techniques are classified into exact and approximate methods. The first has been widely used. However, the probability of finding an optimal solution decreases when the problem size increases. Genetic algorithms have been the approximate method used for designing flotation circuits when the studied problems were small. The Tabu-search algorithm (TSA) is an approximate method used for solving combinatorial optimization problems. This algorithm is an adaptive procedure that has the ability to employ many other methods. The TSA uses short-term memory to prevent the algorithm from being trapped in cycles. The TSA has many practical advantages but has not been used for designing flotation circuits. We propose using the TSA for solving the flotation circuit design problem. The TSA implemented in this work applies diversification and intensification strategies: diversification is used for exploring new regions, and intensification for exploring regions close to a good solution. Four cases were analyzed to demonstrate the applicability of the algorithm: different objective function, different mathematical models, and a benchmarking between TSA and Baron solver. The results indicate that the developed algorithm presents the ability to converge to a solution optimal or near optimal for a complex combination of requirements and constraints, whereas other methods do not. TSA and the Baron solver provide similar designs, but TSA is faster. We conclude that the developed TSA could be useful in the design of full-scale concentration circuits

Experimental Uncertainty Analysis for the Particle Size Distribution for Better Understanding of Batch Grinding Process

Uncertainty in industrial processes is very common, but it is particularly high in the grinding process (GP), due to the set of interacting operating/design parameters. This uncertainty can be evaluated in different ways, but, without a doubt, one of the most important parameters that characterise all GPs is the particle size distribution (PSD). However, is the PSD a good way to quantify the uncertainty in the milling process? This is the question we attempt to answer in this paper. To do so, we use 10 experimental grinding repetitions, 3 grinding times, and 14 Tyler meshes (more than 400 experimental results). The most relevant results were compared for the weight percentage for each size (WPES), cumulative weight undersize (CWU), or the use of particle size distribution models (PSDM), in terms of continuous changes in statistical parameters in WPES for different grinding times. The probability distribution was found to be changeable when reporting the results of WPES/CWU/PSDM, we detected the over-/under-estimation of uncertainty when using WPES/CWU, and variations in the relationships between sizes were observed when using WPES/CWU. Finally, our conclusion was that the way in which the data are analysed is not trivial, due to the possible deviations that may occur in the uncertainty process

Toward the Operability of Flotation Systems under Uncertainty

The purpose of this work was to analyze the requirements for the operational feasibility of flotation systems as well as the effects of the selection of flotation equipment and metal price uncertainty. A procedure based on mathematical optimization and uncertainty analysis was implemented to achieve this aim. The optimization included flotation and grinding stages operating under uncertainty, whereas the uncertainty analysis considered the Monte Carlo method. The results obtained indicate a small number of optimal flotation structures from the economic point of view. Considering the relationship between the economic performance and metallurgical parameters, we established that these structures exhibited favorable conditions for operating under uncertainty. Such conditions were proportional to the percentages representing each structure in the optimal set; i.e., a higher percentage of a structure implied a greater capacity to face operational and metal price changes. The set of optimal structures included configurations implementing cell banks, flotation columns, or both, indicating the influence of the flotation equipment type on the optimal structures. We also established the influence of metal price on the number of optimal structures. Therefore, the results obtained allowed us to separate the design of the flotation systems into two stages: first, a set of optimal structures exhibiting favorable conditions for facing uncertainty is determined; second, the optimal operation is established via resilience/flexibility approaches after the previous determination of the equipment design parameters

Control Structure Design Using Global Sensitivity Analysis for Mineral Processes under Uncertainties

Multiple-input and multiple-output (MIMO) systems can be found in many industrial processes, including mining processes. In practice, these systems are difficult to control due to the interactions of their input variables and the inherent uncertainty of industrial processes. Depending on the interactions in the MIMO process, different control strategies can be implemented to achieve the desired performance. Among these strategies is the use of a decentralized structure that considers several subsystems and for which a SISO controller can be designed. In this study, a methodology based on global sensitivity analysis (GSA) to design decentralized control structures for industrial processes under uncertainty is presented. GSA has not yet been applied for this purpose in process control; it allows us to understand the dynamic behavior of systems under uncertainty in a broad value range, unlike approaches proposed in the literature. The proposed GSA is based on the Sobol method, which provides sensitivity indices used as interaction measures to establish the input–output pairing for MIMO systems. Two case studies based on a semi-autogenous grinding (SAG) mill and a solvent extraction (SX) plant are presented to demonstrate the applicability of the proposed methodology. The results indicate that the methodology allows the design of 2 × 2 and 3 × 3 decentralized control structures for the SAG mill and SX plant, respectively, which exhibit good performance compared to MPC. For example, for the SAG mill, the determined pairings were fresh ore flux/fraction of mill filling and power consumption/percentage of critical speed

Assessment of the Supply Chain under Uncertainty: The Case of Lithium

Modeling the global markets is complicated due to the existence of uncertainty in the information available. In addition, the lithium supply chain presents a complex network due to interconnections that it presents and the interdependencies among its elements. This complex supply chain has one large market, electric vehicles (EVs). EV production is increasing the global demand for lithium; in terms of the lithium supply chain, an EV requires lithium-ion batteries, and lithium-ion batteries require lithium carbonate and lithium hydroxide. Realistically, the mass balance in the global lithium supply chain involves more elements and more markets, and together with the assortment of databases in the literature, make the modeling through deterministic models difficult. Modeling the global supply chain under uncertainty could facilitate an assessment of the lithium supply chain between production and demand, and therefore could help to determine the distribution of materials for identifying the variables with the highest importance in an undersupply scenario. In the literature, deterministic models are commonly used to model the lithium supply chain but do not simultaneously consider the variation of data among databases for the lithium supply chain. This study performs stochastic modeling of the lithium supply chain by combining a material flow analysis with an uncertainty analysis and global sensitivity analysis. The combination of these methods evaluates an undersupply scenario. The stochastic model simulations allow a comparison between the known demand and the supply calculated under uncertainty, in order to identify the most important variables affecting lithium distribution. The dynamic simulations show that the most probable scenario is one where supply does not cover the increasing demand, and the stochastic modeling classifies the variables by their importance and sensibility. In conclusion, the most important variables in a scenario of EV undersupply are the lithium hydroxide produced from lithium carbonate, the lithium hydroxide produced from solid rock, and the production of traditional batteries. The global sensitivity analysis indicates that the critical variables which affect the uncertainty in EV production change with time

On the use of Na2SO3Na_2SO_3 as a pyrite depressant in saline systems and the presence of kaolinite

The effect of Na2SO3 as a pyrite depressant in NaCl and KCl saline media and the presence of kaolinite were evaluated by zeta potential tests. Chalcopyrite was also included in the study, because pyrite usually accompanies this mineral. Subsequently, the floatability results of both minerals in the NaCl solution were optimized based on the design of experiments (DoE). The Box–Behnken DoE was applied considering the percentage of kaolinite $(X_1)$, collector dose $(X_2)$, and depressant dose $(X_3)$ as factors. The results were modeled using artificial neural networks (ANNs) to construct contour plots and to determine the optimal conditions. In particular, maximization of the mass recovery of chalcopyrite and minimization of that of pyrite were sought. The particle swarm optimization algorithm was used as an optimization technique. The results indicated that the optimal conditions to maximize the floatability of chalcopyrite were kaolinite 6.85%, collector dose $3.58 × 10^{–3} mol$ / $dm^3$, and depressant dose $3.49 × 10^{–5} mol$ / $dm^3$. On the contrary, the optimal conditions to minimize the floatability of pyrite were 5% kaolinite, collector dose $5 × 10^{–4} mol$ / $dm^3$, and depressant dose $6.4 × 10^{–5} mol$ / $dm^3$. Under these conditions, the mass recoveries of chalcopyrite and pyrite were 66.1% and 14.0%, respectively. The results also indicated that the presence of kaolinite negatively affects the flotation of chalcopyrite, while the effect of $Na_2SO_3$ is not significant. In general, the findings suggest that $Na_2SO_3$ is a viable alternative to consider as a pyrite depressant in saline environments

Response Surface Methodology for Copper Flotation Optimization in Saline Systems

Response surface methodology (RSM) is one of the most effective tools for optimizing processes, and it has been used in conjunction with the Analysis of Variance (ANOVA) test to establish the effect of input factors on output factors. However, when this methodology is used in mineral flotation, its polynomial model usually performs poorly. An alternative is to use artificial neural networks (ANNs) in such situations. Within this context, the ANOVA test is not the best option for these model types; moreover, it requires statistical assumptions that are difficult to satisfy in flotation. This work proposes replacing the polynomial model of the RSM with ANNs and the Sobol methods to determine the influential input factors instead of the ANOVA test. This proposal is applied to two porphyry copper ores with a high content of pyrite, clay, and dilution media. In addition, this study shows how other computational intelligence techniques, such as swarm intelligence, can be incorporated into this type of problem to improve the learning process of ANNs. The results gave an adjustment of over 0.98 for R2 using ANNs, in comparison to values of around 0.5 when the polynomial model of RSM was utilized. On the other hand, the application of Global Sensitivity Analysis (GSA) identified the aeration rate and P80 size as the most influential variables in copper recovery under the conditions studied. Additionally, we identified significant interactions that affect the recovery of copper, with the interactions between the aeration rate, frother concentration, and P80 size being the most important

Response Surface Methodology for Copper Flotation Optimization in Saline Systems

Response surface methodology (RSM) is one of the most effective tools for optimizing processes, and it has been used in conjunction with the Analysis of Variance (ANOVA) test to establish the effect of input factors on output factors. However, when this methodology is used in mineral flotation, its polynomial model usually performs poorly. An alternative is to use artificial neural networks (ANNs) in such situations. Within this context, the ANOVA test is not the best option for these model types; moreover, it requires statistical assumptions that are difficult to satisfy in flotation. This work proposes replacing the polynomial model of the RSM with ANNs and the Sobol methods to determine the influential input factors instead of the ANOVA test. This proposal is applied to two porphyry copper ores with a high content of pyrite, clay, and dilution media. In addition, this study shows how other computational intelligence techniques, such as swarm intelligence, can be incorporated into this type of problem to improve the learning process of ANNs. The results gave an adjustment of over 0.98 for R2 using ANNs, in comparison to values of around 0.5 when the polynomial model of RSM was utilized. On the other hand, the application of Global Sensitivity Analysis (GSA) identified the aeration rate and P80 size as the most influential variables in copper recovery under the conditions studied. Additionally, we identified significant interactions that affect the recovery of copper, with the interactions between the aeration rate, frother concentration, and P80 size being the most important